Differential.equations.and.Control.theory.ebook EEn

differential equationsand control theory

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differential equationsand control theory

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Additional Volumes in Preparation

Preface

This volume is based on papers presented at the International Workshop on DifferentialEquations and Optimal Control, held at the Department of Mathematics of Ohio University inAthens, Ohio. The main objective of this international meeting was to feature new trends in thetheory and applications of partial differential and functional-differential equations and theiroptimal control. The workshop can be viewed as a follow-up to the March 1993 Ohio UniversityInternational Conference on Optimal Control of Differential Equations, whose proceedings wereedited by N. H. Pavel and also published by Marcel Dekker, Inc., as volume 160 of the series:Lecture Note in Pure and Applied Mathematics.

A large variety of related topics is covered in this volume, both theoretical and applied,deterministic and stochastic. The topics include: nonlinear programming and control with closedrange operators, stabilization of the diffusion equations, flow-invariant sets with respect to theNavier-Stokes equations, numerical approximation of the Riccati equation, telegraph systems,dispersive equations, viable domains for differential equations, almost periodic solutions toneutral functional equations, Wentzell boundary conditions, parabolic phase-field models withmemory, Kato classes of distributions, optimal control and algebraic Riccati equations,identification problems for wave equations via optimal control, integrodifferential and otherfunctional equations, stochastic Navier-Stokes equations, Lavrentiev phenomena, volatility forAmerican options via optimal control, obstacle problems, necessary conditions of optimality forsemilinear problems, Lyapunov stability, least action for ,/V-body problems, and more.

The workshop was sponsored by the College of Arts and Sciences, the Department ofMathematics, and the Research Office of Ohio University. We gratefully acknowledge theirfinancial support, which made this workshop possible. We are also very indebted to allparticipants and contributors. Finally, our thanks go also to Marcel Dekker, Inc., for undertakingthe publication of this volume.

Sergiu AizicoviciNicolae H. Pavel


Contents

PrefaceContributors

1. Existence and Uniqueness of Solutions to a Second Order NonlinearNonlocal Hyperbolic EquationAzmy S. Ackleh, Sergiu Aizicovici, Michael Demetriou, and Simeon Reich

2. Fully Nonlinear Programming Problems with Closed Range OperatorsSergiu Aizicovici, D. Motreanu, and Nicolae H. Pavel

3. Internal Stabilization of the Diffusion EquationLaura-Iulia Anita and Sebastian Ani/a

4. Flow-Invariant Sets with Respect to Navier-Stokes EquationV. Barbu and Nicolae H. Pavel

5. Numerical Approximation of the Riccati Equation via Fractional Steps MethodTudor Barbu and Costica Moro§anu

6. Asymptotic Analysis of the Telegraph System with Nonlinear BoundaryConditionsL. Barbu, E. Cosma, Gh. Moroganu, and W. L. Wendland

1. Global Existence for a Class of Dispersive EquationsRadu C. Cascaval

8. Viable Domains for Differential Equations Governed by CaratheodoryPerturbations of Nonlinear m-Accretive OperatorsOvidiu Carjd and loan L Vrabie

9. Almost Periodic Solutions to Neutral Functional EquationsC. Corduneanu

10. The One Dimensional Wave Equation with Wentzell Boundary ConditionsAngela Favini, Gisele Ruiz Goldstein, Jerome A. Goldstein, and Silvia Romanelli

11. On the Longterm Behaviour of a Parabolic Phase-Field Model with MemoryMaurizio Grasselli and Vittorino Pata

12. On the Kato Classes of Distributions and the BMO-ClassesArchil Gulisashvili

13. The Global Solution Set for a Class of Semilinear ProblemsPhilip Korman


14. Optimal Control and Algebraic Riccati Equations under Singular Estimatesfor eAlB in the Absence of Analyticity. Part I: The Stable CaseIrena Lasiecka and Roberta Triggiani

15. Solving Identification Problems for the Wave Equation by Optimal ControlMethodsSuzanne Lenhart and Vladimir Protopopescu

16. Singular Perturbations and Approximations for Integrodifferential EquationsJ. Liu, J. Sochacki, and P. Dostert

17. Remarks on Impulse Control Problems for the Stochastic Navier-StokesEquationsJ. L. Menaldi and S. S. Sritharan

18. Recent Progress on the Lavrentiev Phenomenon with ApplicationsVictor J. Mizel

19. Abstract Eigenvalue Problem for Monotone Operators and Applications toDifferential OperatorsSilviu Sburlan

20. Implied Volatility for American Options via Optimal Control and FastNumerical Solutions of Obstacle ProblemsSrdjan Stojanovic

21. First Order Necessary Conditions of Optimality for Semilinear OptimalControl ProblemsM. D. Voisei

22. Lyapunov Equation and the Stability of Nonautonomous Evolution Equationsin Hilbert SpacesQuoc-Phong Vu and Siu Pang Yung

23. Least Action for TV-Body Problems with Quasihomogeneous PotentialsShih-liang Wen and Shiqing Zhang


Contributors

Azmy S. Ackleh University of Louisiana at Lafayette, Lafayette, Louisiana

Sergiu Aizicovici Ohio University, Athens, Ohio

Laura-Iulia Anija University "ALL Cuza", lasi, Romania

Sebastian Anifa University "ALL Cuza", lasi, Romania

L. Barbu Ovidius University, Constanja, Romania

Tudor Barbu Institute of Mathematics of Romanian Academy, lasi, Romania

V. Barbu University of lasi, lasi, Romania

OvidiuCarja "ALL Cuza" University of lasi, lasi, Romania

Radu C. Cascaval University of Missouri, Columbia, Missouri

C. Corduneanu University of Texas at Arlington, Arlington, Texas

Michael Demetriou Worcester Polytechnic Institute, Worcester, Massachusetts

P. Dostert James Madison University, Harrisonburg, Virginia

Angelo Favini Universita di Bologna, Bologna, Italy

Gisele Ruiz Goldstein University of Memphis, Memphis, Tennessee

Jerome A. Goldstein University of Memphis, Memphis, Tennessee

Maurizio Grasselli Politecnico di Milano, Milan, Italy

A. Gulisashvili Ohio University, Athens, Ohio

Philip Korman University of Cincinnati, Cincinnati, Ohio

Irena Lasiecka University of Virginia, Charlottesville, Virginia

Suzanne Lenhart University of Tennessee, Knoxville, Tennessee

J. Liu James Madison University, Harrisonburg, Virginia


J. L. Menaldi Wayne State University, Detroit, Michigan

Victor J. Mizel Carnegie-Mellon University, Philadelphia, Pennsylvania

Costica Morosanu University "ALL Cuza", lasi, Romania

Gh. Moroganu Stuttgart University, Stuttgart, Germany

D. Motreanu University of lasi, lasi, Romania

Vittorino Pata Politecnico di Milano, Milan, Italy

Nicolae H. Pavel Ohio University, Athens, Ohio

Vladimir Protopopescu Oak Ridge National Laboratory, Oak Ridge, Tennessee

Simeon Reich The Technion-Israel Institute of Technology, Haifa, Israel

Silvia Romanclli Universita di Bari, Bari, Italy

Silviu Sburlan Ovidius University, Constantza, Romania

J. Sochacki James Madison University, Harrisonburg, Virginia

S. S. Sritharan U.S. Navy, San Diego, California

Srdjan Stojanovic University of Cincinnati, Cincinnati, Ohio

Roberto Triggiani University of Virginia, Charlottesville, Virginia

M. D. Voisei Ohio University, Athens, Ohio

loan I. Vrabie "ALL Cuza" University of lasi, lasi, Romania

Quoc-Phong Vu Ohio University, Athens, Ohio

Shih-liang Wen Ohio University, Athens, Ohio

W. L. Wendland Stuttgart University, Stuttgart, Germany

Siu Pang Yung University of Hong Kong, Hong Kong, China

Shiqing Zhang Chongqing University, Chongqing, China


Existence and Uniqueness of Solutions toa Second Order Nonlinear Nonlocal

Hyperbolic Equation

AZMY S. ACKLEHDepartment of Mathematics, University of Louisiana at Lafayette,

Lafayette, LA 70504, USA

SERGIU AIZICOVICIDepartment of Mathematics, Ohio University,

Athens, OH 45701, USA

MICHAEL DEMETRIOUDepartment of Mechanical Engineering, Worcester Polytechnic Institute,

Worcester, MA 01609, USA

SIMEON REICHDepartment of Mathematics, The Technion-Israel Institute of Technology,

32000 Haifa, ISRAEL

We establish existence and uniqueness of weak solutions to a class of secondorder distributed parameter systems with sudden changes in the input term.Such systems are often encountered in flexible structures and structure-fluidinteraction systems that utilize smart actuators. A Galerkin finite dimensionalapproximation scheme for computing the solution of these systems is developedand its strong convergence is proved. Numerical results are also presented.

1 Introduction

In this paper we consider the nonlinear, nonlocal partial differential equation

wtt + KT.WZXXX + Kzwxxxxt = [/3(x,t)g(y)]xx + f ( x , t ) , (1.1)

with boundary and initial conditions given by

wx(0,t)=w(0,t) = 0, wx(l,t)=w(l,t)=Q,


In equation (1.1) the function y satisfies

= \Jo

where X[xi,x2] denotes the characteristic function on the interval [x i ,X2J , with 0 < x\ <

£2 < 1. The constants KI, KI and ks are positive and g is a Lipschitz continuous function.

There is an extensive literature on linear and semilinear second order (in time) evolution

equations (e.g., [1, 2, 3, 4, 6, 7, 12, 13, 14, 16]). For example, the existence-uniqueness

results presented in [4] apply to the system (1.!)-(!.2) for the case g = 0. However, to

our knowledge, no existence-uniqueness results for the system (1.1)-(1.2) (with a nontrivial

nonlinear function g) are available.

Equation (1.1) is a general form of the model developed by Demetriou and Polycar-

pou [9, 10]. Indeed, in the context of the flexible structure encountered in Demetriou and

Polycarpou [9], KI denotes the stiffness parameter, KZ the damping parameter and ks the

sensor piezoceramic constant; see Banks et al. [5] and Dosch et al. [11]. When the actuator

(input) failure term /3(x,t)g(y) is written as

P(x,t)g(y) = AW (kaX(xim](x)e(t)) g(y)

with the time profile (Polycarpou and Helmicki [15]) of the failure given by

f 0 i f t < T ffl(" = {l-^», „ « > ! - , • A > ° ' <L3>

and the nominal forcing (actuator) term given by

f ( x , t ) = [kaX[Xl,X2](x)e(t)]xx, ka > 0,

then equation (1.1) has exactly the same form as the beam equation considered in Demetriou

and Polycarpou [9]. The time T; denotes the unknown instance of the failure occurrence

and the signal e denotes the input voltage to the patch. Similarly, ka denotes the actuator

piezoceramic constant; see Banks et al. [5]. Therefore, this model describes the dynamics

of a flexible cantilevered beam before (t < T/) and after (t > Tf) the occurrence of an

anticipated actuator failure commencing at an unknown time Tf. In view of the above, the

plant equation (1.1) can now be written as follows:

wtt

Our efforts here are a continuation of an earlier work [8]. There, the following Galerkin

approximations for solutions of the system (1. !)-(!. 2) were considered:

where {V'j}^! are the eigenfunctions corresponding to the eigenvalues {Xj}^ of the strictlyd4

positive self adjoint operator A — -— with the dense domain in L2(0, 1) given byax*

-D(A) = {<t> 6 #4(0, 1) : 0'(0) = 0(0) - 0, 0'(1) - 0(1) = 0}.

It is well known that the eigenvalues Xj are simple and that the set of eigenfunctions

{tjjj} forms a complete orthonormal system in L2(0, 1). A priori bounds which are based on

energy estimates were established for these Galerkin approximations. Furthermore, in order

to detect, diagnose and accommodate the actuator failure, a model-based fault diagnosis

scheme was presented. Our scheme consisted of a detection/diagnostic observer and an

estimator of the actuator failure term. Since the proposed scheme is infinite dimensional,

a finite dimensional approximation was considered for computational purposes.

The present paper is organized as follows. In Section 2 we give the definition of weak

solutions to problem (1. !)-(!. 2) and establish existence-uniqueness of such solutions using

a Galerkin approximation technique. The strong convergence of this approximation is also

proved. In Section 3 we use the Galerkin method to give a numerical solution to a model

problem.

2 Existence and Uniqueness of Weak Solutions

We begin this section by letting H — L2(0, 1) and V = HQ(O, 1), so we have the Gelfand

triple V ^-> H V with V = #~2(0, 1). We denote by {- , - ) the inner product in H,

while (•, -}v*,v stands for the usual duality product. Let || • ||, || • ||v, and || • \\y* denote the

norms of the spaces H, V, and V , respectively. Assume that the parameters in (1. !)-(!. 2)

satisfy the following conditions:

(A/}) The function j3 e L°° (Q,T,H), with ||/?||LOO(0>T.ff) < L.

(Ag) The nonlinear function g satisfies the following Lipschitz condition:

\9 (6) - 5(6)1 < 16 - 61 , for all 6,6 € R,Ks

where C\ < K^/L.

(A;) The forcing term f e L2 (0, T; V*) .

To establish the existence-uniqueness of solutions we use a Galerkin type method which

is comparable to the one employed in the study of well-posedness for other second order

(in time) evolution equations (see, e.g., [1, 2, 3, 4, 12, 13, 14]). To this end, we define the

space of functions

UT = (u : u 6 Wl>2(0, T; V), utt e L2(0, T; V*)}

with norm

UUT = i.2(0,T;V)

We now define the notion of a weak solution to the problem (1. !)-(!. 2).

Definition 2.1 We say that a function w G UT is a weak solution of (1. !)-(!. 2) if -it

satisfies

>)v,V + Ki(wxx(t),(j>xx) + K2(wxxt(t),(j>xx)

(2.1)= </?(*Ml/W).0xx> + </(*), 0)v,v, V 0 e V

and

w(0) = WoeV, wt(0) = Wl e H. (2.2)

Next we state the existence-uniqueness theorem which is the main result of this paper.

Theorem 2.2 The problem (1. !)-(!. 2) has a unique weak solution.

Proof. Let {^ISi be any linearly independent total subset of V. For each m, let

and let w™, w™ £ Vm be chosen so that w™ -> w0 in V, w™ — >• w\ in H as m — >• oo. For each

TO we define an approximate solution to the problem (1. !)-(!. 2) by wm (t) = X^i C™(i)il)i,

where wm is the unique solution to the m- dimensional system

«(*), ) + Kl(w?x(t), ^jxx) + «2«S4(<), ^xx>

(2.3)= {/3(i)5(ymW), ^xx> + </(*), ^>v-,v, j = 1, 2, . . . , m,

with initial conditions

iom(0) = 10™, iotm(0) - wf. (2.4)

The function ym in equation (2.3) satisfies

ym(t)= f ksX[xi,X2](x)w™t(x,t)dx.Jo

Multiplying the equation (2.3) by ^C*j"(t) and summing up over j we obtain

(f(t),w?(t))v.,v.

Hence,

Upon integrating this equality we obtain

+ 2(2.5)

/o JoNow, using the assumption (Af), the fourth term on the right hand side of (2.5) can be

bounded as follows:

2 [\f(T),W?(T))V.,VdT<6 [t\\W™XT(T)\\2dT+l-Jo Jo ° Jo

for any 6 > 0. Furthermore, note that from assumption (Ag) it follows that

\g(ymm<\g(ym(t)}- 5(0)!(2.6)

for some GI > 0. Hence, the third term on the right hand side of (2.5) satisfies the following

estimate:

<2/W)|||s(ym(r))|||<IT(r)||dr

- _ \-\ym(r)\+C2}\\l3(T)\\\\w2

(LC^dr

+6

\dr

Now choose 5 such that

1 ~

Thenrt

„ .m / _ \ l | 2 r _ *• i k . m i i S

0

+ «i IhSLII2 + K _2L~ (LC2)2T + ^d ll/ll^o,^.).

Recalling that w™ —> WQ in V, w™ —) w\ in H as m —> oo, we conclude that there exists a

positive constant C independent of TO such that

+ K! |K (t) |2 + «2 - LC, ||<ir (r)||2dr < C.v y 7o

It follows that {<•} is bounded in C([0,T];H)and in L2(0,T;K), and that {wm} is

bounded in C([0, T]; V). Furthermore, from (2.6) and the boundedness of w™ in L2(0, T; V)

it follows that there exists a positive constant M such that

Hence, there exists a subsequence {wmk} of {wm} and limit functions w € W /1 '2(0,T;y)

and g 6 L2(0, T) such that

wmk -> w weakly in W1'2 (0, T; F)

5(y™ fc) ->• 5 weakly in L2(0,T).

Note that w(0) = w0-

Following [13] we fix j < TO and let r\ £ Cl[0,T] with rj(T) = 0 be arbitrarily chosen.

Set rjj(t) = r)(t)ip,j and multiply both sides of (2.3) by the function rj(t). Integrating over

[0, T] we obtain

- Ki(w™x(t),r)jxx(t)) + K2(w'xlxt(t),rjjxx(t))}dt

mt)g(ym(t)),r]JXX(t)) -

Using integration by parts for the first term and letting f)j = ^TJJ we get

•T

0

{(/3(t)g(ym(t)},rjJXX(t)} + (f(t)^(t))v.y} dt +o

Using the above weak convergences and taking subsequential limits as m = mk -> oo in

the previous equation we get

(t),T)j(t)) + Ki(wxx(t),rijXX(t)} + K2(wxxt(t),rijxx(t))}dt(2.7)

o

Recalling that r]j(t) = r)(t)wj and further restricting r\ so that 77 6 Q°(0,T), we get

fT/ {r, (t) (-w^t),^) + Klri (t) (wxx(t),^jxi} + K2r1(t)(wxxt(t'),tjjjxx)} dt

CT

= I r j ( t ) { ( / 3 ( t ) g ( t ) , ^ x x ) +(f(t),^j)v,v}dt.Jo

This implies that for each ipj,

— (w^t),^) + Ki(wxx(t),ipjxx) + K2(wxxt(t),t/j:jxx) = (P(t}g(t),tl)jxx) + (f(t),tl)j)v,v.(2.8)

Since ipj is total in V we thus have that wtt e L2(0, T; V) and for all <j> £ V,

(wtt(t), 0) + KiKi(t), 0xx> + K2(wxxt(t), (f>xx) - (/3(t)g(t), 4>xx} + (f(t),4>)v>,v- (2.9)

We already have w(0) — WQ and to argue that wt (0) = w\, we return to (2.7) which holds

for all rjj(t) = r](t)'ipj, T? e C^O,^, 7/(T) = 0. Integrating by parts the first term in (2.7)

and using (2.8) we obtain

<-wt(t),^(i)>|^ = {«;!, 77X0)).

From this it follows that wt (0) = wi. To prove that the limit function is indeed a weak

solution left to be shown that g(t) = g(y(t}} for a.e. t 6 [0, T]. Recall that we already

proved that g(ym) —> ~g weakly in Z/2(0, T) (along a subsequence). Our next goal is to show

that this weak convergence is actually a strong one.

To achieve this step we follow ideas developed in [7] for linear second order (in time)

evolution equations, and adopted for other nonlinear second order problems in [1, 12]. we

let zm(t) = wm(t) — w ( t ) , where wm is the unique solution to the finite dimensional system

(2.3)-(2.4) and w is the limit function which solves the linear problem (2.9) with w(0) = w0

and Wt(Q) = w\. Now, use the test function wm in (2.3) and the test function w in (2.9)

and add and subtract terms to obtain

\\z?(t)f + Kl \\z™(t)\\2 + 2K2 \\z^ (r)f dr

+2 / ( f t ( r ) ( g ( y m ( r ) ) - g ( y ( r ) ) , z ^ x ( r ) ) d rJo.

+2'°,t

+2 (Jo

Here,

(t) = 2 \-(Wt(f),W?(t)) - Kl(Wxx(t),W™(t)) ~ 2K2 [\Wxxr(T),W™T(T))dTI Jo

) + Kl(w0xx,w™x) + I (/3(r)g(ym(T)),wXXT(T))drJo

(f3(T}g(r}^XT(r)}+'2 I </(r),«;T(r)>v.,vdr] .^o J

The third term on the right-hand side of (2.10) satisfies the following estimate:

2t

< 2 f \\/3(r)\\ \g(ym(r) - g(y(r))\ \\z™xx (r)\\ drJo _

< 2 T \ym(r) - y(r)\ ||/3(r)|| \\z?xx(r)\\ drJO <*s

< 2 r^fcs||zT^(r)||i]k^(r)||rfrJo ™s

<2LC1 \\z™x(r)\\2 dr.Jo

From this estimate we get the following inequality:

0

Letting m = m* —>• oo, we clearly get \\w™k — w\\\ + KI \\w^.kx — WQXX\\ —>• 0. Recalling

that zmk = wmk -W-+Q weakly in Wl'2(0, T; V) and that g ( y m k ) ->• 5 weakly in L2(0, T),

we see that Tmk(t) —> 0 because w satisfies the integrated form of (2.9). Furthermore, we

also see that the third and the fourth terms on the right-hand side of the above inequality

converges to 0. Hence, we have that wmk —>• w strongly in C([0,T]; V) and that w™k —)• wt

strongly in C([0, T]; H) n L2(0, T; V). This implies that

/•T i^\2 rTT

/ \9(ym"(t))-9(ym2dt< (^ fJo \ s / ^°

<(C02 fJoo

Since g(ymk) -+ g weakly in L2(0,T) also, we get that g(y(t)) = g(t) for a.e. t <E (0,T).

Now that we have proved the existence of a weak solution to problem (1. !)-(!. 2) we

would like to point out that a solution to this problem satisfies the additional regularity

w e C([0,T]- V) and wt € C([0,T]; H) (cf. [13, p. 273], [14, Chap. 3j). To see this, observe

that for any ~g e L2 (0, T) the linear problem (2.9) has a unique solution w with w(ff) — WQ,

wt(0) = wl satisfying w 6 C([0,T}; V) and wt 6 ^([O.T]; H) (for details see [4]). Since for

#(') = d(y('}) £ L2(0,T) the unique solution to the linear problem w coincides with the

solution w to the nonlinear problem (1. !)-(!. 2), the result follows.

The uniqueness of solutions can be easily established. Indeed, assume that w\ and w-i

are two solutions to (1. !)-(!. 2) and define z = w\ -w2. Then by calculations similar to the

ones employed above we deduce that for each t e (0, T),

Kl\\zxx(t)\\2+( 2K2-2LC!) / \\ztxx(r)\\2dT<0.^ ' Jo

Thus z = 0 and this completes the proof of Theorem 2.2. D

We conclude this section by pointing out that from the uniqueness of solutions it fol-

lows that the Galerkin approximations wm converges to the unique solution w strongly in

C([0, T]; V). In the next section we present a numerical solution to a model problem using

the Galerkin approximation developed here.

3 Numerical Results

In this section we summarize the numerical implementation scheme and present some of

our numerical findings. For the purpose of the numerical results we set

P(x,t)g(y) = ft (*) (kaXlXlM](x)t(t)) g(y) and f ( x , t ) = [kaX[xi,X2](x)e(t)]xx.

Using a standard Galerkin scheme, we discretize the problem (1. !)-(!. 2) in terms of spline

expansions. More precisely, we use modified (for essential boundary conditions) cubic

splines on the interval (0,1) with respect to the uniform mesh {0, — , — , . . . ,1} to ap-

proximate (1. !)-(!. 2). Using the notation of Section 2 we denote the 1-D cubic splines

that are used to discretize (1.1) by {V'i™}^1 and the approximating subspace by V™ —

Choosing the approximate beam solution to be

m-l

and restricting the infinite dimensional system to the space Vm, we arrive at equations

(2.3)-(2.4) with the test function replaced by if; € V. When the test function ^ is chosen

in Vm, we obtain the finite dimensional system

MmCm(t) + KlKmCm(t) + K2KmCm(t) = Pi(t)Bmg(ym)e(t) + Bme(t),

ym(t) = Dmw?(t), (3.1)

KmCm(Q) = w^, MmCm(0) = w?.


Here the coordinate vector representation of wm(t,x) with respect to the basis {^f1}^1

is Cm(t) = [C?(t) Cf(t) . . . C™_l(t)]r , with Cm(t] represented analogously. In (3.1) the

mass and stiffness matrices are given by

dx, K™ = ^T(x}\xx(^(x}}xx dx, i,j = 1,2,..., m- 1,Jo

(3.2)

respectively, and the input and output vectors by

ks[^(x)}xxdx, i = l,2,...,m-l. (3.3)X I

The initial conditions w™, w™ are given by

r1 r1= / w0xx(x}[^(x)]xx dx [<], = /

Jo Jodx,

o JoWe first tested our numerical scheme for various values of the discretization index m =

4,8, 16,32,64 and 128. To do so, we set /3 = 0 and choose the forcing function in (1.1) to

be

f ( x , i) = - sin(tfx2(x - I)2 + 24/ti sin(i) + 24K2 cos(t).

Then one can easily verify that w(x,i) = sin(t)x2(x — I)2 solves (1. !)-(!. 2). A finite

dimensional system can be obtained for this choice of forcing function / using the same

technique which led to (3.1). The approximation errors and the percentage values of the

approximation errors are depicted in Figure 1. It can be observed that the approximate

solutions wm converge to the true solution w with order 1/m2 for the L2 norm and 1/^/rn

for the H$ norm.

The system with a nonzero j3 and e(t) = sin(2i) is simulated for m = 128 in which the

nonlinear function g(y) is chosen as

and where the plant parameters are given by n\ — 1, K? = 0.001, ka = 3.5 x 10~3,

ks = 3.5 x 10~4, Xi — 0.45, x% = 0.55 and Tf = 0.1. The above choice satisfies the

% of L, nom ol jcrfutxxi onctf\_ noim ot solution v.

20 40 60

(a)100 120 140 ' 0 20 40 60

(b)!00 120 140

20 40

(c)100 t?0 140 0 20 4O 60 80 100 12O

(d)

Figure 1: Evolution of (a) L2 and (c) H$ approximation error norms, and (b) L2 and (d) H$

percentage approximation error norms vs. m.


L norm of displacement

0.05

0.04

0.03

0.02

0.01

t.5 2

L2 norm of vekxity, wt(x.t)

Figure 2: The L2 norm of w(t) (upper) and wt(t) (lower).

Displacement w(x,1)

Figure 3: 3-D mesh of (a) w(x,t) and (b) wt(x,t).


assumptions (Ap) and ( A g ) . The L2 norm of the solution w(t) and its derivative w t ( t ) are

depicted in Figure 2 and the 3-D graphs of w and wt are presented in Figure 3.

Acknowledgments: The work of S. Reich was partially supported by the Fund for

the Promotion of Research at the Technion and by the Technion VPR Fund - E. and M.

Mendelson Research Fund.

References

[1] A.S. Ackleh, H.T. Banks and G.A. Pinter, On a nonlinear beam equation, Applied

Mathematics Letters, to appear.

[2] H. T. Banks, D. S. Gilliam, and V. I. Shubov, Well-posedness for a one dimen-

sional nonlinear beam, in Computation and Control, IV (Bozeman, MT, 1994),

Progr. Systems Control Theory, vol. 20, (Birkhauser, 1995), pp. 1-21.

[3] H. T. Banks, D. S. Gilliam, and V. I. Shubov, Global solvability for damped abstract

nonlinear hyperbolic systems, Differential and Integral Equations, 10 (1997), 309-332.

[4] H. T. Banks, K. Ito, and Y. Wang, Well-posedness for damped second order systems

with unbounded input operators, Differential and Integral Equations, 8 (1995), 587-606.

[5] H. T. Banks, R. C. Smith, and Y. Wang, Smart Material Structures: Modeling,

Estimation and Control (Wiley-Masson, 1996).

[6] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach

Spaces (Noordhoff International Publishing, 1976).

[7] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods

for Science and Technology, Volume 5, Evolution Problems I (Springer, 1993).


[8] M. A. Demetriou, A. S. Ackleh and S. Reich, Detection and accommodation of second

order distributed parameter systems with abrupt changes in the input term: existence

and approximation, Kybernetika, 36 (2000), 117-132.

[9] M. A. Demetriou and M. M. Polycarpou, Fault accommodation of output-

induced actuator failures for a flexible beam with collocated input and out-

put, in Proceedings of the 5th IEEE Mediterranean Conference on Control

and Systems, Phaethon Beach Hotel, Paphos, Cyprus, July 21-23, 1997, CD-ROM

publication.

[10] M. A. Demetriou and M. M. Polycarpou, Fault diagnosis of output-induced ac-

tuator failures for a flexible beam with collocated input and output, in

Proceedings of the IFAC Symposium on Fault Detection, Supervision and

Safety for Technical Processes: SAFEPROCESS'97, University of Hull, Hull,

United Kingdom, August 26-28, 1997.

[11] J. Dosch, D. J. Inman, and E. Garcia, A self-sensing piezoelectric actuator for collo-

cated control, Journal of Intelligent Material Systems and Structures, 3 (1992), 166-

185.

[12] J. Ha and S. Nakagiri, Existence and regularity of weak solutions for second order

semilinear evolution equations, Funkcialaj Ekvacioj, 41 (1998), 1-24.

[13] J. L. Lions, Optimal Control of Systems Governed by Partial Differential

Equations (Springer, 1971).

[14] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and

Applications (Springer, 1972).


[15] M. M. Polycarpou and A. J. Helmicki, Automated fault detection and accomodatwn: A

learning systems approach, IEEE Trans. on Systems, Man and Cybernetics, 25 (1995),

1447-1458.

[16] J. Wloka, Partial Differential Equations (Cambridge University Press, 1987).


Fully Nonlinear Programming Problems withClosed Range Operators

Sergiu Aizicovici Ohio University, Athens, Ohio

D. Motreanu University of Ia§i, Ia§i, Romania

Nicolae H. Pavel Ohio University, Athens, Ohio

AbstractThe paper provides necessary conditions for the optimality of a

pair (y,u~) with respect to a locally Lipschitz cost functional L(y,u~),subject to Ay = Cu + B(y, u). Here A and C are closed range, denselydefined linear operators on some Banach spaces Y and X, while B isa (Gateaux) differentiable map on Y X X. This extends the resultin [1], where the case B(y,u~) = B(u) — F(y), with B and F Frechetdifferentiable, was studied.

1 Introduction and Main ResultsLet X, Y and E be real Banach spaces. Let A : D(A] C Y — > E andC : D(C] C X — > E be (possibly unbounded) closed linear operators withdense domains D(A) and D(C) in Y and X, respectively. We also considera (Gateaux) differentiable map B : V x U — > E, where U and V are opensubsets of X and Y, respectively, and a locally Lipschitz function L : W — > IRon an open subset W of Y x X containing V x U .

This paper is devoted to the following nonlinear programming problem:

(Locally) Minimize L(y, u)

subject to Ay = Cu + B(y, u).


The set of constraints for our problem (P) is

M = { ( y , u) e (D(A) n V} x (D(C) nU):Ay = Cu + B(y, u)}. (1)

We assume M ^ 0. The tangent cone T(y,u)M of the set M in (1) at a point(y, u) G M is given by

T(y,u)M = {(z,w)£D(A)xD(C):3p(t)-+QmYand q ( t ) -> 0 in X as * -> 0+ such that (2)

(y + t(z + p ( t ) ) , u + t(w + q ( t ) ) ) G M}

(see Aizicovici, Motreanu and Pavel [1] or Motreanu and Pavel [5], Pavel [6]).In what follows, the superscript * denotes the adjoint of a linear operator.The generalized directional derivative of L(y,u) , the generalized gradient ofL and the partial generalized gradients of L(y,u) with respect to the vari-ables y and u (in the sense of Clarke [3]) are denoted by L°(y,u),dL(y,u),dyL(y,u) and duL(y,u), respectively (unless otherwise specified). Finally,B'(y,u), By(y,u) and Bu(y,u) denote the Gateaux derivative and the par-tial Gateaux derivatives of B(y,u), respectively.

The basic hypotheses are the following:

(#1) If ( y , u ] e M, z e D(A) and w e D(C) satisfy

Az = Cw + B'(y,u)(z,w), (3)

then (z,w) e T(y>u)M.

(Hi) For all ( y , u ) G M, R(A - By(y,u}} and R(C + Bu(y,u)) are closed inE and either

R(A - By(y, u)) C R(C + Bu(y, u)) (4)

orR(C + Bu(y, u ) ) C R(A - By(y, u ) ) . (5)

Remark 1 For our main result (Theorem 1) it is sufficient to assume that(Hi) and (Hi) hold for optimal pairs ( y , u ) of problem ( P ) , only.

The following simple Lemma will be essentially used in the proof of our mainresult.


Lemma 1. Let F : U —* R be a locally Lipschitz function on an open subsetof a Banach space X,Xi a subspace of X and let G be a linear functional onX\ such that:(1°) G(v)<F°(x-v), VvtXiwhere F°(x;v) is Clarke's generalized directional derivative of F at x in thedirection v. Then there is an extension G> G X* of G, i.e. G#(v) =G(v), Vu G Xi , with(2°) G*(v)<F°(x;v), VvEX,i.e. G» G d F ( x } .Proof. Recall that v —>• F°(x; v) is positively homogeneous and subadditiveon X, with F°(x;v)\ < l\v , Vv G X, where / is the Lipschitz constant ofF near x. By the Hahn-Banach theorem, there is a linear extension (7* of Gfrom Xi to all of X satisfying Inequality (2°). But F°(x;v) is continuous atv = 0 and we clearly have

-F°(x- -v) < G,(v) < F°(x- v), VvtX

which implies the continuity of G* at zero. PWe now state our main result concerning the necessary conditions of

optimality for problem ( P ) .THEOREM 1. Under assumptions (Hi) and (H2), for every (locally)

optimal pair ( y , u) G M of problem (P) there exists p G D(A*} n D(C*) suchthat

(-(A - By(y, u)Yp, (C + Bu(y, u))*p) € dL(y, u). (6)

If, in addition, L is regular at ( t / ,u) (in the sense of Clarke [3], p. 39), thenone has

(-(A - By(y, u))*p G dyL(y, u) and (C + Bu(y, u))*p G duL(y, u). (7)

PROOF. First we show that if (z,w) G D(A] x D(C) satisfies (3), then

L°(y,u;z,w)>Q. (8)

Indeed, we know by assumption (Hi) that (z,w) G T(ytU)M. Consequently,the functions p ( t ) and q(t) exist as stated in (2). The (local) optimality of( y , u ) in problem (P) ensures that

L(y + t(z + p(t)),u + t(w + q ( t ) ) ) > L(y, u)

for all i > 0 small enough. Then we derive that

0

< lim swp-[L(y + t(z + p ( t ) ) , u + t(w + q ( t ) ) ) - L(y + t p ( t ) , u + t q ( t ) ) }

lim sup -[L(y + tp(t)),u + tq(t))) - L(y, u)}

for the constant / > 0 in the proof of Lemma 1. Here we have used thedefinition of L°(y,u;z,w) and the fact that L is locally Lipschitz near (y ,u)in Y x X. The claim in (8) is thereby verified.

On the basis of Lemma 1, Inequality (8) implies the existence of (£,77) €Y* x X* such that

(£,ri)€dL(y,u), and {£, z)Y, iY + (n, w)x.,x = 0 (9)

whenever (z,w) G D(A) x D(C) satisfies (3). Indeed, set Xi = {(z,w) €D(A) x D(C), satisfying condition (3)}.By (Hi), Xi C T(y]U)M, so (9) follows from Lemma 1 with G - the nullfunctional on X\.

If we set w = 0 in (9) we find

(t,z)Y.,Y = 0,Vze D(A) with Az = By(y,u)z,

which can be expressed as

teN(A-By(y,u))^. ( fO)

By (^2), R(A — By(y,u)) is closed in E and therefore

R((A - B y ( y , u))*) = N(A - By(y, u))^ (11)

(see, e.g., Brezis [3], p. 29). Then (10) and (11) imply

teR((A-By(y,u))*). (12)

Setting now z = 0 in (9), we obviously get

{»?, w}x-,x = 0, Vw e D(B] with Cw + Bu(y, u)w = 0.

Reasoning as above we see that

r}£R((C + B u ( y , u ) ) * ) . (13)

By (12) and (13) there exist p G D(A*} C E* and p e D(C)* C £* such that

(A-By(y,u))*p = t, (C + 5u(y,U))*p = 7/, (14)

so (9) can be rewritten as

(p, (A - By(y, U))Z)E;E + (P, (C + Bu(y, U))W}E*,E = 0 (15)

for all (z,w) e D(A) x D(C) satisfying (3).Assume that Condition (4) in (H2) holds. Then for each z G D(A) there

is w e D(C] such that (3) holds. Therefore (15) leads to

(16)

Equivalently, (16) expresses that

p + p^R(A-By(y,u))J-. (17)

Making use of the formula

R(A - By(y,u))L = N((A - By(y, «))*),

(see, e.g., Brezis [3], p. 28), from (17) one obtains that

p + p € J V ( ( A - j B y ( y , u ) n , (18)

so, by the first equality in (14),

(A - By(y, u))*p =-(A- By(y, u))"p = -£. (19)


This, in conjunction with (14) and (9), ensures that

(-(A - By(y, U))*p, (C + Bu(y, «))*p) = & r,) G dL(y, u) (20)

i.e., the conclusion (6) of Theorem 1 is satisfied.Assume now that (5) is fulfilled. We proceed following the same argu-

ments as in case (4). We note that relations (9-15) remain valid because theyare independent of the assumptions (4) and (5). Now, in view of (5), for eachw G D(C) there is z G D(A) such that (3) holds. Then (15) guarantees that

(21)

Notice that (21) means that

p + p&R(C + Bu(y,u)}L. (22)

Combining (22) and the equality

R(C + Bu(y, u))1 = N((C + Bu(y, u))*),

we find thatBu(y,u))*). (23)

By (23) and (14) it follows that

(C + Bu(y, u}}*p = -(C + Bu(y, u))*p = -r/. (24)

Using (9), (14) and (24) we arrive at

(-(A - By(y, «))"(-p), (C + Bu(y, u))"(-p)) = (£, //) G dL(y, u). (25)

It is clear that (25) proves (6) with p = —p.Finally, if the regularity of L at (y ,u ) is satisfied, then

d L ( y , u ) C d y L ( y , u ) x duL(y,u)

(see Clarke [3], p. 48). Thus property (7) follows from (6) which completesthe proof of Theorem 1 . n

We present now a sufficient condition to have optimal pairs for problem


THEOREM 2. Assume that X,Y,E are real Banach spaces, with Xand Y reflexive, A : D(A) C Y — > E is a closed, densely defined linearoperator, C G L(X, E) and B : Y x X — > E is weakly-weakly sequentiallycontinuous and (Gateaux) differentiable. Next suppose that the set M :={ ( y , u ) € D(A) x X : Ay = Cu + B(y,u}} is nonempty. Let L : Y x X — >• IRbe a functional which is weakly lower semicontinuous and bounded from belowon M , and for which there exists an e > 0 such that the set

{ ( y , u ) G M : L(y,u}<\$L + e} (26)M

is bounded. (In particular, the last condition is satisfied if L is coercive onM). Then problem [P] admits a solution, that is, there exists ( y , u ) (E Msuch that L(y, u) = inf^f L.

PROOF. Let (yn,un) E M be a minimizing sequence for problem (P),which means that

Ayn = Cun + B(yn,un) (27)

and(28)

The boundedness of the set introduced in (26) and the convergence in (28)imply that (yn, un) is a bounded sequence. Then the reflexivity of the spacesX and Y ensures that along a subsequence one has

2/n — *• y weakly in Y and un — >• u weakly in X, (29)

for some element (j / , i /) G Y x X. Then, from (29), we derive that C(un) — »C(u) weakly in E and B(ynjun) — »• B(y,u) weakly in E. We see from (27)that

Ayn -> Cu + B(y, u) weakly in E. (30)

Since the operator A is linear and closed, it follows that its graph is weaklyclosed. This enables us to conclude from (29) and (30) that (y ,u ) G M. Theweak lower semicontinuity of L (at ( y , u } } and property (28) complete theproof of Theorem 2. n

2 An ExampleLet f] be a bounded domain in IR , with a smooth boundary <9fi. We considerthe nonlinear Dirichlet problem

?/ = 0 on

where / is a Cl diffeomorphism of IR satisfying the conditions

(a) \(f-l)'(ty< KO, vtem.Here K and A'i are positive constants and AI stands for the first eigenvalueof -A on (fi).

To apply Theorem 1 we take E = X = Y = U = V = £2(0), /I = -A :#1(0) n #2(O) C £2(ft) -> £2(ft), C = 0 and define B : L2(O) x L2(tt) ->£2(0) by jE?(y, w) = f(y-\-u). The mapping 5 is well defined since by condition(i) above one has

| /(t) |<A^| + /il5 VttIR, (31)for some constant K\ > 0.

Let us check that B is Gateaux differentiate. Indeed, it is seen that

- f ' ( y + u)(z^ \ \ f ( y + u + t(z + w))-f(y

\(L fl Lf(y + U + t T ( Z +

Jo dr

2 \ 2

w) dx

(z f ' ( y tr(z + w)} - f(y + u))dr dx

for all ( y , u ) , ( z , w } € ^2(0) x L2(tt) and t e ffi \ {0}. The Lebesgue Domi-nated Convergence Theorem and assumption (i) ensure that the right handside of the above equality converges to 0 as t —> 0.

Moreover, the map (z,w) <= L2(ft] x L2(fi) h-» f ' ( y + u)(z + w) e L2(tt)is linear and continuous. The continuity follows from the inequalities below

f ' ( y + u)(z K

Therefore the mapping B : L2(Q) x L2(Q.) —* L2(fl) is Gateaux differentiable,with B'(y, u ) ( z , w) = f ' ( y + u)(z + w).

We now show that Condition (Hi) is satisfied. Observe that in our setting,the set

M = {(y, u) e (^(ft) n #2(ft)) X Z2(0) : -Ay = f(y + u)}

is nonempty (cf. [4, Chapter 3]). Hence, to verify (Hi), let y,z G #oand u,u; € L2(tt) satisfy

/'(y + u)(2 + ™) (32)

and for any e > 0 define p : [0, e] — > L2(Q) by />(0) = 0 and

/»(<) = \(f~\f(y + «) + </'(y + «)(^ + u;)) - y - « - <(z + u;)), ^ £ (0,e].

Then, by the Mean Value Theorem for f~l, one obtains that

i •/*—i /^ -f ( i \ i -/• •/*'/ i ^ ^ i ^ ^ •/*"I *• \ »/ \ t/ ' ) J \!y ' / \ ' )) J

- u) + i/'(y + w)(^ + to) — /(y + w + i(^ + w))\2dx

The Lebesgue Dominated Convergence Theorem yields

p(t) —> 0 in L2(£l) as t —*• 0+.

This is true because

yl/(y + «) + «/'(y + u)(z + w)-f(y + u + t(z + w))f

= \ f ( y + u)(z + w)- -(/(y + u + t(z + w)) - f(y + u))\2

t-> \ f ' ( y + u)(z + w)- f ' ( y + u)(z + w)\2 = 0 for almost all x G O as t -^ 0+

and by the Mean Value Theorem and assumption (i) we have

l-\f(y + u) + t f ' ( y + u)(z + w)-f(y + u + t(z + w))\2

< ( \ f ( y + u}(z + w)\ + K\z + w\)2 < 4A'2 z + w\2 G Ll(ty.

By (32) and the definition of p, we have

-A(y+tz) = f(y+u)+tf(y+u)(z+w) = f ((y + u) + t(z + w + p(t))) , V* G [0, ,

This is because

(y + u) + i(z + w + p ( t ) ) = f - l [ f ( y + u) + t f ' ( y + u)(z + w)]).

Therefore the assertion (Hi) holds true with p ( t ) = 0 and q(t) = p ( t ) .To prove that condition (#2) holds, we first verify that

To this end, we show that the problem

= 0 on

has a unique solution z G HQ($I) fl #2(Q), for every / G I/2(fi). Actually,since

i i M 9 — ^ i > "£ G - n 0 ( S i j ,

from assumption (i) we have the estimate

- K

- KX?\\Vz\\l,(a) = (1- KX^Vzlfaw, Vz eConsequently, we can apply the Lax-Milgram Theorem to deduce the exis-tence and the uniqueness of the solution z as desired.

To complete the proof of condition (^2) it remains to justify that R(C +Bu(y,u)) = R(Bu(y,u)) is closed in L2(0). Actually R(Bu(y,u)) =Indeed, for any g G L2(£l), jrr^—^g G L2(£l) as well. This is because

1

f'(y + a.e. n

Thus,

g = Bu(y,u] ( -i-u)9) ' i-6' 9 ^ R(Bv(y,u)).

As a specific example of a function / satisfying the requirements in ourexample, we indicate

f ( t ) = kt + e g ( t ) , VteIR,

where g (E Cl(IK) is such that max{|g(£)|, |g'(t)|} < c, Vt G -ffi, with constants0 < k\ < \i and c > 0. Then, if e > 0 is sufficiently small, the conditionsimposed in our example for / are satisfied.

Assuming that the hypotheses in our example are fulfilled, Theorem 1enables us to find necessary optimality conditions for the problem:

Minimize /n /Oy(r) g(x, t}dx dt + \ Jn u2dx

subject tof -Ay = /(y + u) i n f i\ y = 0 on 9H.

Here / satisfies (i) and (ii), and g G (7(0 x_R; R) satisfies the growth condition

with CQ >0, 2 <p< 2N/(N-2), if N > 3, and any p > 2, if TV = 1,2.According to Theorem 1, if (y, u] is an optimal pair there exists p G HQ($})

satisfying

We get that-Ap-u = -g(x,y)f ' ( y

From this we deduce that

and the optimal pair (y , u) must solve the system

( -&y = f ( y + u)} -A (777%^) = -g(x,y){ \f(y+u)J -^ i*'

References[1] S. Aizicovici, D. Motreanu and N. H. Pavel (1999). Nonlinear program-

ming problems associated with closed range operators. Appl. Math. Op-timiz. 40:211-228.

[2] H. Brezis (1992). Analyse Fonctionnelle. Theorie et Applications. Mas-son. Paris.

[3] F. H. Clarke (1983). Optimization and Nonsmooth Analysis. John Wileyand Sons, New York.

[4] D.G. De Figueiredo (1989). The Ekeland Variational Principle with Ap-plications and Detours, Springer, Berlin.

[5] D. Motreanu and N. H. Pavel (1999). Tangency, Flow-Invariance forDifferential Equations and Optimization Problems, Marcel Dekker, Vol.219, New York.

[6] N. H. Pavel (1984). Differential Equations, Flow-Invariance and Appli-cations. Pitman Res. Notes Math. 113, London.


Internal Stabilization of theDiffusion Equation

Laura-Iulia Ani^a1 and Sebastian Ani^a2

faculty of Physics, University "ALL Cuza",Ia§i 6600, Romania

and2Faculty of Mathematics, University "ALL Cuza",

Ia§i 6600, Romania

ABSTRACT: In this paper we analyze a stabilizability problem for the diffusionmodel. We provide results of stabilizability based on spatially localized control, i.e.,we show that it is possible to diminish exponentially the density of a diffusive gas, byacting in a nonempty and large enough subset of the spatial domain.

1. INTRODUCTION AND SETTING OF THE PROBLEM

We consider a general mathematical model describing the diffusion of a gas ina bounded domain fi C Rn (n € N*) with a smooth boundary d£l. Let y(x,t) bethe density of the gas in the position x 6 fi at the moment t > 0. The diffusion isdescribed by the following system:

yt — Ay + a(x)y = m(x}u(x,t), ( x , t ) 6 QT = n x (0,T)

y ( x , t ) = Q, (x,t) € Sr = an x (0,T) (1.1)

y(x,Q)=y0(x}, x <E n

(T €E [0,+00]). Here m is the characteristic function of UJ, where a; is a nonemptyopen subset satisfying uj CC fi and u(x, i) is a control function. So, this is the casewhen the control acts only on a subset of fi.

We assume that

a 6 L°°(n); y0 e L°°(fl), yo(x) > 0 a.e. a: € fi.

It is well known that for any T € (0,+00), there exists u G ^2(<2w,r) (Qu,T —u x (0,T)) such that

yu(x,T) = 0 a.e. a; e ft,

where y" is the solution of (1.1)- This means that system (1.1) is exact null control-lable. For this we refer to G. Lebeau and L. Robbiano [8].

On the other hand the exact controllability of (1.1) implies the stabilizability (inL2(fi)) and that there exists a stabilizing feedback control

u(x,t) =-m(x)(Py(t))(x), (1.2)

where P is the solution to a certain algebraic Riccati equation (see J.L. Lions [9]and J. Zabczyk [10]). The problem is that the solution of (1.1) corresponding to thefeedback control given by (1.2) is not necessary nonnegative (as should be the density3/OM)).

For the exact controllability of the semilinear heat equation we refer to [5] and [7].The null controllability of a reaction-diffusion system has been investigated in [2].

The stabilization of a certain reaction-diffusion system was studied by L.I. Ani^ain [1].

Our goal is to find necessary and sufficient condtions for "nonnegative" stabi-lizability (stabilizability with preservation of the nonnegativity of the solution y to(1.1)) and in the affirmative case to indicate a feedback control.

The plan of the paper is the following. In Section 2 we shall present the mainresult. Section 3 is devoted to some final remarks.

2. THE MAIN RESULT

Denote by X^ the first eigenvalue corresponding to the following problem:

+ a(x)(p(x) = \tp(x), x 6 fi \ tJ(2.1)

Theorem 2.1. System (1.1) is "nonnegative" stabilizable if and only z/A" > 0.

Proof. First we shall prove that A" < 0 implies that (1.1) is not "nonnegative"stabilizable. Indeed, if there exists a control acting in ID such that

andyu(x, t)>0 a.e. in Q = fi x (0, +00),

then it is obvious thatyu(x,t)>z(x,t)

a.e. (x, t) 6 (fi \ w) X (0, +00), where z is the solution to

zt - Az + a(a;)z = 0, (x, t) 6 (fi \ w) x (0, +00)

(z,t) = o, (x, t) e (da u aw) x (o, +00)zx,Q = y0x, x 6 fi.

(2.2)

The solution z to (2.2) is strictly positive a.e.. On the other hand, using the Fourierdevelopment for z(t) in L2(fi \w) we may infer that

>M i e-A ' , V i > 0 ,

where M\ > 0 is a constant. The conclusion is now obvious.

If X^ > 0, then we shall prove first that there exists a positive T 6 (0, +00) anda control u €E L?(QU,T) such that

yu(x,t) > 0 a.e. in QT

andyu(x,T) = 0 a.e. x E u.

Indeed, if y0(x) = 0 on a subset of cu of positive measure, then we consider thefollowing system:

yt - Ay + a(x)y = -m(x)p •

y(x,t) =0,

y(x,Q) =y0(x),

x € fi, t > 0

x<Ed$l, t>Q

x e Q

(2.3)

where p > 0 will be precised later and

sgnLi(ul}y = \\y\\ll(a}y, if \\y\\mv) = 0.In what follows we shall use the next auxiliary result:

Lemma 2.2. For any p > 0 large enough, there exists T G (0, +00) such that(2.3) has a unique and nonnegative solution y on the time interval (0,T) and inaddition

y(x,T) — 0 a.e. x € ui.

Proof of Lemma 2.2. The operator defined by

D(A) = {y€Wt'\ty; Ay € L1^)}

Ay = Ay - a(-)y, Vy e D(A),is the generator of a compact Co-semigroup in L1(fi) (see [6]).

For each e > 0 we consider the approximating system

lit — A.y + a(x]ii = —m(x]p • ,, ,.?,,' —r-, x G fi, t > 0a » ' V '* V /f HtfWIIil^+e' '

y(x, 0 = 0, x e an, i > o

2/(a;,0) = yo(^), a; e fi,

Since the application

(2-3)'

m-yy(form L1(fi) to L1(J7)) is locally Lipschitz we conclude that problem (2.3)' has aunique mild solution ye € C(R+; LJ(n)). Using the comparison result for parabolicoperators we conclude that if 0 < e\ < £2, then

o<ysi(x,t)<a.e. On the other hand, since

— mp •

for almost every t > 0 and using the Baras compactness theorem we get that

for any T > 0. It follows that

inC([0,T]),for any T > 0.Let

f = Sup{T e [0, +00]; |b(£)IU'(-> > 0, Vi e [0,T]}.Passing to the limit (e — »• 0+) in (2.3)' we conclude that y is the solution of (2.3) on[0, T] for any 0 < T < T. Multiplying (2.3) by sgn y and integrating over fi we get

- tp.

Denote by w(t) = \\y\\L1 (fix (o,t)) and by a a constant satisfying a > ||a||L°°(n)- Itfollows that:

A*) = ll3/(*)IU'(n) < \\yo\\v(n) + az(t) - tpand so

(e-atz(W < \\y0\\L^)e-at ~ Pte~at, Vt € [0, T].In conclusion

-(l - e~at) - p fse^dsCx J U

)-(i - e~at) - 4 + Pe'at(~ + HF)-a az a az

Thus

So, for p > 0 large enough, there exists T 6 (0, +00) such that (2.3) has a uniqueand nonnegative solution on the time interval (0, T) and in addition

y(x,T) = 0 a.e. x € u

and

Proof of Theorem 2.1 (continued). We conclude that using the feedback control

u(t) = -p-sgnLl(u}y(t),

we get thatyu(x,t) > 0 a.e. inQT

andyu(x, T) = 0 a.e. in u.

Multiplying (2.3) by y and integrating over QT we conclude that

y(T) e L2(fi).

For t > T we shall use the following feedback control:

u(t] =

where p,y is the measure defined by

Heredy , . y(x

= lim ^— y(x)

dv~MX

where v~ is the outward normal versor to u).So, f a is a measure with support in du C UJ.The following system

' yt-&y + a(x}y = fj,y(t), (x, t) e fi X (T, +00)

(ar, t) = 0, (x, t)edttx (T, +00)

(x,T) = h(x) x<Ett

(h € L2(fi)) has a unique weak solution y & C7([T,Ti];L2(n)) n J4Cr([T,T1];L2(fi \aJ))nAC([T;r1];L2(a;))nL2(T,T1;^(0))nL2

oc(T,T1;//2(n\cU))nLL(T,TVTj e (T,+oo) (see [3], [4]) and it satisfies

y(x, t) = 0 a.e. x E w, i > 0

and the restriction of y to (O \ a;) X [T, +00) is obviously the solution to

' yt - Ay + a(x)y = 0, (z, t) 6 (fi \ w) X (T, +00)

y(x, t) = 0, (x, t) £ (flfi U <9w) x (T, +00)

y(x,T)=h(x) x e O .

Now it is clear that

W, Vi > 0

and we get the exponential stabilization of the solution y.

3. FINAL REMARKS

The main conclusion of the previous section is that the feedback control

-p-sgnLl(u})y(t), te [0,TJu(t] =

stabilizes the system (1-1) if and only if A" > 0.

If we denote by AI the first eigenvalue of the operator

D(A) = #01(O)n#2(n),

Ay = -Ay + a(-)y, Vy € D(A),then we have

AI < A?.

If AI > 0, the system (1.1) can be stabilized by the trivial control u = 0.If A! < 0, then the control u = 0 does not stabilize (1.1) (for related results see

[1]), but the system can be stabilized if A" > 0.

In the same manner as in Section 2 can be proved that (1.1) is not exact "non-negative" controllable.

REFERENCES

[1 ] L.I. Ani^a, Asymptotic behaviour of the solutions of some reaction-diffusion processes,International ,]. Appl. Math., submitted.

[ 2 ] S. Anilja and V. Barbu, Local exact controllability of a reaction-diffusion system,Diff. Integral Eqs., to appear.

[ 3 ] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Acad-emic Press, Boston (1993).

[ 4 ] V. Barbu, Partial Differential Equations and Boundary Value Problems. KluwerAcad. Publ., Dordrecht (1998).

[ 5 ] V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Op-tim., to appear.

[ 6 ] H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures asinitial conditions, J. Math. Pures Appl, 62 (1983), 73-97.

[ 7 ] E. Fernandez-Cara, Null controllability of the semilinear heat equation, ESAIM:Control, Optim., Gale. Var., 2 (1997), 87-107.

[ 8 ] G. Lebeau and L. Robbiano, Controle exact de 1'equation de la chaleur, Comm.Partial Diff. Eqs., 30 (1995), 335-357.

[ 9 ] J.L. Lions, Controlabilite exacte, stabilisation et perturbation de systemes distribues,RMA 8, Masson, Paris (1988).

[10 ] J. Zabczyk, Mathematical Control Theory: An Introsuction, Birkhauser, Boston(1992).


Flow-Invariant Sets with Respect toNavier-Stokes Equation

V. BARBU* AND N. H. PAVEL**

* University of lasi, Department of Mathematics, 6600 lasi, Romania

**Ohio University, Department of Mathematics, Athens , Ohio 45701, USA

1. Introduction. A new result (Theorem 2.1) on the flow-invariance of a

closed subset with respect to a differential equation associated with a nonlinear

semigroup generator on Banach spaces is given(the proof will be given in [1]).

Applications to the flow-invariance of controlled flux sets ( the Enstrophy and

Helicity sets) with respect to Navier-Stokes equations are presented.

Recall that K is said to be a flow invariant set with respect to a differential

equation y' = Ay, if every solution y starting in K (i.e. y(0) = x 6 A') remains

in K as long as it exists (i.e. y(t) 6 K. for all t in the domain of y ] . In our

cases here, we are using the strong solutions, so actually we deal with the flow

invariance of A* D D(A). We think that the existing results on this topic are not

applicable to our cases treated here. Indeed, the general result of R.. H. Martin

Jr. [8] requires the right hand side A of (2.9) to be continuous and dissipative

on K. None of these key conditions (on A) are required here.

Note also that the subsets considered here are closed, but not necessarily

convex. A different approach to flow-invariance of such sets with respect to

Navier-Stokes equations was given by Barbu and Sritharan [2].

Our general framework is V <—»• H <—> V' algebraically and topologically with

V and H real Hilbert spaces, V—densely and compactly embedded in .//, V—

the dual of V.

Section 3 is devoted to the structure of the contingent cone Tjf(f) to A' at

/ 6 K in the topology of H. Therefore, applications of Theorem 2.1 are given

to the semilinear equations of the form:

y' = Cy + Dy, (1.1)

where C is a Co-semigroup generator So(t) in // and D is a nonlinear perturba-

tion of C such that C + D = A is a generator of a nonlinear semigroup S(i) as

indicated in (2.4).

Recall the general results of Pavel [7] for the flow-invariance of a closed subset

K with respect to (1.1). "If D is continuous and dissipative from K C X into

X" , or UD is only continuous on K but So(t) is compact for t > 0", then K is

invariant with respect to (1.1) iff:

Td(SQ(h)x + hDx: K) -» 0 as h | 0 (1.2)a

for all a; 6 K, where d(z\ K) stands for the distance from z to K.

Again, even this result which is more general than Martin's result above, is

not applilcable to our cases here, simply because here D is neither continuous

on K nor dissipative.

2. Flow-invariance results in reflexive Banach spaces

Let X be a Banach space of norm || • || and let X'. Recall that a. (possible

multivalued) operator A : D(A) C X — » 2A is said to be w-dissipative (for some

u G E), if for every A > 0 with 1 — Aw > 0, and for every Xj G D(A), there are

j/j G AXJ, j = 1,2 such that:

yi - (,TJ - Aj/ 2) | | (2.1)

or equivalently:

(j/i - y-2,J(x\ ~ ^2)} < ^Iki - £ 2 [ | 2 , (2.2)

where J : X —> X' is the duality mapping of X. If J is multivalued, one replaces

J(x\ — x 2 ) in (2-2), by some x* G J(.^i — Xi)-

Suppose in addition to the inequality (2.1) the following range condition holds

D(A)C R(I-XA), (2.3)

for all sufficiently small A > 0 (precisely, for \ui < 1).

The fundamental result on the generation of nonlinear semigroups S'A(t) =

S(t) is the following one (known as the exponential formula of Crandall-Liggett

[6])-Let A be w-dissipative (i.e. (2.1) holds) satisfying the range condition (2.3).

Then

lim (/ - -A)~nx = S(i) G D(A) (2.4)

for all x 6 D(A) and t > 0. Moreover \\S(t)x - S(i)y\\ < etw\\x - y\\, Mt > 0,

x, y 6 D(A), S(t + s) = S(t)S(s), 5(0) = /—the identity on X, limuo S(t)x = x,

Vx 6 D(A), \\S(t)x - S(s)x\\ < \t - s \Ax exp(2^-0(t + 5)), t,s > 0, for all

x G D(A), where uj0 = max{0,w} and \Ax = inf{||t/||, y e Ax}. If s -^ S(s)x

is differentiable at s = t, then u(t) = S(t)x G D(A) and it is the only (strong)

solution to the Cauchy problem

u'(f) = Au(t), u(0) = a;, x 6 D(A), t > 0 (2.5)

In our example applications here, the following hypotheses are fulfilled.

(HI) For every x £ D(A), t — *• S(t}x is differentiable at every t > 0.

So:

lim ^^^ ~ ^ = Aa;,Va; € D(A) (2.5J '

and u(i) = S(i)x is the only strong solution to the problem (2.5). Let K be a

closed subset of X. The basic hypothese on the relationship between A and K

are given below:

(H2) The projection Pjf(y} on K exists for all y 6 D(A). Moreover,

PK(D(X))cD(A)nK. (2.6)

This means, that for every y £ -D(A), there is y0 — Pi{(y} £ D(A) fl K such

that the distance d(y; K) from y to K satisfies

d(y- K) = mi{\\y -z\\,z£ K} = \\y - y0\\ = d(y; K n D(A}) (2.7)

for some y0 £ Kr\D(A). Recall also the definition of the tangential (contingent)

cone TK(X) to K at x £ K in the sense of Bouligand [4, Ch.l]

TK(x) = {v £ X, lim \d(x + hv; A') = 0} (2.8)HO n

= {v £ X; 3r(h) £ X with r(h) -*• 0 as h j 0 and <c + /i(v + r(h}} £ K}

Note that TK(X] is a closed cone, even if K is not closed [4. p. 2].

The main result of this section is

Theorem 2.1. Suppose that A and K satisfy Hypotheses (HI) and (H2). Then

a necessary and sufficient condition for K 0 D(A) to be a flow-invariant set with

respect to

(2.9)

s

lim-^/(.T + / iA;r ;A'nD(A)) = 0, Va: 6 A' n £(A). (2.10)ft 4-0 h

The proof will be given in [1]. Note that Necessity is immediate: Indeed,

suppose that the strong solution y ( t } = S(t)x, of (2.9) with x £ K n D(A)

remains in A', for all t > 0. As 5(t)D(A) C D(A), it follows that actually S(f)x £

Kr\D(A)so±d(x+hAx;KnD(A)) < ^\\x+hAx-S(h)x\\ = \\s(h)*~x -Ax\\ -* 0

as h I 0, so (2.10) is a necessary condition for the flow-invariance of K n D(A)

with respect to (2.9).

The following simple known lemma will be useful in the next section.

Lemma 2.1. Let X be a real Hilbert space of inner product {•, •} and let (/> :

X -^ R be a Frechet differentiable functional. Consider the closed set:

(2.12)

Then the tangent cone T/<-(/) to K at f £ K is given by:

( X, if<t>(f) < 0T ( f ) = < (213)M ; ~ I v £ X ; < t > ' f , v < Q = M, if </>/ = 0. V ' ' ;

Proof: The inclusion T/<-(/) C My is immediate. Viceversa. the interior M9 of

My is:

My0 = {v 6 X; {<£'(/), V) < 0} C T K ( f ) .

This is because y £ M9 implies / + tv £ M9 for all £ > 0 sufficiently small,

which is easy to check. Taking into account that the closure My = My and that

!/<•(/) is closed, we get My C T/<-(/), which completes the proof.

3. Flow-invariance of flux sets

We will present several consequences of Theorem 2.1 to the particular case

X = H — a real Hilbert space of norm • | and inner product ( - , •}, and K C F,

with K — closed in H . Precisely, we will assume that

V ^ H <-> V (3.1)

algebraically and topologically, and the embedding of V in H is dense and com-

pact (i.e. the bounded subsets in V are relatively compact in H]. V' is the dual

of V, and H' is identified with H . Denote by Pjf(y} the projection of y on K in

(the norm of) H , and by T^(x) and Tji(x) the tangential cones of K at x £ K

in V and H, repectively. Denote by <C •, • > and || • || the inner product and

the norm of V, respectively. Let also A : D(A) C H -* H satisfy (2.1)+(2.3).

Denote by S(t) = etA the semigroup generated by A (see (2.4)). The basic

hypotheses of this section are:

(C3.1) K is a subset of F, which is closed in H. Every subset of K which is

bounded in H is bounded in V, too.

(C3.2) Pg(D(A)} C D(A).

(C3.3) Ax G Tjf(x), V.T e K n D(A), i.e. /la; is tangent (in H) to K at a;, in

the sense of (2.8).

We first note that under (Cl), the projection of y on K exists, i.e. there is

2/o G K, such that d(y; K) — \y — y0 . This is because the minimizing sequence

zn G K', lim^-Kx, \y — zn\ = d(y; K) is bounded in // so by (Cl) is also bounded

in V, so it is relatively compact in H, i.e. zn contains convergent subsequences

znk in H. Say znh —» 2/0- Then Pjl(y) — 2/0-

A direct consequence of Theorem 2.1 is

Corollary 3.1. Under Hypotheses (C1)-(C3), K n D(A) is a Row invariant set

with respect to y' = Ay (i.e., etAx 6 K n D(A), Vx G K n D(A)).

Hypotheses (C3.1) and (C3.2) are satisfied in the case of subsets of the form

K = {f £ V- H / l l 2 < yjd/ l 2 ) + /.} = {/ € F; (/) < 0} (3.2)

where

-P, P>® (3-3)

and

( /?eC1(R+), v'(O) = 0, v?'(r) > 0, Vr 6 R+. (3.4)

Clearly, A' is not necessarily convex.

In applications to heat propagation, we will use

V - H*({1) H = r2(ft), V' = H-l(tt] (3.5)

Here fi C IK", is a bounded open subset with smooth boundary T = d£l. In this

case

H / l l = |V/| = |grad/|L2(ft) (3.6)

so K represents a constraint on the flux q = V/, i.e.,

/ q 2 ( x ) dx <(p(! e2(.T) da;) + p (3.7)Jn Jv,

where e(x) = f ( x ) is regarded as the energy of the flow / = /(a;), in fi. In

order to verify the tangential condition (C3) we need to find the tangential cone

Tjf(f) to K as in (3.2) at / 6 K, in the norm of H . Precisely, this is given by:

Lemma 3.1.

f //, f lW)<orrH H< "TJl(f) = l ' '

( if(f>(f) = Q a n d f € D ( J ) .

Here J is the duality mapping ofV (the cannonical isomorphism from V into

V'}.

Finally

D ( J ) = {/ 6 V;./(/) e H}. (3.9)

Clearly, in the case of V = //o(0) and // = i2(fi), Lemma 3.1 becomes:

Corollary 3.2. Let

V/|2 < ^(|/|2) + p} (3.10)

with p as in (3.4) and </)(/) = |V/|2 - </>( | / |2) - p. Then

, i

< 0= 0 and (3.11)

Proof: Indeed, in this case J(/) = -A/ with .D(A) = ,ff01(fi)n//2(fi). Clearly,

in this case, / 6 K and f£H2(£l) means f<=D(A).

Remark 3.1. In the proof of Lemma 3.1 it is essential to observe that:

T % V ( f ) c T g ( f ) , V / 6 A - . (3.12)

This is because (with the notations in (2.8)) r(h] —> 0 in V implies r(h) —> 0 in

H, as h I 0. Now, with K as in (3.2), it follows from Lemma 2.1 that:

= 0.{v 6 V-< f , v > -^(|/|2){/,w> < 0} = M]f if

Indeed, <?!> is continuous from V into R and

(3.14)

so (3.13) follows from (2.17).

Proof of Lemma 3.1: If <£(/) < 0, then by (3.12) and (3.13) we have

V = T%(f) C If (/) C H. (3.15)

This, the density of V in H and the fact that Tjf(f) is closed in //, yield

Tg (/) = H.If <£(/) = 0 and f £ D ( J ) , then

M°f = {v£V-^f,V> V(l/|2){/,»> = 0}(3.16)

J-l(f},v >= 0}

is dense in #. Indeed, if by contradiction, this were not the case, then the

closure M j would be strictly included in H . This implies the existence of an

TJ G H, TI ^ 0 with 77 orthogonal on My, i.e. (r/,v) = 0, for all v £ M9, so

< J ~ l ( r ] ) , v >= 0, Vv £ M°. This and (3.16) (the orthogonal space of M° is

one dimensional) imply that

/ - v'(\f\*)J-\f] = tJ-^rj), for some t 6 E

so / £ D(J), which is absurd. On the other hand, in view of (3.13)

M°f C M] = T%(f) C TJKf) C H

which implies Tg(f) = H for <£(/) = 0 and feD(J). Finally, if / 6 D(J), than

we can replace < /, u >= ( J ( f ) , v } in (3.13) so for </>(/) = 0 and / G D(J),

T%(f) = {veV- « - J(f) + '(|/|2)/, V) > 0} C If (/)

which yields Mf C T7f (/).

For proving the converse inclusion let v £ T j l ( f ) , i.e. there is r ( t ) —> 0 in H

as / J, 0 such that / + t(v + r ( t ) ) = f + tvt £ A', with vt = v + r ( t ) —>• v in

H. We derive (for 0(/) = 0) < f,vt >< ^'(|/ |2)(/,Wf} + a(i) with a(i) ^ 0

as t I 0 and < f,vt >= ( J ( f ) , v t ) which implies (J(/) ,«) < </>'( | / |2){/,o) i.e.

Tjf(f~) C M¥ and the proof is complete.

4. Flow invariant sets for Navier-Stokes equations.

We will discuss the invariance of Enstrophy and Helicity sets with respect to

the Navier- Stokes system:

yt - v&y + (y • V) y = Vp, in (0, T) x fi

V • y = divy(f, a:) = 0, i n ( 0 , T ) x Q(4.1)

J/(O, .T) = 2/0 (z), m 1)

2/ ( t , z ) = 0, o n ( 0 , T ) x < 9 1 ) ,

where fi, is a bounded domain of E", with smooth boundary dfl, yo 6 (L'2(Q))n.

V = ( a f ^ - - - ; gf^) is tne gradient operator, and T/(X) = ( ? 7 i , . . . , ryn) is the

outward normal to d$l at .r € 9fi, r/ is positive and p = X^x) ^s ^ne pressure

[2].

The basic functional spaces are

V = {y 6 (tfoH^))"; div j/(a?) = 0, in fi} (4.2)

, div y(a;) = 0, in 0, ^.T) • ij(x) = 0, on 5fi}. (4.3)

For y e (Hl(tt))n, y=(yi,..., yn), by definition

(4.4)

Denote by

(4.5)

the inner products and norms in V and //, respectively.

Finally, (y • V)y is the following n-component function of

(4.6)

with

, k=l,...,n. (4.7)ox jj=L J

Recall also the formula of "integration by parts:

/ V/ • ydx = f y - r,da - I /div ydx, f G HJ(O), y e (Hl(Q})n (4.8)7i7 Jdil J f l

In view of (4.8), the gradient Vp(t,x) (with respect to a;) is orthogonal on

H (Jn Vp('i, a;) • j/ = 0 for all j/ G H, with p(i) G H1^), (p(t))(x) = p ( t , x ) ) .

Therefore, multiplying formally (4.1) by w G V", in the sense of the inner product

of H, we also have (Vp(f), w) = 0, so (4.1) can be represented in H as:

= 0, < G ( 0 , T )(4.9)

where

, w) = b(y, y, w), Vy,w£V (4.10)

withn

iZjtiWjdx (4.11)b(y,z,w}= V / yiDiZjWjdx = Vi, j=1^n

and A e £(F, F')> with

n

Vtt t • V^da;, Vu ,u £ F (4.12)

In order to show that vA-\-B is a semigroup generator, one proceeds by (trunchi-

ation) of B, i.e. the quatization

„ . . I By, X \\y\\ < t f , y t D ( A )

It follows that for a = a^ sufficiently large, vA + BN + a^I is ?n-accretive in

H (see [2]). Therefore, for y0 6 -D(-4), the Cauchy problem

24 + //Ay7V + BN(yN) = 0, < 6 (0, T)(4.14)

2/jv(0) = t/o

has a unique solution

yN € W1'00([Q,T]-1H)nL°°(0,T;D(A)).

Moreover, by standard estimates([2],[10],[ll], there is an interval T* with T* —

T*(||j/o||) < T if n = 3, and T* = T if n = 2, such that

\\yN(t)\\<N, v < e [ o , T * ]

for N sufficiently large. This means that y = y^ satisfies B^(yN) = By, so

y = yN is the strong solution to (4.9). In other words it suffices to check the

condition of Theorem 2.2, with — vA — Bj\ in place of A. In what follows we

shal assume that n = 2,3.

Flow invariance of the Enstrophy sets

Let K be so called the Enstrophy set [2].

, p>0 (4.15)

where V X / = curl/(a;). It is true (although not immediat!) that the (

norm of curl/, satisfies

| V x / | = |V/| = ||/||. (5.16)

The tangent cone TJi(f) is given by (3.8) with /(/) = Af. Clearly, K is closed

in H . Therefore, the invariance of K n D(A) is equivalent to the tangency of

v = —vA — BNf to K to K n D(A), i.e. to the inequality in (3.8), i.e.

E ( f ) = (Af - ¥/(|/|2)/, vAf + BNf) > 0 (4.17)

with / 6 D(A) n K, I I / H 2 = |V/|2, H / l l 2 - ^(|/|2) -p = Q. Indeed, E ( f ) can be

estimated as follows:

E(f) = V\Af\2 + (Af, B N f ) - zV(l/|2)(^/, /} (4.18)

as (SNfJ) = 0 due to {/,/} = &(/,/ , /) - 0 (as b ( f , g , w ) = -b(f,w,g)).

Similarly to inequalities in (4.6),

(4.19)'M

where X± is the first eigenvalue of A. Finally

( B N f , A f } < \(Bf,Af}\ = \b(JJ,Af)\ < c\\f\\\Af\'2 (4.20)

for some c > 0. Therefore, with r = |/|2, we have

E(f) > (v - c f\\ - JV(r))|A/|2 > 0 (4.21)

if// . ,„

> 0 (4.22)

as

To conclude, we have proved the following result on the invariance of A' in

(4.15):

Theorem 4.1. Let p : R+ -> R+ be of class C<1(R+), and

)T^< 1

where AI is the first eigenvalue of A, and c > 0 is the best constant in (4.20).

Then the subset K n D(A) is flow invariant with respect to the Navier-Stokes

equation (4.9).

Flow invariance of Helicity sets.

With the same technique as in the previous subsection, let us study the invari-

ance of the "Helicity Set" (see also [2],[9])

K = {/ e V- | / f ( x ) • V x f(x)dx\'2 + A2 f \ V f ( x ) \ 2 d x < p} (4.24)Jn Jn

where V X / = curl/. The helicity set has an important role in fluid mechanics

and in particular it is an invariant set of Euler's equation ([9]). With the previous

notations, A" can be written as:

K = {/ £ V- </>(/) = </,curl/)2 + A2 | |/ | |2 -p2 < 0} (4.25)

where A and k are positive constants. Clearly, K is closed in H. Set y(/) =

{/, curl/}. Taking into account that

(y, curb) = (curly, v), y,v£V, (4.26)

./(/) = Af, and

{<£'(/),!>} = <4cX/)curl/,u) + 2A2 < f,v >

for u 6 V. we derive as in the case of (3.8) (as < /, v >= (A/, v) for /

H, if < £ / < 0

<; e F, <-A2A/ - 2^(/)curl/, «} > 0} if 0(/) = 0, and / e D(/l).(4.28)

Therefore, the only fact we have to check for the invariance of A', is the last

inequality in (4.28) with v = —vAf - BNf, i.e.

E ( f ) = (\2Af + 2^(/)curl/, vAf + BNf) > 0 (4.29)

for </>(/) = 0 and / 6 D(A).

First let's note that (4.25) (i.e. the definition of A') implies

We also have (see (4.20))

< A2cj | / | | |A/ |2 < c\p\Af\2

(4.30).

< 2HA/IH/H < 2^A-1/2|A/|2

due to (4.19) and |curl/| = ||/|| < AJ / 2 Af\.

Combining these inequalities, we clearly get

E(f) > (A2 / / - Xcp - -2Pv\-l/2)\Af\\ (4.31)

Thus, if

\2v > Ac/> + 2pz/A~1 / 2 , (4.32)

we have E(f) > 0, i.e. —vAj — B^f 6 T'jf(f). Therefore we have proved:

Theorem 4.2. Suppose that (4.32) holds. Then the helicity set defined in

(4.24) is a flow mva.ria.nt set with respect to the abstract Navier-Stokes equation

(4.9). More precisely, the solutions starting in K n D(A) remain in K n D(A)

as long as they exist in the future.

The proof of formula &(/, /, curl/) = 0 in (4.30) is given in [1].

References

[1] V. Barbu and N. H. Pavel, Flow-Invariant Closed Sets with Respect to

Nonlinear Semigroup Flows, to appear.

[2] V. Barbu and S. Sritharan, FJow-in variance preserving feedback controllers

for the Navier-Stokes equation,]. Math. Anal. Appl.( to appear).

[3] N. H. Pavel, Invariant sets for a class of semilinear equations of evolutions,

Nonlinear Anal. TMA, 1(1977), pp. 187-196.

[4] H. Brezis, On a characterization of now-invariant

sets, Comm. Pure Appl. Math., 23(1970), pp. 261-263.

[5] D. Motreanu and N. H. Pavel, Tangency, flow

invariance and optimization problems, Pure and Appl. Math., A series

of monographs and textbooks, Marcel, Vol. 219, Marcel Dekker. New York,

Basel, 1999.

[6] V. Barbu, Nonlinear Semigroups and Differential

Equations in Banach Spaces, Noordhoff, Leyden 1975.

[7] N. H. Pavel, Semilinear equations with dissipative

time-dependent domain perturbations, Isreal J. Math., 46(1983), pp.

103-122.

[8] R. H. Martin, Differential equations on closed

subsets of a Banach space, Trans. Amer. Math. Soc., 179(1973), pp.

399-414.

[9] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics,

Springer-Verlag, 1998.


[10] P.Constantin and C.Foias, Navier-Stokes Equations, The University of

Chicago Press.Chicago,1988.

[11] R. Temam, Navier-Stokes Equations; Theory and

Numerical Analysis, North Holland, Amsterdam. 1984.


Numerical Approximationof the Riccati Equation

via Fractional Steps Method

Tudor BARBU1 and Costica MORO§ANU2

1 Institute of Mathematics of Romanian Academy,6600 Ia§i, Romania

2Department of Mathematics, University "Al.I.Cuza",6600 Ia§i, Romania

Abstract. In this paper we prove the convergence of an iterative schemeof fractional steps type for the nonlinear Ricatti equation. The use of thismethod simplifies the numerical algorithms due to its decoupling feature. Anumerical algorithm and numerical results ore presented.

I. INTRODUCTION

Consider the Ricatti equation

( P'(t) + A*P(t) + P(t)A + P(t}BN~iB*P(t) = Q, t(= (0, T]

lP(0) = P0,

where A, Q e L(lRn,lRn), B E L(Mr\lRm), N e L(IRr\JRn}, N = N*


Q = £* > 0.For any integer M > 1, let us associate to the time-interval [0, T] the

equidistant partition 0 < e < • • • < Me = T of length e — T/M and,corresponding to this, the following approximating scheme (the Lie- Trotterscheme)

( P'E(t] + P£(t}BN-lB*Pe(t] = 0 t e [ie, (i + l)e]

\ Pe(ie) = Z£((i + l}e] + ee-Af£Qe~A£

Z'e(t) + A*Ze(t) + Ze(t}A = 0 te [ie, (i + l)e]

for i = 0, 1, • • • , M - 1, where P~ (ie) = lim Pe(t).t/*is

Taking into account Theorem 3 in [1], we may view Pe(t), a.e. t e [0, T],as an approximate solution for the unknown P(t) in (1.1). So, in the sequel,we will deal with the computation of the approximate solution for Pe in

(1.2)-(1.3). The advantage of this approach consists in simplify the numeric

calculus of the approximation to P(i).

2. THE MAIN RESULT

In this section we will concern with the nonlinear algebraic equation

(1.2). Firstly, we will introduce the discrete state equation. So, using astandard implicit method, (1.2) becomes

(2.1) Pl+l +ePl£+1BN-1B*pi+1 = Z*+1, i = 0, 1, • • • , M - 1,

where we have denoted Pe(ie] = Pj, Z*+1 = Ze((i + l)e) + ee~A*£Qe-AE .Because B and jV are self-adjoint then BN~l B* can be equivalently written

as BB*.

At any time level i + 1, i = 0,1, • • • , M — 1, we associate to equation(2.1) the following iterative linear process

(2.2) £PejkBB*PE,k+l = Z£tk.

The main result, Theorem 1 below, is concerned with the convergence

of the scheme (2.2) for k —> oo.

Theorem 1. // Z\ > 0, i = 1, 2, • • • , M, and P£jk > 0, for all k GIN, then the sequence {P£,k} is convergent to P£

+1 in L(Mn,JRn], i =0 , 1 , - - - , M - 1.

Proof. From (2.2) we have

<| |B< \\B*P£,k+iX\\,

Then

(2-3)

We have also

(2.4)

\B*P£,k+1X\\2<C, VX = 0.

So

i.e..

\B*(P£;k+1 - P£,k}\\ < e \B\\

(2.5)

From (2.5) we may conclude that

(2.6) {5*(Pe,fe} — > 0 ask —— oo,

for e > 0 small enough.

From (2.2) it follows

(2.7) P£,fc+1 + eP£,kT = Zl+l + eRe,k,

where Re^ — > 0 as k — > oo and T — lim BB* P£,k- Then, we have thek —— K>o

inequality

-Pe,*||<e||Pe,*-Pe,*-l|| T\ +£\\Retk\\

and taking into account (2.6), we can derive that P£jk is convergent asclaimed.

3. NUMERICAL APPROACH

This section is devoted to numerical approximation of the solution

corresponding to problem (1.2)-(1.3).

As regards the equation (1.3), we know that the solution is given by

(3.1) Zl£+1 =e~£A*p-(ie)er£A+e-£A*Qe~£A, i = 0,M-l.

Next, for convenience we re-write the iterative linear process (2.2) associated

to the nonlinear equation (1.2), namely

(3.2) P£,k£,k+1

Algorithm FRACJc

1. Initialize parameters

e, M, kmax, A, B, Q, N, P0;

P£-(0) = Po;for i = 0 to M - 1 do

2. Compute Zl£+1 from (3.1);

3. Compute P£i+1 solving (3.2);

for k — 0 to kmax do

End-forpi+l — p ,r

e — re,kmaxi

End-for;

End.

The MATLAB code listed below can be used in computing the approx-

imate solution Pe(T).

A=[1.4 -.208 6.175 -5.676; -.581 -4.29 0 .675;

1.067 4.273 -6.654 5.893; -.048 4.273 1.343 -2.104];

Q=[l 0 1 -1; 0 1 0 0; 1 0 1 -1; -1 0 -1 1] ;

B=[0 0; 5.679 0; 1.136 -3.146; 1.136 0] ;

P0=100*eye(4) ;

M=ll;

e=0.5/M;

N=700*eye(2);

kmax=l ;

PM=PO ;


The graph of Z is

1.95

The graph of PK is

2.25


for i=0:MZ=exp(-e*A')*PM*exp(-e*A)+e*exp(-e*A')*Q*exp(-e*A);

PK=PM;

for k=l:kmaLXPK=(eye(4)+e*PK*B*inv(N)*B')'*Z;

end

PM=PK;

enddisp(Z);disp(PK);plot(Z);

plot(PK);We have obtained the following numerical results for Z and PK- They are

presented exactly as they appear in the MATLAB output screen

The matrix Z

/1.6134 1.5870 1.6703 1.6979 \

1.4973 1.4728 1.5501 1.5758

1.8670 1.8365 1.9328 1.9648

\ 1.5041 1.4795 1.5571 1.5829 /

The matrix PK

Z = l.0e + 118*

PK = l.Oe + 230 *

/ 0.0 0.0 0.0 0.0

1.6972 1.6695 1.7570 1.7861

3.2879 3.2341 3.4038 3.4601

V 2.3082 2.2705 2.3895 2.4291)


REFERENCES

[1] V. Barbu, Approximation of the Hamilton-Jacobi equations via Lie- Trot-ter product formula, Control- Theory and Advanced Technology, vol. 4(1988), No. 2, pp. 189-208.

[2] V. Barbu and G. Da Prato, Hamilton-Jacobi equations in Hilbert spaces,

Research Notes in Mathematics, 86, Pitman, London.


Asymptotic Analysis of the Telegraph Systemwith Nonlinear Boundary Conditions

L. Barbu*, E. Cosma*, Gh. Moro§anu", and W.L. Wendland**

"Ovidius University. Math. Dept.. Bd. Mamaia 124. 8700 Constanta, Romania

'Stuttgart University. Math. Inst. A. Phaffenwaldring 57, 70569 Stuttgart. Germany

Abstract. The boundary layer function method of Vishik and Lyusternik is used to investigate the

behavior of the solution of the problem (1.1). (1.2). (1.3) below as the small parameter s tends to zero.

1. Introduction

In the domain Dj-: = {(x. t); 0 < x < I. 0 < t < 7'} we consider the telegraph system

f -:ut + vx + ru = /,,(1-1)

[ v, + vx + gv = /2.

with the following initial and boundary conditions:

u(x,0) = «0(;r). v(x,0) = v(>(x), 0 < x < I. (1.2)

. ( 0 ; f ) = -A(i '(0 !f)) !

( l , t ) = f a ( v ( l , l ) ) ; Q<t<T.

where g > 0. r > 0 are constants. f\. ji : DT —» R. 0\. ft-i : IR —> IR are given functions,and £ is a positive parameter.

'The third author's work has been supported by the German Research Foundation DKG (Deutsche["orschungsgcmeinschaft). under the project DFG We 659/35-2.

This problem (which will be called P?) is a model for an electrical circuit with nonlinearresistors at the ends x = 0 and x = 1. where u represents the current flowing in the line,and -i' is the voltage across the line. The parameter £ represents the specific inductance,

while the specific capacitance, which usually multiplies i>t in the telegraph system, isassumed to be equal to 1. This does not restrict the generality of the problem. For moredetails concerning the physical model, we refer the reader, e.g.. to [Brl]. [Br2], [BrM].[CK2], [Mo].

In the case of integrated circuits, we have a system of In equations instead of (1.1),but the treatement of the corresponding problem Pf: is essentially similar.

We will assume that e > 0 is, a'small parameter. If we put 5 = 0, then our problemP, becomes a parabolic boundary value problem, say P0. (Notice that such a parabolicmodel, extended to the case of n telegraph systems which are connected by means of someappropriate boundary conditions, is applicable to on-chip and interchip interconnectionsfor many digital systems, where the distributed inductances of the conductive layers arenegligible [MaN]).

However, from a physical point of view, the parameter e appearing in our problem Pc

is just small, not zero. So. we ask ourselves if. by putting s = 0. the resulting parabolicmodel PQ still describes the physical phenomenon properly. In other words, the question iswhether the solution of the reduced model. PQ, is "sufficiently" close to the solution of thehyperbolic model Ps. Using the Vishik-Lyusternik method, an asymptotic expansion ofthe solution of P, will be constructed. The presence of some corrector in that asymptoticexpansion shows that the reduced parabolic model Pf> is not valid in DT- with respect tothe C'-norm (i.e., the norm of uniform convergence in DT}-

In Section 2 we will indeed derive a formal zeroth order asymptotic expansion for thesolution (u£,ve) of problem PE. The segment {(x,0); 0 < x < 1} is a boundary layer andthe expansion of UE contains a corresponding corrector (boundary layer function), withrespect to the C'-norm.

In Section 3 we will state some sufficient conditions for the existence, uniqueness,and higher regularity of the solutions to problems P? and P0 (see Theorem 3.1). Theseconditions are independent of s.

Finally, in Section 4. we obtain some estimates for the remainder terms of the expan-sion established in Section 2. with respect to the C-norm. This justifies our asymptoticexpansion.

Notice that, if we consider the //-norm (1 <p< oo) instead of the C-norm. then ourcorrector (see (2.2). (2.4). and (2.9) below) is not necessary any more (actually, it can beincluded in the remainder), and this means that our problem is a regularly perturbed onewith respect, to the //-norm (cf., e.g.. [E, Chapter 1, §1.1]). On the other hand, takinginto account the form of the corrector, we can see that the solution of the reduced problemPO approximates uniformly the solution of Pt in any rectangle [0.1] x [8.T] with 6 > 0.

A'lany similar, singularly perturbed boundary value problems can be studied by thesame technique, in particular, the case of a nonlinear partial differential system insteadof (1.1) as well as the case when the boundary conditions contain time derivatives of theunknowns (see [BaMl], [BaM2], [BaM3], and [MoP]).

The theory of singular perturbations is a huge field, with many different subdivisions,and many considerable efforts have been done in this area. For general or more advancedinformation we refer the reader to the books: [CK1]: [D], [E], [I]. [JF]. [L], [MaNP], [VBK].and the list may continue. In particular, a large amount of work has been dedicated tohyperbolic-parabolic singularly perturbed problems. See, for instance. [HW1], [HW2],[HW3], and the references therein. However, our problem Pt with the nonlinear boundaryconditions (1.3) is considered here for the first time. One of the main difficulties weencounter here is to prove results of higher regularity for the solutions to problems Pf andPO- This is necessary for the existence of our asymptotic expansion as well as for provinguniform estimates for the remainder components.

2. A formal asymptotic expansion for the solution of Ps

Let f / j = (us.vs) be a smooth solution of problem Ps. The existence of such a solutionwill be discussed in the next section.

Our problem P, is a singularly perturbed one. with respect to the C-norm. To thisend. let, us remark that if the solution IL of Ps converges uniformly in D-j- to the solutionof P0 then we necessarily have

4(x) + ruo(x) = /, (x, 0), 0 < x < 1. (2.1)

If (2.1) is not satisfied then that uniform convergence cannot be true and Us has a singularbehavior in the neighbourhood of the segment {(:r.0).0 < x < I } . This segment is our

boundary layer.According to the Vishik-Lyusternik method (see, e.g.. [VBK]), wee seek a solution

V, = (u,,v,) of P, of the form

U s ( x , t) = U0(x, i) + V0(x, T) + R(x. t.;). (2.2)

whereT = t/s is the boundary layer variable (or rapid variable);UQ(X.I) = ( X ( x ^ t ) . Y ( x . t ) ' ) is the first term of the regular series:VQ(X,T) = (XI(X,T),YI(X, T)) is the corresponding boundary layer function;R(x,t,e) = (Ri(x.t.£).Rz(x.t.c:)) is the remainder (of order zero).

In order to find VQ we will take into account the fact that VQ should be "smallenough" far from the boundary layer; more precisely VQ ——> Q as r ——> oo. for every x.Now. we will formally substitute (2.2) into (1.1) and then equate the coefficients of thelike powers of ek(k = —1,0 ) , separating those depending on t from those depending on T.First, we can see that Y\ satisfies the problem

oc and for every x. 0 < x < 1.

Therefore.Yi = 0. (2.3)

Also. X\ satisfies the equation

Xlr+rXl = 0,

and. hence,X1=a(x)e-r\ (2.4)

where the function Q will be found later. Furthermore, assuming that the remainder R is"small enough", we can see that X and Y satisfy the so-called unperturbed system

Y,+rX=fl.(2.5)

Yt + Xx+gY = J 2 , in I)T.

Clearly. (2.5) can be written in the form

X = ( l / r ) ( J \ - Y.,.). (2.6)

Yt - ( l / r ) Y X I + gY = /2 - (1/r)/,.,.. (2.7)

Finally, the remainder R satisfies the system

f sRu + R,2x + rRl = -sXt,(2.8)

[ Ru + Rix+gR* = -Xte.

Now, let. us formally substitute (2.2) into (1.2). We remark that X\ has to compen-sate for the initial condition for X which is not necessary any more because X satisfiesan algebraic equation. So we get

a(x) = u0(x) + (l/r)(v'0(x) - f^x, 0)),0<x<l, (2.9)

and the following initial conditions

Y(x, 0) + Yi(x, 0) = v0(x) & Y(x, 0) = v0(x), 0 < x < I , (2.10)

Rl(x,0,e)=0, R2(x,0:£) = 0. (2.11)

Analogously, by substituting (2.2) into (1.3), we obtain

A'(0, t) + ,/?i(y(0, f ) ) = 0, X(l,t) - f a ( Y ( I : t)) =0, 0 < t < T, (2.12)

, t, s) + Y(Q, 0) - A (Y(Q, t)) = 0.(2.13)

)) = 0, 0 < t < T.

From (2.6) and (2.12) we get

-(l/r)Yx(0,t)

0<t<T,

XI(O,T) = X^l.r) =0 ^> a(0) = o(l) =0. (2.15)

Summarizing, the reduced problem. (Po), is defined by (2.7) — (2.10) — (2.14). while theproblem satisfied by the remainder terms is given by (2.8) — (2.11) — (2.13).

3. Existence and higher regularity for the perturbed and for the

reduced problem

The aim of this section is to state some sufficient conditions for the existence and higherorder regularity of the solutions to problems Ps and PQ. This requires higher regularity ofMO. f'o, fi-, fi-, /?i, fit as well as higher order compatibility conditions. We need higherregularity not only for the existence of our asymptotic expansion (i.e., the existence ofall functions involved in this expansion), but also for proving estimates for the remaindercomponents.

The main difficulties in the study of these problems are due to the nonlinearity ofthe boundary conditions (1.3) and (2.14).

Theorem 3.1. Assume thai

r>0, g > 0, A, /?2 6 Wf0f (K), ft > 0, ft > 0;

h e W-2 '1(0,T;L2(0,l))nC'1([0 )r];C[0,l]),

/i(-,0) 6 #3((U), /2(-,o) e If (0,1) /i (o , - ) , /2(i,0 e•UoeC1^,!]), t'o e //4(0, 1);

that the following compatibility conditions are satisfied,

I «o(0)+/? 1 ( i ' 0 (0))=0 !

I wo(l) - A("o(l)) = 0,

/ 1 (0 ! 0)-ruo(0)-z / 0 (0)=0,

= 0,

^0(0) + /2x(0,0) - r l j

gVo(Q) + /2(0, 0) - r llx(0^ 0)) = /1((0, 0)

„.(!. 0) + »-$(

(3.1)

(3.2)

(3.3)

(3.4)

(3.6)


Tlien problem P, has a unique, solution IL = (u,, v£] £ Cl(D-r)'i, and also problem l\thas a unique solution Y e W2(0,T: Hl(0,1)).

For the proof of Theorem 3.1 and for other related details the reader is refered to [AMH],

[BaM3], and [BaM4].

Remark. from the conditions (3.4). (3.5), and (3.6) there follows (2.15).

4. Estimates for the remainder terms of the asymptotic expansion

The main result of this section is formulated in the following theorem.

Theorem 4.1. Assume that (3.1) — (3.6) are satisfied. Then, for every s > 0. problem,PC has a unique solution, Us of the form, (2.2). where UQ is determ.ined from, (2.6) and(2.7) - (2.10) - (2.14), VO(X,T] = (a(x)e~rT ,Q), with a(x) given by (2.9). The remainderR satisfies (2.8) - (2.11) - (2.13).

Moreover, we have the following estimate

where M depends on r, g, l3\, /?2, WD, i'o- f i , fz, and T.

Proof. We can basically use the technique of [BaMl], [BaM3]. and [MoP]. How-ever. for completeness, we will give here the full proof. By Theorem 3.1, the problem(2.8) — (2.11) — (2.13) has a unique, smooth solution,

R(x, t,£) := (/?, (x, t, £),R2(x, t, s)) for (x, t) € T)T.

Let //: = (L2(0. 1)) which is a Hilbert space, with the scalar product

r\ M

(p,.q}:=£ pi(x)pi(x)dx + qi(x)q<2(x)dx for any p.qell.Jo Jo

P'-=(pi,Pi), (1 := (9i,92.).

Denote by || • j| the corresponding Hilbert norm. We define the operator

Bs(t) : D(B.(L)) C II -^ II.

q) e ( / - / ' («• l)f. p(0) + ,fl

It is known that Bs(t) is a maximal monotone operator for Q<.t<T [Mo. ChapterIII. §4]. and the problem (2.8) — (2.11) — (2.13) may be written as the following Gauchyproblem in //.

= F,(t), 0 < t < T,

=0,

where Rs(t):=R(-,s), Fs(t):=(-Xt(-,t),-Xla!(-,r)), 0<t< T.

Because BE(t) is a maximal monotone operator and (0.0) € D(Bs(t)), we obtainthat /?£ is the unique strong solution of problem (4.1).

Let us multiply (4.1)i by Rs(t) and then integrate over [0, t] :

(S))ds, (4.2)

where ( - . - ) denotes the scalar product of L2(0. 1). Because (Be(s)R.s(s), R*(s))>Q for0<s <rr (notice that I3\. fa are both nondecreasing), we get

a t \ 1/2 / rf \ 1/2t \ / rf \\\Fs(s)\\*ds) ^\Rs(s)fdSJ . (4.3)

On the other hand.

and therefore\\rs(,)fd,<M,£. (4.4)

Denote ks(f.):=fo\\R£(s)\\2ds. From (4.3), we can deduce that h's(s)<2^/M^s ^/hs(s),and integrating with respect to s over [0. t], we get

/?.,(/.) < ;W25 and /4(/.) < 2v/M^?v/^£ = A/:j--. (4.5)

From (4.5)2 : we can deduce that

= -=\\lil(-..Ls)\^+\\R^-,L.-=)\\z0<^3s.. (4.6)

where [| • | jo denotes the norm of />'2(0, 1).

As '3\, 3'2 are nondecreasing. we have

*(..s:e)\\'20ds + r£ \\R,(^s^) \\lds. (4.7)

From (4.2), (4.4), (4.6), and (4.7) we can deduce that

- , S , e ) \ \ l ) d S < M.e, 0<t< T. (4.8)

Because re#2(0,T; //J(0,1)) (cf. Theorem 3.1 above), we obtain from (2.6) thatAe-//2([0,7'];f;2(0,l)) , so

Q£(f) := f f s ( t ) =

is the unique solution of the Cauchy problem in H:

(4.9)Qe(0) = F£(0),

where Ss(t):D(£E(t))cH^H,

D(£E(t)) = {(p,q) e /^((U). p(0) +0'l(H2(0,t.£) + Y ( 0 , t ) ) ( q ( 0 ) +

+ Y,.(0, 0) = ft (Y(0, t))Yt(0, t), p ( l ) - &(R*(l,l, £) + Y(l, t»[q(I) + Y t ( l . t ) } =

es(t)((p,q)) = Bs(t)((p,q)) for 0 < t < T.

Now, we multiply (4.9) i by Q£(i) and then integrate over [0,i] :

+2 (ft (^(0, a, e) + Y(0, a)) - ft (Y(0, a))) K,(0, s)Rva(Q, s. -:

< ||Q5(0)||2 + 2 |/^(a)||||g£(a)||da. (4.10)

Obviously, ||Q,(0)||<,A/5.

In order to continue the proof, we need the following auxiliary result:Lemma 4.2. If the assumptions of Theorem 4.1 hold, then \\v(-. •,S)\\C(J)T-I< C,

where C is independent of s.

Proof. We define the operator Bi£:D(Bl£)cH->H by

D(Ble) := {col(p,q) e Hl(Q, I). p(0) + A(g(0)) = 0, p(l) - /32(g(l)) = 0}.

Bls(col(p, q)) = (£-'9' + re'1?, p' + gq).

It is obvious that problem P, may be written as the following Cauchy problem inH.

' U's(t) + BlE(UE(t)) = Gs(t), 0<t<T(4.11)

UE(0) = (ua:v0).

where U£(t):=(u(.,t,S),V(.,t,S)), GE(t): = (S-i />(• ,t)J,(-,t)).

Now. it is easily seen that

\\U'.(t)\\ < ||Ge(0) - Bl£((^,i^)\\ ++ [' \\G',(s)\\dS < Ci-;-1'* for 0 < I < T.Jo

Therefore.-=\\ut(.,t,^0+\\vt(^,-=)\^<^-\ (4.12)

From (4.6) we derive

I K , * , £ ) l l o < \\Y(-,f)\\0+\\^(-^)\\0<C^ (4.13)

\\u(...t,s)\\0<\\X(.,t)\\0 + \\Xl(.,T)\\0+

+ \ \ R l ( - , t , s ) \ \ Q < C3, 0 < /. < T, e > 0. (4.14)

On the other hand, from (l.la), rx=fl—sut—ru. in />/-. and, further, according to(4.12) and (4.14). we can see that

||-".T(-, t.. ;)||0 < C\ for 0 < t < T and -: > 0. (4.15)

Using now (4.13) and (4.15) it follows that \ \ v ( - . •. ;)||c/7j v < 6Y, as asserted (noticethat // '(0. 1) '— > C[0. 1]). which completes the proof of the lemma.

We now continue the proof of Theorem 4.1. By Lemma 4.2 it follows that

(i .(0,s,c)) - f ? l ( Y ( Q , s ) ) Y a ( Q , s ) ( v a ( Q , s , £ ) - Ya(Q,s))ds >

> (,0'l(v(0,s,s))v,(0.s.s)}Ys(0.s)ds - K'f-70 v '

- ft0'1(Y(0,s))Ya(0,s)va(0,s,£)ds >Jo

> A MO, s, £))Ya(0, a)?,, - fl(K(0, s ) ) Y a ( 0 , s)v(0, s, s)^-

- ft/31(v(0,3,£))Yaa(0,s)ds+Jo

+ f (ft (Y(®, s))Y,(0, s))v(0, s, e)ds - A'f > - A"22 for 0 < t < T.

Analogously,

From (4.16), (4.17) and (4.10) we can deduce that

< A/6 + 2 /' \\F^s)\\\\Q4s)\\ds for 0Jo

< t < T.

On the other hand,

2^\\F'E(s)\\\\QE(s)\\ds < 2S£\

(because

'2(r/e}K '--r f(s)\ds<

(4.16)

(4.17)

(4.18)

(4.19)

where K- = | | a ' | | ( ) . j / ( , s ) l= !!/?,.,(.., s, 5) H 0 ) .From (4.18) and (4.19) we obtain that

\\Qs(t)\\2 < A/8 + Al + r/e)e rs/£\\Qs(.s)fds for 0<t<TJo

and so. bv Gronwall's lemma. j |Q £ ( ' ) l l — MQ-

Therefore.

£| |f lu(. ,M)llo + p2 f(-,M)llo < AAo, (4.20)

(4.21)

so. using (4.8) and (4.21), we get

l lf tM^H^AW/2 , t = l ,2 . (4.22)

From (2.8). (4.6) and (4.21) we can obtain

| | /4,M.c)H0<A/1 4 , k = l,2. (4.23)

Let i-e[0,T]. From (4.22) we obtain that there exists a4e[0, 1] such that /if(:cj,J,£)<A/i ( !c1 / 2 . Because

But

! i = L 2 ,

we have that ^?(.T; t. -)< Mus1/4, i = 1,2, so

g.

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Perturbation Problems. SIAM, Philadelphia. 1995.


Global Existence for a Class ofDispersive Equations

Radii C. CascavalUniversity of MissouriColumbia, MO 65211

AbstractWe study the existence of solutions for a class of disper-

sive equations which includes the generalized Korteweg-de Vriesequation and the Benjamin-Ono equation. An explicit methodfor generating solutions is provided by a semilinear Hille-Yosidatype semigroup theory, allowing to obtain - at the same time- estimates for different norms of the solution. The semilinearHille-Yosida theory has been developed in a general setting by S.Oharu and T. Takahashi and extended by J. Goldstein to coverequations such as the generalized KdV equation, for the real linecase [23] [25]. One of our purposes here is to show that this semi-group approach extends to a larger class of dispersive equations,including the generalized Benjamin-Ono equation, and appliesto the real line as well as to the periodic case.

1 IntroductionIn this paper we study the global existence of smooth solutions and

the well-posedness of the following Cauchy problem (written in generalform):

du— -Mux + F(u}x = 0, (NDE)

u(0) = u0,


with the linear dispersive term — Mux, M being the linear operator-given by a multiplication operation in the Fourier space: Mv(£) =|£|2^u(£), /3 > |. The nonlinearity is of the form F^i},,,; = F'(u)ux, withF : R —> R a (sufficiently) smooth function satisfying the followinggrowth condition:

F'(r)lirn sup ——p- < oo for some p < 4/3. (C)

|r|—t-oo I '

In many situations it is instructive to consider the particular nonlin-earity F(u}x — upux, for some power p > 1, (with the conventionup = \u\p~lu for non-integer powers p). For p = 1, (NDE) general-izes the KdV equation (/3 = 1), the Benjamin-Ono (/3 = \ ) and thefifth-order KdV (/? = 2). For other integer powers p = k, one obtainsthe (KdVfc) equations (see below), which exhibit many interesting fea-tures. We address the question of global existence of solutions andwell-posedness in the Sobolev spaces HS(T). Our results are also validin the context of //S(R), but we shall not bother to focus on this casesince it is similar to and slightly easier than the HS(T) case.

We begin by mentioning some of the previous work done in theliterature. Many authors have studied the initial-value problem for thegeneralized KdV equation:

and, in particular,

uxxx + F(u)x = 0, (GKdV)u(0) = u0,

+ uxxx + ukux = 0,•u(O) = UQ.

Here x G E (real line case) or x G T (periodic case). The first resulton global well-posedness for (GKdV) was proved by Bona and Smith[8] in 1978, for the classical KdV equation, i.e. F(u) = y. Later, Kato[29], in 1983, developed a theory for quasi-linear evolution equations,and applied it to equation (GKdV). This was the first place where itwas realized that global existence of solutions depends in a precise wayon the nonlinearity. In a series of papers, e.g. [1], [7], Bona et alhave studied the more general forms of this equation, with generalizeddispersion relations. See also Dix [16], Ginibre-Velo [20], [21], Saut [42].

It is worth noting that the signs of the dispersion term and of thenonlinearity play an important role in the global existence of the so-lution u(t). As it will be seen in Chapter 4, global existence in timeis guaranteed by the uniform boundedness of the H@(T} norm of thesolutions. If condition (C) is assumed, this uniform bound can be auto-matically derived from the conserved quantities. The condition p < 4/3imposed in (C] is essential to guarantee the boundedness of the normmentioned above. For /3 = 1 and p > 4, Bona et al, [5], obtained nu-merical evidence of blow-up of ux in finite time. It is widely believedthat the Cauchy problem is not globally well posed for j3 = I and p > 4,but no complete proof is available. In general, for p > 4/3, one can stillget uniform bounds for the H^-noim of the solution if the initial dataare small enough (in a sense to be made precise later).

One interesting problem that has been recently addressed in the lit-erature is that of obtaining optimal results with respect to the smooth-ness of the solution. That is, one is interested in solving the Cauchyproblem for "rough" data. This direction has been pursued in the lit-erature through the leading work of Kenig, Ponce, Vega (see [32], [33],[34], [35]), Bourgain ([10]), Staffilani ([44]) and many others. Most ofthe methods used are applied also to other dispersive equations, suchas the nonlinear Schrodinger equation and the Zakharov system. Arelated question is the growth of the Sobolev norms of solutions of suchsystems. Results have been obtained in this direction, for example, in[44].

Despite the legitimate interest in these directions of research, inthe present work we do not concentrate on (optimal) small values of s.Instead we plan to study the relationship between the dispersion effectsand the strength of the nonlinearity in the usual functional setting ofthe Sobolev spaces.

The main result in this paper is:

Theorem 1. The Cauchy problem (NDE) is globally well-posed in HS(T)with s = max{2/3, | + e} for some e > 0, provided that the growth con-dition (C) is satisfied and we are in any of the following cases:

• /3 = 1 and s = 2;

• j3 = \ and s <E ( f , 2] and F'(u) = u;

• /3 > | and s = 2/3.

For | < 13 < \, 13 ^ I , (NDE) is locally vjell-posed m HS(T), withs = 2 / 3 f o r / 3 > | ands 6 (f, 2/3+1] for \ < /3 < f.

In particular, we obtain well-posedness results for (GKdV), when/ ?=! .

Theorem 2. The Cauchy problem for the periodic (GKdV) equationis globally well-posed in H2(T}.

Theorems 1 and 2 also hold in the real-line case, i.e. on the spaces.£P(IR), where s = max { 2/3, | + £} and £ chosen as in Theorem 1. Inboth cases, the growth condition (C) on the nonlinearity is assumed.

The plan of the paper is as follows. Section 2 describes the abstractsemilinear Hille-Yosida theory used in the proof of Theorem 1. Theproof of the abstract theorem is contained in Section 3. Finally, insection 4 we prove that the operators and functionals assiciated with(NDE), satisfy all the assumptions that make the abstract semilineartheory work. We conclude with a list of open problems.

2 Semilinear Hille-Yosida theoryThe abstract semilinear Hille-Yosida theory has been developed in

[39], [40], and adapted in [25] for some situations not covered by thestandard Crandall-Liggett theory. Here is the setting of the abstracttheorem.

Let X be a real Hilbert space with norm ||-|| and inner product ( - , •).In the applications we may consider a complex Hilbert space (H, { - , - } )to be a real one (H, (•, •)) by taking (x, y} = Re (x, y) .

C is a subset of X, u0 G C. ip — (y?i, . . . , <^v) are N lower semi-continuous functionals on X, with D((f^} = C, for k = 1, . . . , N. Fora = (0:1,... , a;v) C R^, define Ca = {u G C\(f>k(u) ^ ak for k =

Of concern is the abstract Cauchy problem in X

u(0) = UQ.

Throughout the rest of this chapter we make the following assumptions.(Al) A : D(A) C X — •> X is the linear generator of a Co-semigroup

{T(t)}t>0 with

\\T(t)\\ < ewot for some LOO 6 R.

(A2) In addition, assume that D(A) fl Ca C C, for all a.B : C — > .X" is a nonlinear operator satisfying:(Bl) 5 is weakly locally sequentially continuous, i.e.

{un} C Ca for some a > 0 and un —> u implies Bun — >• 5w. (2)

(Here — > [resp. — >• ] means norm [resp. weak] convergence in X .)(B2) B is locally quasi-dissipative in the sense that:For all a > 0, there exists u:a such that

(Bu — Bv, u — v) < u}a\u — v\ , for all u, t> 6 (7a (3)

Then the following theorem (due to Goldstein, Oharu, Takahashi [25])holds.

Theorem 3. Let a&, 6^ > 0 6e /jxec? numbers, k G 1, . . . N. Assume thefollowing condition holds true:

For v (E (7, £ > 0; i/iere exists 0 < A0 = A o ( f , £ ) swc/i iftaifor all 0 < A < AO, i/iere exists u = u\ G -0(^4) H (7 satisfying

\u- \Au- \B(u)-v\ < Ae, (4)V?A;(^) + (^ + £)A

W«) < ——— 1 _ a A ——— , « = 1, • • • , N. (5)

Then there exists a nonlinear semigroup {S(t)}t>0 of continuous map-pings from C to C with the following properties:

I , (6)/•* T(t - s)B(S(s)u0)ds, (7)

Jobkt},t>0, k = Q...N-l, (8)

and the mapping u0 i— > «(•) = 5"(-)wo is continuous from C into (7(R+, ^T)locally Lipschitzian for UQ 6 -D(-A), i.e.

|u(i) - v ( t ) \ < eM u0 - v0\ , for some fi = ft(T, a)

whenever w0, u0 G -D(A) n CQ,, i G [0, T] .

Note that the growth condition (8) has as special cases linear growth(when a,k = 0) as well as exponential growth (when fej. = 0). The proofis given in a slightly more general setting in the next section. Thetheorem actually holds if X is a general Banach space. In this case, (3)should be replaced by:

B — uial is dissipative on Ca.

Condition (3) is correctly stated if ( • , • ) is interpreted as a semi-innerproduct (in the sense of Lumer) on X, i.e., for x,y G X :

x}, where <f> G X* is such that <j)(y) = y\\\ =(9)

This fact is basically contained (but not proved) in [23], [25]. For aformulation of our result in general Banach spaces with N functionals,see Remark 2 at the end of Section 3. In the next section, we give theproof of our theorem, generalizing the estimates on the functional <$j.

3 Abstract theorem. Existence of solu-tions

Of concern is the abstract Cauchy problem in X

duB(u], (10)

«(0) = x0 6 X.

We assume all the hypothesis on A, B, C and (p from the precedingsection, i.e. (A1),(A2), (Bl) and (B2), hold throughout this chapter.

Let g = g(r) > 0 be a Cl increasing function and m = m(t, a) themaximal solution of the initial value problem

m'(t)=g(m(t)),t>Q, (11)m(0) = a,

where a G K. For the sake of simplicity, we will assume that the max-imal solution is defined for all t > 0. Otherwise, when we say "fort > 0" we mean "for all t G [0, Tma;E)". When necessary, we will writem(t, a) = mg(t, a).

Here is the main abstract theorem, formulated for a single functional(/? (TV = 1). The general case is very similar and will be explained later.

Theorem 4. Assume that, for each XQ E C , the following hypothesisholds:

For every e > 0, there exists 6 — 8(xQ,e) such thatfor all h £ (0,£), there exists Xh G D(A) D C satisfying

-r[f(xh)nThen there exists a nonlinear semigroup {S(t) : C — >• C}t>0 of contin-uous operators on C with the following properties:

/** T(t-s)BS(s)ds, (12)Jo

m(t,v(x0)), (13)

for all XQ G C and all t > 0.Most of Theorem 3 is obtained from Theorem 4 when we use N

functions gk(r~) = akr + &&, for k = I,... ,N. Extra arguments areneeded for the conclusion involving local Lipschitz conditions.

The idea of the proof follows the work of Kobayashi, [36], and con-sist of constructing, for each x0 e (7, the map t — > S(t)xo as the limitof a sequence of approximations {un(t)}n , where un(t) are step func-tions constructed by means of an implicit difference scheme. The onlyaddition here is the estimates using the functionals i£>j.

Let XQ 6 Cro (for some fixed r0) and T > 0 be fixed. Choosealso TI > 1 (close to 1) and R = R(r0, T) such that

R > mTO(r0, T) = mg(r)r0, r,T}.

Denote ui — UJR the dissipativity constant for the operator A + BR onCR. Throughout this section, BR is the restriction of B to CR. Let / bea lower bound for <p on CR. Let K = max{u;, sup g'(m)} and denote

l<m<R

Lemma 5. Assume the hypothesis (H] holds. Then, for x0 G Cro and0 < £ < e", the family

Txo,s := {(xk,hk)nk=l (xk)n

k=lcD(A)nCR,Q<hk<e,n—\

k=\ k=l

xk-xk-i-hk(Axk + B(xk)}\<ehk, (14)

\(f>(xk} - v(xk-i}\ < g((p(xk}} + £, for k = 1, . . . ,hk

is nonempty.

Proof. The existence of a sequence (x^, f t^) satisfying the requirementsabove follows inductively, using the hypothesis (l~t), which can be con-veniently written as a "tangential condition":

For all £ > 0, x G CVsuch that there exists 8 > 0, such that

h G (0, o) implies dist(x, Range(I — h(A + BR)) < eft, (16)

or,

lira -dist(x,Range(I - h(A + BR))) = 0, for all x G Cr, (17)h->0 ft

where r < R must satisfy mw?(r, T) < R (see Proposition 6 below).Let £ < (?be fixed and denote 8 = 8(x, e) the supremum of 8,8 < s,

satisfying (16).The induction argument is as follows. Assuming we have generated

%i € Cn,i = !, . . . ,& — 1, ^2hi < T, we can find 8 = 8(xk-i,£) ,

hk G (0,<5) and xk = Xhk as in (7i). We choose ft^, in each step, suchthat

- 5 ( x J t _ i , e ) < A f e < e . (18)

fc-iOur first claim is that x^ G CR for as long as J^ ft; < T. Indeed, we

can prove that

lf(xi) ^ mrige(tp(xi-'i}i f t i )> ^Or a^ Z = ^ •") ^i (19)

here </£ :=</ + £. This follows from the following

Proposition 6.

(/ — hg)~lz < mrig(z^ f t) , for any z < /?, ft < £". (20)

To prove the above inequality, it is enough to rewrite it as

z < mrtg(z: ft) — hg(mrig(z, f t )) =: $(/i).

Recall that g is increasing function. Then $'(ft) = r/g(m) — (m) —hg'(m)rjg(m) > (TJ — 1 — hK-q}g(m} > 0, for ft < £", which implies$(ft) > $(0) = z, if 0 < h < £~. This proves the claim of Proposition 6.

Using (20) with ge = g + e instead of g and the fact that (f>(xt) <(I — /i t-(y'e)~1(/?(x,_1) (by construction, see (15)) we get

(f>(xi) < (I- ^e)~V(^-i)< mr,gc(V>(Zi-l),hi)

and therefore (19) holds. Iterating (19), we obtain

As mentioned above, this guarantees the induction can continue untiln^hk > T for some n. We claim that this is indeed reached in a fi-k=lnite number of steps, which is the conclusion of Lemma 5. We argueby contradiction. Suppose that we can indefinitely generate (x;,/i;),

k oowith "^hi < T, for every integer k > 1. Then s0 := ^hk < T < oo.

Without loss of generality, we can assume x0 G ^(A). One reaches acontradiction by showing that the sequence (xk) C CR has a limit pointx* 6 CR which fails to satisfy the hypothesis (H).

In a first step, one proves by double induction (see [36]) that for alli>j> 0,

Xi -Xj\< e^'+^Kti - t3] \Bx0\ + e(tt + t 3 ) ] . (21)

or, more generally,

Xl -Xj\< e^'+^-^[(tt - tj) \Bx0 + e(ti + tj- 2tp)} (22)

for every i > j > p. As we take z , j — > oo , ti,tj — >• s0 in (22) and weconclude

limsup \Xi -Xj\< 2£(s0 - tp)e27?w(s°-iP)

for all p. Now take p — > oo, tp — > s0, which leads us to the conclusion

lim \Xi — Xj = 0I,] — »OO

i.e., the sequence {xk}k>i is Cauchy, therefore convergent to some x* GCR.

By the tangential condition, there exists 8* = S(x*, |) such that forall 0 < A < <*)*, there exists 6 € D(A) fl (7 such that

| £ A _ X * _ A ( A 6 + £(6) ) I<A| . (23)

oo _

Because hk — »• 0 (as hk < T < oo ), and |£(xfc, e) < /ijt (by our choicefc=i _

of hk, see (18)), it follows that 6 ( x k , £ ) — > 0 as A; — > oo. Thus there existsA;* such that 6(o;fc,£) < 4j-, for all A; > k* . This guarantees the existenceof an infinite sequence (A/t) k G [y, <5*] and (£,xk)k G P(A)n(7fi satisfying,for all A < 5*,

|6 - xk - A,(A6 + 5(6))| > eA f c , for all A; > F.

If A* G [y, <?*] is an accumulation point for (\k)k , then we have

6* -xk- A fe(A6* + 5(60) > eA^, for all A; > F,

and, in the limit A; — >• oo,

which contradicts (23).

The contradiction we reached shows that our assumption made ear-lier, that we can generate an infinite sequence (xk, hk)k, is false. There-fore the set TXQt£ is indeed nonempty. This concludes the proof ofLemma 5. D

Lemma 5 allows us to construct an approximate solution to theinitial value problem

( }

More precisely, for a finite sequence (xk^hk)nk=l G T r0 iE, we define the

step function ue : [0, T] -> CR fl T>(A], such that

ue(t) = xkiit£[tk--i,tk') (25)

for k = 1, ...,n. What remains to be proved is that for £ = en — > 0, thesequence {u£(i)} converges uniformly to a continuous function t*(i),which turns out to be the desired S(t}xQ, satisfying

u(t) = T(t)x0+ T(t-s)B(u(s)}dsJo

for all t G [0, T], i.e. u is the mild solution for the initial value problem(24).

The convergence of the approximate solutions constructed in theprevious section follows from the following estimate (see [36]).

Lemma 7. Let 0 < 'e < e and { ( x k , h k ) } k G rx0iS and {(2^, /&j)}j GF r0 j£ be finite sequences constructed as in the proof of Lemma 5. Then,for' all u G D(B),

x0-u\ + [(tk - 7j)2 + etk + etj}* \Bu\ + etk + etj](26)

for all k,j.

Based on this estimate and given e — en — >• 0, the sequence ofapproximate solutions un(t] = uEn(t] constructed in (25) converges toa continuous function u : [0,T] — > CR, which turns out to be a mildsolution for the initial value problem (24).

Indeed, for each n, we choose ( x j j ™ , h£ ) G ^Xo,en and denote

4n) = EA!n) • Let u G V(B] arbitrary but fixed. Fix also t G [0,T].1=1

Let m,n G N. One can find k and j such that t G [4- i>4n ))n [*$-!' m))'We then apply Lemma 7 with £ = £„ ,£"=£ ,„ :

un(t)-um(t) < e " w * > { | X o _ u | + (27)

[(4"} - <5m))2 + en4B) + emt^ \Bu + eB4n) + emfH}.

Letting n,m —> oo, we get

limsup un(t) — um(t)\ < e2r?wt \x0 — u , for all u G "D(B}.n,m—>oo

Because 'D(B) — 'D(A) fl CR is dense in CR we obtain

lim |itn(i) — um(t)\ = 0 , uniformly for t G [0,T].n,m—>-co

This implies that (wn)n is convergent to a function u : [0,T] —> CR :

l imw n ( t ) = w(i) , uniformly for t G [0,T].n—>oo

Reasoning in the same way, but for t ^ s, we get

\u(t) - u(s)\ < e2wT[\x0 -v\ + \t-s\ \Bv\], for all v G V(B). (28)

Because X>(H) is dense in Cjj, it follows from the above inequality thatu : [0, T] — > CR is continuous.

Define S(t)xo '.= u(t). It is important to note at this point that,when XQ G T^(B} — T>(A)r\CR, we have more than continuity. Choosingv = XQ in the inequality (28),

\u(i) - u(s}\ < (?wT \t-s \Ax0 , (29)

for t, s G [0, T1], i.e. u is Lipschitz continuous on [0, T], for XQ 6 T>(B}.What is left to prove is that, for any x0 G (7^, u(i) satisfies

/•*u(t) = T(t)x0+ T(t-s)B(u(s))ds, (30)Jo

, for < G [0, T]. (31)

Let un(t) be the approximate solution constructed above and letr ^- T->/ ^ * N Ti- i j_ (") (1) (1) 7 (n) / A (n) . j-w ( n ) \ \/ G I?(A*J. If we denote f.\. ' := x\ ' - x\^ - h\ '(Ax\ ' + B(x\ ' ) ) ,then we know (see (/H)) that nh(^. Then

which implies

(un(t)J) = <z0 , /) + [ MS), A*f) + (B(un(s)), /) + {£„(*), f)]dsJo

for i G [tk-ntk ]• Letting n — » oo, we deduce

{«(<), /} = <*o, /} + "[(«(*), A*/> + {£(«(*)), />]&. (32)Jo

(r>} (r>\ (T)\In the formulas above, tn(t) = ek ' for t G [^_i, tk ] and

£ntk ^- 0 as n ^- oo. The integrand in (32) is continuous in s, becauseu is continuous in X and, consequently, B(u) weakly continuous in X(see (2)). Therefore (w(t), /} is differentiable in t and

f). (33)

Let w(t) := T(i)xQ + J0 T(^ — s}B(u(s)]ds. We want to prove thatu(t] = u(i).We have, for / € V(A*},

( u ( t ) , f) = (T(t)x0, /} + f (T(t - s}B(u(s}}, /) ds-Jo

thus

This shows that (u(t}-u(i)J} = (w(0) - u(0),/) = 0 for all / €), and consequently (30) holds, i.e.

/ T(t-s)B(u(s))ds. (34)</o

To prove inequality (31), let un = u£jj be given the approximatesolutions. We know from the estimate (19) that

whenever en < !t=J- for some constant M, independent of n. Thus, forsequences 77 — > 1 and n — >• oo, we infer that

<p(u(t)) < limmf(f>(un(t)) < mg(p(xo),t). (35)n— s-oo

i.e. (13) holds. Here we used the lower semicontinuity of </?.Note that all the above results hold true for all XQ £ C = U Cr. To

r>0complete the proof of Theorem 4, it remains to prove the uniquenessof the solution u(i] = S(t)x0, with x0 G C. This is the claim of thefollowing

Proposition 8. The mild solution of the Cauchy problem

u\t] = Au(t) + B(u(t}},t>Q, (36)«(0) = XQ <G C,

that is, the strongly continuous function u satisfying (34) and (35), isunique.

Proof. Let Xo,yo G CT and T > 0. Denote R := m(r, T).Assuming there are two functions, u(t) and f( t ) , both satisfying

(34), we have

u(t),v(t) e Cfl, for all* e [0,T].

As 5 — fi/ is dissipative operator on CR with some constant fi, we canapply the standard techniques to obtain that

\\u(t)-v(t)\\<efit\x0-y0l f o r < 6 [ 0 , T ] .

This clearly implies the uniqueness of the Cauchy problem (36). Theproof of Theorem 4 is now complete.Remark 1. In Theorem 4, only the lower-semicontinuity of the func-tional^) (f was needed. The converse of Theorem 4 holds in the specialcase when the set C and the functional(s) 99 : X — >• R is (are) convex.This was proved in [39] for the case of a single functional.Remark 2. Theorem 4 can be generalized by considering X a generalBanach space and A^ functionals (/?1,(^2, • • • , VAT : -X" — » I&, instead ofjust one functional. In this case C = C\T> (<£>,•) . The proof follows the

zsame lines as in the Hilbert space case, with the inner product replacedby the semi-inner product (in the sense of Lumer) that was defined in(9). We omit the details, which are tedious but routine.

Remark 3. Our proof of the abstract theorem is related to the ideasof Kobayashi [36], who was inspired by the historic Crandall-Liggettpaper [15]. L.C. Evans ([17], [18]) extended Kobayashi's constructionfrom the context of du/dt = Au to the time-dependent operator contextof du/dt = A(t)u. This enables us to generalize our abstract result todu/dt = A(t)u + B(t)u, and to, for instance,

ut - a(t}Mux + F(t,u)x = 0,

a nonstationary version of (NDE). We have no specific applications inmind for this equation so we do not give a precise formulation here.

4 Proof of the Main Theorem 1We will fit our concrete nonlinear dispersive equation (NDE) in the

general framework of the semilinear Hille-Yosida theory presented inthe preceding section.

The linear operator A = D2?dXJ with domain V(A) = #2/3+1(T),is skew-adjoint on any of the spaces HS(T], and it generates a groupof isometries {T(t)\t G B£} on each HS(T). This is easy to see sinceAu(£) = i£\£\2/3u(£] for u G HS(T}. The same argument applies to therestriction of A to #2/3+1(T) = V3, and {T(t)\t € R} is also a group ofisometries on HS(T}. In particular,

k = v i for v C Vk,k = 0, 1, 2, 3.

Define a nonlinear operator B on Hl(T) by

Bu = -dF(u] = -F'(u}du.

Clearly, B : H1^) -> L2(T) and, more generally, B : Hk(T} -^Hk~l(T). Note that the only thing needed so far is for F to be suffi-ciently smooth.

Then the following properties hold (for a proof, we refer to [12]:

Lemma 9. (i) If wn — > w in L2(T) and supn\wn\H2/3 < M < +00,then(I - XA)-lB(wn) -> (/ - XA}-1B(w) m H2? , for all real A 0.

(ii) If wn — >• w in L2(T) and supn wn\H2f>+i < M < +00, thenB(wn] -+B(w) in L2(T).

Lemma 10. (i) For each real A ^ 0; the operators ±A\ = ±(7 —\A]~1B are locally quasi-dissipative on H2t3(T), i.e. for all r > 0,there exists u = u(r) G M. such that

- \A) B(v] — (I — \A) B(w), v — w) 2/3 < —\v — wA

2H2?

whenever .^z'zj ±5 are locally quasi-dissipative in L2(T) on bounded sets in

) , i.e. for all r > 0, there exists u = cu(r) G R

\ ( B ( v ) — B(w)Jv — w)L2 1 < u\v — w L

whenever \v\H2^+i , |w| #2/3+1 — r-Let us introduce the following functionals:

1 on M) (37)T

«) = I t \D^U\2 ~ f G(u), on V, (38)^ JT JT

„ , on y2(39)

\ on V3- (40)

Here G(u) = ft F(r)dr and /(•) satisfies /"(«) = (F'(u))2, 7(0) =/'(O) = 0. In the integrable cases, Korteweg-de Vries (/3 = 1) andBenjamin-Ono (/? = |), with F'(u] = u, these are just three of aninfinite list of invariant functionals.

These functionals are lower semi-continuous on the correspondingspaces and, in particular, on H2@+1. They are also bounded on boundedsets in Vj, more precisely:

B is a bounded set in Vj, for 0 < j < k and k = 0, 1, 2, 3, implies: / € B} < oo for all 0 < j < k.

Let g = g ( r ) > 0 be a Cl increasing function and m = m(t,a) themaximal solution of the initial value problem

m'(t) = g(m(t)),t>0, (41)ra(0) = a,

where a G IR. When necessary, we will write m(t,a) = mg(t,a).Our main result concerning the existence of solutions for the non-

linear dispersive equation (NDE) is stated here:

Theorem 11. Let /3 > |. Under assumption (C) on the growth of thenonlinearity, the Cauchy problem (NDE) is well-posed in HS(T], withs = max{2/?, | + e}, (e > 0 is as in Theorem 1), for arbitrary initialdata UQ. Moreover, there exists a Cl function g such that the followingestimates hold when u0 G HS(T):

(42)(43)

< mg(t, ¥>2(wo)), for t > 0, (44)

where mg is the solution of the initial value problem (41)- If, then there exists CUQ depending only on a bound for u HS on

[0, T] so that

VsKO) < e^VsM, /o r*€ [0 ,T ] ,

for any T > 0.

The choice of the function g mentioned in the theorem depends onf3 and on the nonlinearity F. Our well-posedness theorem is global intime, provided that the solution m of dm/dt = g(m) exists for all t > 0.We have shown this to be true whenever /3 > ^ and when j3 = 1. Italso holds for /3 — | and F any quadratic polynomial. For any othercase of a pair (/?, F) for which one can show that m exists globallyin time, then (NDE) is globally well-posed. Thus, local well-posednessassertions in our theorems are actually global in all such cases wherem exists globally. Since (/?, F) determine many possible choices of g,it seems likely that our result are of a global nature in cases where wedo not assert this.

From now on it is sufficient to assume -F(O) = 0 and to restrict allour calculations to initial data belonging to Hs, i.e. with zero mean,instead of the whole Hs . This is possible (without loss of generality)due to the fact that the mean of the solution u(t) is constant in time,so F(u) in the equation can be replaced by F(u) = F(u + c) — F(c),c= fu.

The proof of Theorem 11 will be a consequence of the abstracttheorem. If p > 4/3 in the assumption (C) on the growth of F, global

existence of solutions is still guaranteed, provided that the H/3 normof the solution is uniformly bounded. This can be accomplished, forexample, by requiring the smallness of the initial data. We refer to [12]for details.

The next lemma assters that, under assumption (C), boundedness ofthe functionals <y?j implies boundedness of solutions in the appropriateSobolev norms.

Lemma 12. (i) For 0-0,0-1 > 0, there exists Oi — $1(0-0,01) such thatw G V\,(po(w} < ao,(f>i(w) < 01 implies jii^ < 9\.

(ii) For a0,0i,o2 > 0, there exists 9-2 = $2(0-0,0-1) such thatw 6 V<2,tpQ(w} < a0,Lpi(w) < ai,v?2(u>) < o2 implies \w\2 < 02-

(Hi) For OQ, ai, o2, 03 > 0, there exists $3 = #3(0:0,01) such thatw e y3, <^j(iy) < Oj, ; = 0,1, 2, 3, implies \w 3 < #3.

Proof. Recall the growth condition (C) imposed on F. We assumedthat there exists p < 4/3 such that

lim sup •\T

< co.

Hence, there are constants (7, C' such that

F'r < C rp + C'.

(45)

(46)

Because F(0) = 0 and G(0) = 0, where G(w) = $™ F ( £ ) d £ , it followsthat (with different constants K, A"')

G(r) < K \r p+2 + K' \r\2 for all r e R.

Thus,

rI G(w}dx < K

JT< K

P+2 . rrl Irr,^ + K \W

ML I™< it; 2f}> 2/3

4/3-tt;

,

Here we used the Gagliardo-Nirenberg inequatity and Young's inequal-ity

, (aY bs 4/3 4/3 . I f ,ab< —— + —, r = ———, s = — ( r , 5> 1,- + - = 1).r s 4/3 - p p r s

The assumption p < 4/3 was essential in the previous calculation (toguarantee that r, s > 1). As a result of the previous estimates we get

2 __ 2(p(2/3-

< K w w

which implies (i).To prove (ii), note that

-(F(w),D2^w) = (dxF

< C w

u) = f F'(w)d-lD2f3wdx</

\d~lD2l3wdxw +C' fJT

w

w

and also

u |2p+2

Thus, from the definition of </?2,

^ 4/?, 4/3 + 1

w

i.e. (ii) holds.The proof of (iii) is straightforward:

\D2/3+l wC W 2 / 3 -

D

We need one more result, which will be used in the proof of Theorem14.

Lemma 13. Consider the following expression.y , 3 ( z } • = (D2f3 (F'(z)dz) - F'(z)D^dz - (2/3 + l ) d F ' ( z ) D 2 f 3 z , D2f3z) .(i) For f3 > \ we have that, for all z G H2f3+1 ,

^ft(z)<C\DzUD2^2. (47)

where C depends only on a bound for \Dl3z\.(ii) For (3 = | and (3 = 1, one can improve the estimate above, inthe sense that î(z) < Cil^Dzj4 when /3 = |, with C\ depending onlyon \D1/2z\, and ^2(z) < C\D2z\3/2 in the case (3 = 1, for some C2

depending only on \Dz .

Remark. This result can be reformulated as follows. One can con-struct a function g = g(r] (depending on /?), which, in general, hassuperquadratic growth as r — > oo, such that

z ) ) , (48)

where C depends only on |.D^z|. For /3 > |, g can be chosen to belinear. For /3 = 1, g can be chosen to be sublinear and for /3 = |, g canbe chosen to be quadratic. What is remarkable is that in the specialcase F'(u) = u and (3 = I or /? = | we get /3(z) = 0, for all z.

The proof of (i) relies on estimates obatined from the product ruleand chain rule for fractional order derivatives, for which we refer to[21], [28], [35]. The improved estimates mentioned in (ii) can be foundin [12], so we will not reproduce them here.

The last step in the proof of the main theorem is to solve the resol-vent equation. Here is the result needed, which makes the hypothesisof the abstract theorem.

Theorem 14. Let v 6 Vs satisfy \\v\H2f>+i < r, e > 0. There existsô = ô(^?s) such that, for all real \X\ < XQ, there exists an uniqueu = u\ (E Vs satisfying

') = v, (49)

(50)

u})+e), (52)

< r . (53)i — cu0 I A [

Here g is the function satisfying (48), depending on f3 and the nonlin-earity F, and LOQ can be chosen to depend only on a bound for \Dv L™.

Proof. Fix r > 0 arid e > 0 arbitrarily. Let v G Vs, with v < r, andchoose 0:0,0:1 be positive numbers such that (f>0(v)-\-£ < a0,^i ('«) + £ <0.1. By Lemma 12, there exists 9i = #1(0:0, Qi) such that for all z 6 H'8,with 950(2) < 00,^1(2) < o"i implies |z a < #j.

For the function g defined by (48), let ms(i,a) be the maximalsolution of the initial value problem (11). Choose

a2 > mS£(r, (y22(u)) . (54)

Here r is sufficiently small (depending only on ip^v)) such that theright hand side of (54) is finite.

From Lemma 12 we conclude that there exists #2 = #2(0:0,0:1,0:2)such that, for all z € H2^, with y>o(z) < ao,^i(^) < ai,^!^) ^ 0:2implies z|2 < #2.

For later purposes, let

p = sup{\F"(w)dw\,w e V2,\w 2 < #2}

and

sup{\F'(w)dw , w iw 2

(73 = Sup{\DwF(w)\,w£V3,\w3<93},

(55)

(56)(57)(58)

where we choose 93Because F is assumed to be smooth enough (at least in C2f3+1),

there exists 8 = 6(\v 3, s) > 0 such that, for all w v — w < 8 andu;|j < max{9j, v j -\- CTJ+I}, j = 0,1, 2, the following hold true:

\F(v)-F(w)\93<£-,

\D^F(v)-D2f3F(w)\93<£l,+ CT3) < £2,

(59)

(60)(61)(62)

where £.,• are chosen such that 3£i+2£2+£3 < |. Let A" = 3up{|(7'(r)|,r <#3} and denote A0 = min{r, -J-, -, -^}. Consider A G ( — A 0 , A 0 ) , A ^ 0be arbitrary but fixed.

Consider the set

K = {w e V3\ \w - v < \X\ #3, w . < 93 for all j = 0,1, 2, 3}. (63)

Note that K is a compact convex set in L2(T).We seek fixed points for the operator F : K — > X defined by

By linearity of A,

Tw :=

= (I- XA)~lv - XA)~lB(w).

As we saw in Lemma 9, the operator A(J — XA)~1B is L2 -continuous onbounded sets in /f2/3, thus F is a continuous operator on K . In order toapply the Schauder-Tichonov fixed point principle, we have to ensurethat F leaves K invariant, i.e.

F (K) C K.

We now prove (64). With v G #2/3+1(T) fixed, let warbitrary. Denote z = Tw. We will show that z G K. Since

- XD^dz = v- XdF(w)

we obtain

— v\ = (XD2/3dz,z-v)-(XdF(w),z-v)= \(Dwdv,z-v] - X(dF(w),z-v)

z-v\,

or

\z-v\<

From (65) we also obtain

z3 = < \z - v + \dF(w)I'M

< \v\3 + (7 1 + a i < 93.

In order to conclude that z G K we only need to show

(64)

K be

(65)

or, in view of Lemma 12, it is enough to prove

) < "o,

) < a2.

From (65), we obtain

\z\2 < ( v , z ) - ( X d F ( w ) , z )

(66)(67)(68)

i.e.

= (v,z)-X(dF(w)-dF(z),z)< \v\\z +\X\\dF(w)-dF(z)< (\v\ + \ X \ e ) \ z \ ,

z\ ^ \v\ + 1^1 £• We conclude that ifo(z) < <£>o(v) + |A| e <£ < a0.

Next, multiply (65) by D2/3z to obtain

A (D20dz, F(w))

Now we can estimate

 FH) - T / ( I F(TZ + (l~ T}v)di\ (z - v)A JT \Jo /

~\ I ( ! F(TZ + (1 - T)v)dr - F ( v ) ] (z - v) dxA JT \Jo )

z-v\\F(w)-F(v)

T\z-v\dx

Thus we conclude

The only remaining estimate is for ^2(2)- Consider an arbitrarynumber 7 G [0,1] and let

7

(we will obtain the value 7 — |fr| to be the useful one, thus 99Rewrite (65) in the form

Then,

= XD2f3dz-XdF(w).

= (D^z, D^v) + (D^z, D^, D2f3v) -

(z-v + XdF(w), d-1D2^(z - v})

= (D2/3z, D™v) - (dF(w), d~lD2P(z - v))= (D^z, D™v) + (D2(3F(w), (z - v))= (D2/3z, D^v) + (D2/3F(w), XD^dz - XdF(w}}

X (D2f3F(w}, D^dz]

\D2f3v 2 + X (D2<3F(z), D2(}dz) + \X\ £

where e\ is as in (60). Hence

1 1, D2(3dz} + \X\ £l. (69)

On the other hand,

= (F(z) - F ( v ) , D2(3z) + ( F ( v ) , D20z

= (I F'(rz + (1 - T)v)drD2pz, (z - v]\ + (D2(3F(v), z -\Jo /

> (F'(z)D2f3z, z-v}- |A| £2 - \D2(3F(v} - D2f3F(w}\ \z - v+ (D2(3F(w),z-v)

> (F'(z)D2f3z, \D2(3dz - XdF(w)} + (D2?F(w), XD20dz) -> A (F'(z}D20z, D2/3dz] - X (F'(z)'D2(3z, F'(z)dz)

+ ( X D 2 0 F ( z ) , D^dz] - |A| (2£l + 2£2),

where £2 is as in (61).Also,

(/(z),!)-(7(t;) , l) = (I(z)-I(v}}dx

TJO1

£2)

w

I'(TZ .+ (1 — T)v)dr(z — v)dx

\'(rz + (i _ T)v)dT, \Dwdz - XdF(

-\( f I"(zT)(dzT)dT,D2?z-F(w)\\Jo /

where ZT = rz + (I — r)v,-A (I"(z)dz, Dwz - F(w}} + |A| £3

-A (I"(z)dz, D20z) + X (I"(z)dz, F ( z ) ) + X\

where e3 is as in (62).Putting all the above estimates together, we obtain


< (D2pF(z), D2f)dz} - 7 (F'(z}D2f}z, D2(3dz) + 7 (F'(z)D^z, F ' ( z ) d z )-7 (D2?F(z), D^dz) - 7 ( F ' ( z ) d z , F'(z)D^z)

< (1-7) (D2^F(z), D2!3dz] - 7 (F'(z)D2/3z,= -(l-7) (D2(3dF(z),D2f3z) - 7 (F'(

Thus,

< (7-1) [(D2f3dF(z),-(F'(z)D20z,dD2l3z)+e

= ( 7- l ) [ ( D 2 / 3 d F ( z ) , D2f3z) - (F'(z)D2?dz, D2f3z)}

-l) [ ( D 2 l 3 d F ( z ) , D2(3z) - (F'(z)D2l3dz,

= (7-1) (D2f3 (F'(z)dz) - F'(z)D2f}dz - (2/3 + l)dF'(z)D2^z, D2() z] + e,

provided that

which is satisfied precisely for 7 = |f |, when </? =Using Lemma 13 we get the estimate

e.

This implies (using (54) in conjunction with (20)

)) < a2. (70)

This concludes the proof of (64). The Schauder-Tichonov theoremapplied to F : K — > K gives us the desired fixed point, u = Fu, so that

u - \Dwdu - \dF(u] = v. (71)

The estimates for <fj(z) in terms of <fj(v) imply (50), (51), (52).What remains to be proven is the estimate (53) for t p 3 ( z ) . We will

make use of the dissipativity of A + BR — UR, for some LJR > 0.For v and u as above we know that <fj(u) < O j , j = 0, 1,2. Then

> (Au + B(u) — Av — B(v), u — v)- 3F(u} - D^dv + dF(v),u - v}

) \u - v\ .

Thus

+ LOR U — V

But u — v\ = \\\ if>3(u), so we conclude

or, equivalently,

This completes the proof of Theorem 14. D

Having all the hypothesis in place, we can apply the abstract The-orem 3 and conclude the proof of the well-posedness for the CauchyProblem (NDE) in the space #S(T), where s = max{2/3, f + e}, forsome e > 0 as in Theorem 1.

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Viable Domains for Differential EquationsGoverned by Caratheodory Perturbations of

Nonlinear m-Accretive Operators

Ovidiu Carja "ALL Cuza" University of lasi, lasi, Romania

loan I. Vrabie "ALL Cuza" University of lasi, Ia§i, Romania

ABSTRACT. Let X be a real Banach space, A : D(A) C X —> 1X an m-accretive operatorand let S(t) '• D(A) —> D(A), i > 0 be the semigroup of nonexpansive mappings generatedby — A. Let D be a locally closed subset in D(A) and / : [a, 6) x D —> X a Caratheodoryfunction. We assume that either S(t) is compact for each t > 0, or that the inclusion D C. Xis compact and we prove :

THEOREM. Under the assumptions above, a necessary and sufficient condition in order thatfor each (T,£) € [ a , b ) x D there exists at least one local mild solution u : [T, T] —+ D ofu'(t) + Au(t) 3 f(t,u(t)) satisfying U(T) = £ is the tangency condition

There is a negligible subset % o f [ a , b ) so that, for each ( i ,£) £ ([a,b)\Z) x D,

Here u(t + h,t}£, /(<,£)) — y(t + h) where y is the unique mild solution of the problem

! / (< )=«•An interesting application concerning the existence of monotone solutions and some exten-sions to the case / multivalued are also included.

1. INTRODUCTIONOur main goal in this paper is to prove a necessary and sufficient condition in order that agiven subset of a Banach space X be a viable domain for a strongly nonlinear nonautonomousdifferential equation. Namely, let X be a Banach space, A : D(A) C X — > 2X an m-accretiveoperator with —A generating a semigroup of nonexpansive mappings S(t) : D(A) —> D(A),for t > 0, D a nonempty subset in D(A) and / : [a, 6) x D — > X a function. We consider thenonlinear perturbed differential equation

and we are interested in finding necessary and sufficient conditions in order that D be a viabledomain for (T>£).

DEFINITION 1.1. We say that D is a viable domain for (D£) if for each (T,£) € [a, b) x Dthere exists at least one mild solution u : [T, T} — > D, T < b, of (CDS) satisfying the initialcondition

U(T) = e. (36)

We recall that the function u : [ r , T ] — > D is a mild solution of (!>£) and (JC) if usatisfies (36), it renders the function < / ( • ) = f ( - , u ( - ) ) integrable on [r ,T] and it is a mildsolution on [T, T } of the equation

u'(t) + Au(t) 3 g ( t )

in the sense of Definition 1.7.5, p. 25 in [31].The viability problem has been studied by many authors by using various frameworks

and techniques. We start by reviewing the state of the art in the "continuous" case, i.e. inthe case in which / is continuous. First we discuss the semilinear case, i.e., the case in which— A is the generator of a Co-semigroup. In this respect it should be noted the pioneering workof Nagumo [22] who considered X finite dimensional and A — 0. In this context he showedthat a necessary and sufficient condition in order that D be a viable domain for (D£) is thetangency condition below :

lii£infid(£ + /i/(i,0,£) = 0 (LI)

for each (£,£) G [«,&) x -D. Here and thereafter d(x,C) denotes the distance from the pointx G X to the subset C in X. As far as we know, Nagumo's result (or variants of it) hasbeen independently rediscovered several times in the seventies by Brezis [9], Crandall [16],Hartman [19] and Martin [21].

In the infinite dimensional setting with A = 0 and f ( t , •) dissipative, we recall the resultsof Martin [21]. The semilinear "continuous" case, i.e. the case in which —A is the infinitesimalgenerator of a Co-compact semigroup and / is continuous has been studied by Pavel [23]. Weemphasize Pavel's main contribution who, to our knowledge, was the first who formulated thecorresponding tangency condition applying to the semilinear case by means of the generatedCo-semigroup, in such a way to work also for points £ that do not belong to the domain ofA. More precisely, Pavel [23] shows that, whenever A generates a compact Co-semigroup and/ is continuous on [a, 6) x D with D locally closed, a sufficient condition for viability is:

,),D)=0 (1.2)h

for each ( t , £ ) G [a, 6) x D. We notice that, whenever £ G D D D(A), (1.2) is equivalent to

lim±d(t

which is nothing else than the classical Nagumo's tangency condition (1.1) with A-\- f insteadof /. However, there exist situations in which D is not included in D(A), or even Dr\D(A) isempty and in these cases we can use only (1.2). For instance this happens it D is the trajectoryof a nonwhere differentiable mild solution of (T>£). For subsequent developements, allowing/ to be multivalued, see Pavel- Vrabie [24] and [25], Shi Shuzhong [27], Carja- Vrabie [13] andthe references therein.

The fully nonlinear case, i.e. the case in which both A and / are nonlinear, with Aunbounded but / still continuous, has been considered for the first time by Vrabie [30]. Wenotice that Vrabie [30] introduced the suitable tangency condition to apply also for points ofD which do not belong to D(A). Namely, the tangency condition introduced in [30] is

Iin4<f(u(* + h,t,£,f(t,t)),D) = 0, (1.3)h.10 h

where u(t + - , i , £ , y ) — v(-) is the unique mild solution of the Cauchy problem

v'(s) + Av(s) 3 y

More precisely, Vrabie [30] proved that if —A is the generator af a compact semigroup ofnonexpansive operators and (1.3) holds uniformly with respect to (i,£) in [a, b) x D, then Dis a viable domain for CDS). We emphasize that, whenever A is linear, (1.3) is equivalent to

ft+h \/ S(t-S)f(t,{)d3,D) = 0

Jt )which in its turn reduces to (1.2). Subsequent contributions in this context are due to Bothe [5]who allowed D to depend on t as well. In particular, in case D independent of t, Bothe [5]showed that (1.3) is necessary and sufficient for viability. We also mention Bothe [6], Bressan-Staicu [8] and Carja- Vrabie [14] who considered the case in which / is multivalued andsatisfies a certain continuity-like condition. While Bothe [6] and Bressan-Staicu [8] considera tangency condition which reduces to (1.3) whenever / is single valued and an e — 6 uppersemicontinuity condition and respectively a lower semicontinuity on /, Carja- Vrabie [14] allow/ to be strongly-weakly upper semicontinuous but use a tangency condition expressed in theterms of the weak topology on X and which, in certain situations, is stronger than (1.3).

Concerning the Caratheodory case, again when X is finite dimensional and A — 0, wehave to mention the work of Ursescu [29] who was the first able to overcome the difficulties ofdefining a suitable a. e. -tangency condition in order to obtain a corresponding Nagumo-typetheorem applicable in this general frame. More precisely, Ursescu proved that a necessaryand sufficient condition for viability under Caratheodory conditions on / is the Nagumo'stangency condition (1.1) but satisfied only for each (i ,£) € (iaib) \ Z) x D, where Z is anegligible set.

The true semilinear Caratheodory case, i.e. A unbounded and / measurable in t andcontinuous in £ has been considered for the first time in Carja-Monteiro Marques [11] byusing a "lim inf" variant of Pavel's tangency condition (1.2), again satisfied only for each(t, £) 6 ( [ a , b) \ Z) x D, where Z is as above. We note in passing that, whenever / is single-valued, (1.2) and respectively (1.3) is equivalent with the corresponding condition obtainedfrom (1.2) and respectively from (1.3) by substituting "lim" with "liminf".

In the present paper we consider the fully nonlinear Caratheodory case, i.e. the case inwhich A is nonlinear, possible unbounded and / is a Caratheodory function. We assume thatD is locally closed in the sense that for each £ € D there exists r > 0 such that D D -B(£, r) isclosed in X, where, as usual, B(£,r) denotes the closed ball with center £ and radius r andwe prove that under some extra-conditions either on the semigroup generated by —A or onthe subset D, a necessary and sufficient condition for the viability of D is (1.3) satisfied onlyfor each (i,£) G ([a, 6) \ Z) x D, where Z is a negligible subset in [a, b). See Theorems 3.1and 3.2 below. As a matter of fact, it should be mentioned that our main result here is anontrivial nonlinear extension of that in Carja-Monteiro Marques [11] and, inasmuch as thespecific "linear arguments" used in [11] fail in this new context, our proofs herein are distinctfrom their semilinear counterparts and are essentially new.

We also characterize admissibility of a preorder with respect to the differential equation(D£) through a tangency condition of type (1.3). We recall that a preorder ":<" on a subsetM of D is admissible with respect to (T>£) if for each (r, £) G [a, 6) x M there exists at least

one mild solution u : [T, T] —> M satisfying U(T) = £ and «(•$) ^ u(i) for each T < 5 < i < T.We adopt the approach proposed for the first time in Carja-Ursescu [12], where, in the finitedimensional setting, the admissibility of a preorder "^" with respect to (CDS) is described interms of viability of the sets P(£) = {r/ G M ; £ ^ T/} with respect to the same equation.That approach has been extended recently to the infinite dimensional case by Chi§-§ter [15]in the case of a continuous perturbation /. Usually, the authors get admissibility of preordersreconstructing, and thus increasing in amount, the proofs of viability of sets. See, e.g., [18],[20]. The interested reader in this area, as well as in viability problems, is refered to Aubin-Cellina [3].

Finally, as direct applications of our main theorems, we include some results concern-ing the invariance of a given subset with respect to a differential inclusion governed by aCaratheodory multivalued perturbation of an m-accretive operator. We recall that a subsetD is invariant with respect to (CDS) if /, as a function of its second argument, is definedon a subset in X which is larger than D, for instance if / : [a, 6) x D(A) —> X, and foreach (T, £) £ [a, 6) x D, either there is no solution of (CDS) and (3C), or each mild solutionu : [T, T) —> D(A) of the Cauchy problem above satisfies u(i) 6 D for each t G [ r , T ) .

The paper is divided into seven sections, the second one being merely concerned with somenecessary background material. In section three we state our main results, i.e. Theorems 3.1and 3.2, while in section four we present several auxiliary results. Section five contains thecomplete proof of both Theorem 3.1 and 3.2 and section six is devoted to the statement andproof of a necessary and sufficient condition of admissibility of a preorder with respect to(CDS). Section seven includes a necessary condition in order that a given subset be invariantwith respect to a differential inclusion governed by a measurable x lower semicontinuousperturbation of an m-accretive operator. We also show that the condition in question isalso sufficient for viability, and, in certain specific, but important cases, we prove that it issufficient even for invariance.

Acknowledgements. We express our warmests thanks to Professor C. Ursescu for the verycareful reading of a previous version of this paper, for his helpful critical remarks and sug-gestions concerning the presentation and especially for calling to our attention a possibilityto simplify our initial proof.

2. PRELIMINARIESWe assume that the reader is familiar with the basic concepts and results concerning m-accretive operators and nonautonomous differential equations in abstract Banach spaces andwe refer to Barbu [4] and Vrabie [31] for details.

However, we recall for easy references some basic concepts and results we will use in thesequel. Let A : D(A) C X —> 2X be an m-accretive operator, £ € D(A) and / 6 Ll(a,b;X)and let us consider the differential equation

u'(t] + Au(t) 9 f ( t ) . (2.1)

In all that follows we denote by w ( - , a , £ , / ) : [a, 6] — > D(A) the unique mild solutionof (2.1) satisfying u (a , a ,£ , / ) = £ and by S ( t ) : D(A) -> D(A), i > 0, the semigroupof nonexpansive mappings generated by —A, i.e. S(i)£ — u ( t , Q , £ , Q ) for each t > 0 and

€ D(A). We recall that, for each f , g € Ll(a,b;X) each £,77 <E D(A) and each a < t < 6,

we have

\\u(t,a,tJ)-u(t,a,T,,g)\\<\\t-T,\\+ f ||/(r) - g(r)\\ dr.Ja

Moreover, we note that, for each a < f < t <. b, we have

,(,]). (2.3)

See Vrabie [31]. We also recall that the semigroup S(t) : D(A) —» D(A), t > 0 is compact iffor each t > 0 S ( t ) is a compact operator. A subset G in L1(a, b; X) is uniformly integrable if,for each e > 0 there exists 6(e) > 0 such that, for each measurable subset E in [a, 6] whoseLebesgue measure X(E) < 6 ( e ) , we have

[\\9(*)\\ds<e,JE

uniformly for g G G.

REMARK 2.1. It is easy to see that, whenever I € XL^a, 6 ;R+) , the set

Gl = {gtLi(a,b;X); \ \ g ( t ) \ \ < t ( t ) , a.e. for f e [a ,6]}

is uniformly integrable.

We include for easy reference the following two compactness results which are the mainingredients in the proof of Theorems 3.1 and 3.2.

THEOREM 2.1. (VRABIE) Let X be a real Banach space, A : D(A) C X -> 2X an m-accretive operator, £ G D(A) and G an uniformly integrable subset in Ll(a,b;X). Then thefollowing conditions are equivalent:

(i) the set {w(-, a, £ , < / ) ; g 6 G} is relatively compact in C ( [ a , b ] ; X ) ;(ii) there exists a dense subset E in [a,b] such that, for each t e E, { w ( i , a , £ , < / ) ; g £ G}

is relatively compact in X.

See Theorem 2.3.1, p. 45, in Vrabie [31]. A very useful consequence of Theorem 2.1 is:

THEOREM 2.2. (BARAS) Let X be a real Banach space, A : D(A) C X -> 2X an m-accretiveoperator such that —A generates a compact semigroup, let £ 6 D(A) and G an uniformlyintegrable subset in Ll(a,b;X). Then the set {u(-,a,£,g); g G G} is relatively compact inC ( [ a , b ] ; X ) .

See Theorem 2.3.3, p. 47, in Vrabie [31].We are now able to introduce the tangency condition we are going to use in the sequel.

We begin with the tangency concept.

DEFINITION 2.1. Let A : D(A) C X —> 2X an m-accretive operator and D be a nonemptysubset in D(A). We say that y € X is A-tangent to D at £ 6 D if for each 6 > 0 and eachr > 0 there exist t e ( 0 , 6 ) and p € 5(0, r) such that

u(t,Q^,y) + tp£ D.

The set of all /1-tangent elements to D at £ e D is denoted by T£>(£).

REMARK 2.2. It should be noticed that whenever A = 0, T^(^) is nothing else than thetangent cone at £ € D in the sense of Bouligand [7] and Severi [26]. Moreover, if A is linearand A ^ 0, T^ is the tangency concept used by Shi Shouzhong [27] who considered the casewhen the perturbation is a multifunction not depending on t. It is easy to see that

, , ,ftj.0 ft

Moreover, this concept can be defined equivalently by means of sequences. Namely, y 6 Xis A-tangent to D at £ G D if and only if there exist two sequences, ( tn)ngN- decreasing to 0and (pn)ngN* convergent to 0, such that

u(tn,Q,{,y) + tnPn€D (2.4)for each n G N.

We conclude this section by recalling that the inclusion D C X is compact if each boundedsubset in D is relatively compact in X.

3. THE MAIN RESULTSWe begin by recalling :

DEFINITION 3.1. A function / : [a, 6) x D — »• X is called a Caratheodory function if itsatisfies :(Ci) for every x G D, the function /(- ,#) is measurable on [a, 6) ;(6*2) for almost every t G [a, 6), the function /(t, •) is continuous on D ;(6*3) for every r > 0 there exists a locally integrable function tr : [a, 6) — > R such that

[[/(£, x)|| < 4(0 for almost every t £ [a, 6) and for every x £ D n 5(0, r).

We are now ready to state the main results of this paper.

THEOREM 3.1. Let X be a real Banach space, A : D(A) C X — > 2X an m-accretive operatorwith —A the infinitesimal generator of a compact semigroup S(t) : D(A) — > D(A), t > Q, Da nonempty, locally closed subset in D(A), and f : [a, 6) x D — » X a Caratheodory function.Then a necessary and sufficient condition in order that the set D be viable with respect to(D£) is the tangency condition:(T) there exists a negligible subset Z in [a, 6) such that, for each ( t , £ ) G ( [ a ,6 ) \£ ) x D, we

THEOREM 3.2. Let X be a real Banach space, A : D(A) C X — > 2X an m-accretive operator,D a nonempty locally closed subset in D(A) with the inclusion D C X compact and f :[a, 6) x D — > X a Caratheodory function. Then a necessary and sufficient condition in orderthat the set D be viable with respect to (D£) is the tangency condition (T).

Concerning the existence of saturated, i.e. noncontinuable mild solutions of (D£) and(JC), using a standard argument based on the Brezis - Browder's countable version of Zorn'sLemma, see Theorem 4.2 below, we deduce:

THEOREM 3.3. Under the hypotheses of either Theorem 3.1, or Theorem 3.2, a necessaryand sufficient condition in order that for each (, & D there exists at least one saturated mildsolution of (CDS) satisfying (3C) is the tangency condition (7). If D is closed, then eachsaturated solution is global, i.e. defined on [T, 6).

4. SOME A U X I L I A R Y RESULTSWe begin with the following variant of some general "Lebesgue-type" theorem established in[28] and [11].

THEOREM 4.1. Assume that X is a real Banach space, D is a nonempty and separable subsetin D(A) and f : [a, 6) x D — > X is a Caratheodory function. Then there exists a negligiblesubset Z o f [ a , b ) such that, for every t € [a, 6) \ Z, we have

lim- \ \ f ( s , y ( s ) ) - f ( t , y ( t ) ) \ \ d s = Q (4.1)/Uo h Jt

for all continuous functions y : [a, 6) — > D.

To prove Theorem 4.1 we have only to repeat the same routine as that in the proof ofLemma 1 in [28] or that in the proof of Theorem 2.3 in [11], in the latter case by takingS(t) — /, the identity on X, for each t > 0.

The following (perhaps known) extension of a simple remark due to Vrabie will proveuseful later. See [11] Proposition 2.1.

PROPOSITION 4.1. If X is a Banach space for which there exists a family { S ( t ) ; t > 0} ofcompact operators such that

limS(t)tc = xt[0

for each x G X, then X is separable.

Proof. Take a sequence (in)n6N which is decresing to 0. Since, for each n G N, S(tn)B(0,n) isprecompact there exists a finite family of points Dn in B(0, n) such that for every x € 5(0, n)there exists xn G Dn satisfying

\\S(tn)x - S(tn)xn\\ <tn.

Let x G X and e > 0 and choose n G N such that tn < e, \\x — S(tn}x\\ < e and \\x\\ < n.Taking xn E Dn as above, we have

||z - S(tn)xn \<\\x- S(tn)x\\ + \\S(tn)x - S(tn)xn\\ < 2e.

So D = UnS(tn)Dn (which obviously is countable) is dense in X and this completes theproof. D

One of the main tools in the proof of both Theorem 3.1 and 3.2 is the following charac-terization of the tangency condition (T).

PROPOSITION 4.2. Let X be a real Banach space, D a nonempty and separable subset inD(A), A : D(A) C X —> 2X an m-accretive operator and f : [a, b) x D —> X a Cartheodoryfunction. Then, the tangency condition (T) is equivalent to the condition (7) below:(T) There is a negligible subset Z of [ a , b) such that for every ( i ,£) G ( [a , 6) \ Z) x D there

exist two sequences, (hn)n&K* decreasing to 0 and (pn)neN* convergent to 0, such that

u(t + hn,t,tJ(-,t)) + hnPneD (4.2)

for each n £ N*.

Proof. From (2.2) we have

/

t + h\\f(s^)-f(t^)\\ds

for each (i,£) G [a,6) x D and h > 0 with t + h < b. The conclusion follows directly from(2.4) and Theorem 4.1. D

We end this section by recalling a general principle on ordered sets due to H. Brezis andF. Browder [10, p. 356]. It will be used in the next sections in order to obtain some "maximal"elements in an ordered set. The existence of maximal elements is usually derived by usingthe well-known Zorn's Lemma, an ordering principle which is equivalent to the Axiom ofChoice. The Brezis-Browder's Ordering Principle is based on an axiom which is weaker thanthe Axiom of Choice, i.e. the Axiom of Dependent Choices [17]. See also [12, p. 16] for otherapplications.

THEOREM 4.2. Let X be a nonempty set, < a preorder on X and W : X —> M. U {+00} anincreasing function. Suppose that each increasing sequence in X is majorated in X. Then, foreach XQ € X there exists He G X with XQ < x such that IS < x implies $(x) = 4 r(x).

Note that, in the paper by Brezis and Browder, the function S is supposed to be finiteand bounded from above, but, as remarked in [12], this restriction can be easily removed byreplacing the function $ by the function x H-s- arctan(^(x)).

5. POOF OF THEOREMS 3.1 AND 3.2The necessity of both Theorems 3.1 and 3.2 is an immediate consequence of the next resultwhich is interesting by itself.

THEOREM 5.1. Let X be a real Banach space, A : D(A) C X —> 2A an m-accretive operator,D a locally closed and separable subset in X and f : [a, b) x D —> X a Caratheodory function.Then, a necessary condition in order that D be a viable domain for (D£) is the tangencycondition (7).

Proof. Let Z be given by Theorem 4.1, let r e [ a , b ) \ Z, let £ G D, choose a solution uto (23£) and (3C), which is defined on a subinterval [T ,T] of [a, b) and take a continuousfunction z : [a, 6) —> D which coincides with u on [r, T]. We have

u(T + h,r,tJ(.,z(-))eDfor each h G [0,T — T}. On the other hand, by (2.2), we get

Hr + f t , r , € , / ( - , z ( - ) ) - « ( r + A , r , f , / (

for each h 6 [0, T - T }. By (4.1) we know that

and therefore

Hence /(r, £) e 70(£) f°r ea°h (T;£) € ([a, 6] x D) \ Z and this completes the proof ofTheorem 5.1. D

We are now in the position to prove the necessity of both Theorems 3.1 and 3.2. To dothis we have only to observe that in both cases, i.e. S(t) is compact for each t > 0, or theinclusion D C X is compact, D is separable and therefore the hypotheses of Theorem 5.1 aresatisfied. See Proposition 4.1.

REMARK 5.1. The proof of Theorem 5.1 shows that, even in a more general frame than thatassumed either in Theorem 3.1, or in Theorem 3.2, a necessary condition for the viability ofD with respect to (D£) is a tangency condition which, in general is stronger that (T). Moreprecisely, we proved that such a necessary condition is:(S) there exists a negligible subset Z in [a, 6) such that, for each (i ,£) € ([a, 6) \ Z) x D,

we have/(U) £ §£(0-Here

S£(0 - {y e X; l imid(u(M,£,y),Z?) = 0}nJO ft

which is included in T^(£). Indeed, §#(£) C TD(£) simply because the former is defined bymeans of "lim", while the latter is defined in a very similar way but by means of "liminf".See Remark 2.2.

The proof of the sufficiency consists in showing that the tangency condition (T) alongwith Brezis-Browder Ordering Principle, i.e. Theorem 4.2 above, imply that for each (T, £)in [a, b) x D there exists at least one sequence of "approximate solutions" of (!)£), definedon the same interval, vn : [T,T] —> X, satisfying (JC) for each n £ N* and such that (vn)converges in some sense to a mild solution of (D£) satisfying (JC).

The next lemma represents an existence result concerning "approximate solutions" of(D£) satisfying (JC) and it is a nonlinear version of Lemma 3.1 in [11]. Its proof relies on aninterplay between some techniques developed in [5] and [11].

LEMMA 5.1. Let X be a real Banach space, A : D(A) C X —> X an m-dissipative operator,D a nonempty, locally closed subset in D(A) and f : [a, 6) x D —> X a Caratheodory functionsatisfying the tangency condition (7). Then for each (r, £) G [a, b) x D there exist r > Q,to € [a,b)\Z and T € (T, b) such that D n B(£, r) is closed and for each n 6 N* and for eachopen set £ o/R with Z C £ and A(,C) < - (A is the Lebesgue measure), there exist a familyof nonempty and pairwise disjoint intervals: 7 = {[tm,sm); m £ 3} and two measurablefunctions g 6 Ll(r,T;X) and v : [T,T] —> X satisfying

(i) [J [tm,sm) = [r,T) and sm - tm < - for each m e 3;m£3

(ii) i f t m < E L then [tm,sm) C L ;(iii) v(tm) <E D n 5(£,r) for each m 6 3, v(T) 6 D n B(£,r) and v ( [ r , T ] ) is precompact;(iv) g ( s ) - f(s,v(tm)) a.e. on [tm,sm) if tm <£ &, and g ( s ) = f(tQ,v(tm)) a.e. on [tm,sm) if

tm€£;(v) \\v(t)-u(t,tm,v(tm),g)\\ < -^—^ for each m £3 and each t £ [ t m , T ] .

nProof. Let (T, {) 6 [a ,6) x Z) be arbitrary and choose r > 0 so that D n B(£,r) is closedand there exists a locally integrable function l ( - ) such that | |/(i,x)|| < t(t) for almost every

t 6 [a, 6) and for every x £ D C\ B(£,r). This is always possible because D is locally closedand / satisfies (Cs) in Definition 3.1. Fix t0 ^ Z and T1 £ (T, 6) such that

sup \\S(t - T)£ - £|| + K + T - T < r, (5.1)T<t<T

where K = max {(T - T)l(t0} , £ l(s)ds\ .We prove first that the conclusion of Lemma 5.1 remains true if we replace T as above

with a possible smaller number fj, 6 (T, T] which, at this stage, is allowed to depend onn G N* and then, by using the Brezis-Browder Ordering Principle Theorem 4.2, we will provethat we can take fj, = T independent of n G N*.

For n G N* take an open set £ of K with Z C £ and whose Lebesgue measure A(£) < ^.Case 1. In case r G £, since /(to,£) is A-tangent to D at £ it is easy to see that there

exist <5 6 (0, ^-) and p 6 X with ||p|| < ^ such that [T, r + <5) C /C and such that

u(T + £ ,T , f , / (* 0 , 0 )+*PeI> .Now, let us define g : [T, T + <*> ] — > X and u : [T, r + <5] — > X by </(t) = /(to, £) and respectivelyby

v ( t ) = u(t,T,t,g) + (t-T)p (5.2)for each t G [ T, T + (5 ] .

Let us observe that the family 7T+s — {[TiT + <5)} an(i the functions g and v satisfy(i)-(v) with T substituted by T + £•

Case 1. In case T ^ £, we have T £ Z and in view of Proposition 4.2 there exist 5 G (0, )and p e X with ||p|| < i such that

Setting g ( s ) — /(s,^) and defining v by (5.2), we can easily see that, again, the familyyr+f — {[T, T + <5)} and the functions 5 and v satisfy (i)-(v) with T substituted by T + 6.

Next, we show that there exists at least one triplet (CP, g, v) satisfying (i)-(v) with T givenby (5.1). To this aim we shall use the Brezis-Browder Ordering Principle (see Theorem 4.2)as follows. Let U be the set of all triplets (CP^,^, v^) with fj, < T and satisfying (i)-(v) with fj,instead of T. This set is clearly nonempty as we already proved. On U we introduce a partialorder as follows. We say that

where 3>w = {(tkm,skj;m e 8k}, k = 1,2, if

(0i ) Mi < ^2 and if /^ < //2 there exists i e ^2 such that /«! = t,2 ;(O2) for each mi 6 #1 there exists m2 6 #2 such that t^ = t^2 and sjni = s^2 ;(Os) S'MiC5) = 5w(-s) and W M I ( S ) = VM2(5) f°r eacn 5 ^ [ T > ^ i ]•

Let us define the function ty : U — > K by

It is clear that $ is increasing on U. Let us take now an increasing sequence

in U and let us show that it is majorated in U. We define a majorant as follows. First, set

; j <E N}.

If (i* = fij for some j G N, (!PMj , gtlj , u M j ) is clearly a majorant. If /j,j < /.t* for each j G N, letus observe first that the intervals in the family 3V — {[t3

m, s3m); j ' G N, m G 3j} are pairwise

disjoint and so this family is at most countable. For our latter purposes, it is important toemphasize that in fact 3V is countable. Indeed, by (Oi), we have that for each j G N thereexists m G 3j such that /ij = f^. On the other hand, the set {fj,j ; j G N} is clearly countablebecause //* = sup{/Uj ; j G N} and /j,j < /.i* for each j G N. Hence 3V can be written in theform 3V = {[tm,sm);m G N}. We define

for j G N and every t G [T, fij]. Now let us observe that (3Vi SVi'i^v); where 3V j p^. andUM« are defined as above, satisfies (i), (ii), (iv) with T replaced with T + fj,*. Notice that(v) is also satisfied but only on [r, //*). Obviously we have uM*(im) G D n B(£,r) for eachm G N. To see that (3Vi <7M*, UM-) satisfies also (iii) we have to check first that W M . ( [T , /^*))is precompact in X and next to show how to define v^ (fi*). By (ii) and ((7s) we know that<7 G <C I(T, /i* ;Jf) and so, for each j G N, the function u(-, fj,j, vfl»(/j,j), 17) : [ / Z j , / ^ * ] — > /?(^4) iscontinuous. Therefore C3 = u([//j , /i* ] , / / j , i )M . ( /^j) ,^) is precompact. On the other hand, by(iii), for each j G N, we know that Kj — *V([T, ji3 • ] ) is precompact too. By (v) and (0]) wededuce that, for each j G N,

iv([r,/O) C Q U A', + ^(0, 1)-

Let e > 0 be arbitrary and fix j G N such that

Since Cj U A'j is precompact, there exists a finite family {x1; x2, . . . , £n(£)} such that, for eachx G Cj U A'J, there exists fc G {1,2 , . . . , n(e)} such that

,i,From the last two inequalities and the inclusion above, we get ^.([r, //*)) C U^lj B(xk,£)and accordingly v^.([r, ^u*)) is precompact. Now, take any limit point v* of v ^ ( / j , j ) as _/ tendsto +00 and set UM.(^*) = u*. Clearly v^>(/j.*) & D C\ £?(£,r). So, with UM. : [T,/J,*] — > AT,defined as above, we obviously have that (3V, <7j,*,iv) satisfies (i), (ii), (iii) and (iv). It isalso easy to see that (v) holds for each m G N and each t G [tm ,^*). To check (v) for t = /j,*,we have to fix any m G N, to take t = Hj with /j,j > im in (v) and to pass to the limitfor j tending to +00 both sides in (v) on that subsequence on which (VJ(/J,J))J£N tends tov* — ?v (//*). So (3V>3ViSV'j ?v) ig a majorant for ((3"^, 3V,>#J' W J ) ) J£N and consequentlythe set II endowed with the partial order < and the function \& satisfy the hypotheses ofTheorem 4.2. Accordingly there exists at least one element (!?„, <?,,, u,,) in U such that, if(yv,gv,vv) < (Ty,ga,va) then v = a.

We show next that v = T, where T satisfies (5.1). To this aim let us assume by contra-diction that v < T and let £„ :— vv(v) which belongs to D n B(£,r). In view of (2.2) and( i )-(v) we have

lie, - en < \MV) - «KT,^)H + H^T,^^) - .?(" - ^11 + 115^ - - 1\\ <

( r^< ^ + sup \\S(t - T)^ - £|| + max <^ (v - r)l(t0), / l(s) ds

Recalling that v < T", from (5.1) we get

\\t*- t\\<r. (5.3)There are two possibilities: either v £ £, or f ^ -C.

If z; £ £ we act as in Case 1 above with ;/ instead of T and with £„ instead of £. Sofrom the tangency condition (7) combined with (5.3) we infer that there exist S £ (0, -] withv + 5 < T, [ i/, v + S) C £ and p £ A" satisfying ||p|| < i, such that

u(«/ + 6, v, &,, /(*o, 6)) + Sp £ D n B({, r).If f ^ L we act as in (7ase 2 above with ^ instead of T and with £„ instead of {. So from

Proposition 4.2 combined with (5.3) we infer that there exist S € (0, ^] with i/ + 5 < T andp 6 X satisfying ||p|| < ^, such that

«(«/ + 6, {„, /(-, 6)) + 6P£Dr} B(£, r).We define 7v+s — fu U {[ v, f + <5)}, ^+4 : [T, // + ^] — > X and u^+j : [T, v + ^] — » AA by

( f (Jj6) if

in case v 6 £ and

in case v £ £ and respectively by

(i,t/ ,^, gv+s) + (t-v)p iff € ( ! / , « / + «].

Since wv +«(z/ + <5) £ D and by (2.2) and (5.1)

\\vv+S(t) - f \ \ < \\vv+s(t) - u(t, r, t,gv+s)\\ + \\u(t, r, £, gu+s) - S(t - r)£\\ + \\S(t

)-+ [t\\gn JT

+s(s)\\d3+ sup \\S(t-T)£\\<

< supT<t<T

for each t G [v,v + 6], (yv+s,gv+St^v+s) satisfies (i), (ii), (iii) and (iv). with T replaced byv + S. Clearly (v) holds for each tm and t satisfying tm < t < v, or tm — v < t. The only casewe have to check is that in which tm<v<t<v + 6To this aim, let us observe that, byvirtue of (2.3) and (2.2) and (v), we have

<.

(t - L > ) p - u ( t , L > , u ( i / , t m ,,tm,vv+5(tm},gv+6)\\ + (t- v

t - tm

n n nSo (v) holds for each m € N and each t £ [ < m , j / + 6].

Thus, (yv+s,gv+s,vv+6) G U, (7v,gv,vv] < (yvJrS,gv+s,vv+s} and v < v + 8. This contra-diction can be eliminated only if v = T and this completes the proof of Lemma 5.1. D

DEFINITION 5.1. Let (r,£) G [a, 6) x D, n G N* and the set £ as in Lemma 5.1. A triplet(IP, g,v) satisfying (i)-(v) is called an n-L,- approximate solution of (D£) and (JC) on [T, T].

We are now prepared to complete the proof of the sufficiency of Theorems 3.1 and 3.2.

Proof. Let (£„) be a decreasing sequence of open subsets in E such that Z C <Cn and X ( L n ) <^ for every n G N*. Take L := Hn>i£n and a sequence of n-£n-approximate solutions((yn,gn,vn))neN' °f (33£) and (3C) on [ r , T ] . From (ii i) and condition (Cs) we know that{gn; n G N*} is a uniformly integrable subset in Ll(r, T;X). See Remark 2.1. Under thehypotheses of Theorem 3.1, since the semigroup S(t) : X — > X, t > 0, is compact, byTheorem 2.2 it follows that there exists v G C*([T, T]; X) such that, on a subsequence atleast, we have

,£,gn) = v ( t ) (5.4)

uniformly for t G [T, T]. We shall prove now that, under the hypotheses of Theorem 3.2,(5.4) still holds true. First, let us remark that D n B(£,r) is compact. Next, by (v) we have

for each k G N* and m £ 3k- For n G N* let us denote

Cn = {u(t, r ,£ ,<7 , )5 fc = l^, t G [r,T]} and tf = {«*(£); fc e FT, m € 0fc}and let us observe that, in view of the inequality above, we have

rr~i

{"(C^flfc); f c € N % r o € f l t } C C n U t f H - - 5 ( 0 , 1 ) (5.5)

for each n G N*. But, for each n G N*, Cn and K are precompact, the former becauseeach function u(-,r,£,gk) is continuous on [r,T] for k — l ,n , and the latter as a subset ofD H B(£, r) which in its turn is compact as already mentioned. This remark along with (5.5)shows that {w(^, T, £,g k) ' , k G N*, m G 3k} is precompact too. Now let us observe that,by (i), for each t G [r,T) and each A; G N* there exists m G 3k such that t G [ ^ m > s m ) an(^sm ~~ ^m ^ £• As a consequence {<^; A; G N*, m G 3/t} is dense in [r,T]. Thus we are inthe hypotheses of Theorem 2.1 and accordingly there exists v G C ( [ T , T}\ X) such that, ona subsequence at least, we have (5.4). So, in both cases, i.e. under the hypotheses of eitherTheorem 3.1, or Theorem 3.2, (5.4) holds. By (v) and (5.4), on the same subsequence, wealso have

\imvn(t) = v ( t ) (5.6)n

uniformly for t G [ r , T ] . Next, recalling once again that the set {t^', k G N*, m G 3k} isdense in [r,T] and vk(tk

m) belongs to D n B(£,r) for each A; G N* and m G 3k, from (5.4)and (5.6) we conclude that v(t] G D 0 B(£,r) for each t G [T, T } . Indeed, this is clearly thecase \it-r. So take t G (T, T ] , k G N*, m G 3k and let us denote (for the sake of simplicity)s = tk

m. Assume that s < t and let us observe that, by virtue of (v) and (2.2), we have

IKO - v k ( s ) \ \ < \\v(t) - u(t,s,vk(s),gk)\\ + | u(t,s,vk(s),gk) - S(t - s ) v k ( s ) \ \ +

+ \\S(t - S ) v k ( s ) - vk(s)\\ <~+ [ \\gk(0)\ dO + sup \\S(t - s)r, - r,\\ <

where 1(9) = ma,x{e(t0),C(6)} a.e. for 0 e [ r , T ] and

C = { v k ( t ) ; A: eN* , te(Since, due to (5.6), C is precompact in X, we have

limsup \S(S)r] — n\\ = 0.sl° 7,ec

Recalling that t e L1(r, T ; K + ) , that s denote a generic element tkm with tk

m < t, and usingthe relation above, we easily deduce that

where the latter is included in D fl B(£,r). So, we necessarily have u(i) € £) n J3(£,r) foreach f € [r,T].

Now, let us observe that if s £ L there exists n(s) 6 N* such that, for each n > n(s),s (£ Ln. Hence, by (i) and (iii) we have gn(s) = f(s,vn(t^n)) for each n > n(s) and for somem € 3n with |s — <^| < ~. Therefore by condition (C^) we get

\imgn(s) = f ( s , v ( s ) )n

for almost every s 6 [ r , T ] . /.From (6*3) and Lebesgue Dominated Convergence Theorem wededuce that

limgn = / ( - ,« ( • ) )n

in L I(T, T; A"). In view of (2.2) we then have v ( t ) = w( i , r , ^,5) for each t € [T,r] , where (/satisfies 5(5) = /(S,M(S,T, £ ,<? ) ) for almost every s G [r, T]. But this means that v is a mildsolution of (T>£) and (JC) and this completes the proof. D

REMARK 5.2. We notice that the above proof may be easily adapted to handle a slightlymore general result obtained from Theorem 3.2, by replacing the hypothesis "i/ie inclusionD C X is compact" with KD is locally compact and separable".

6. ADMISSIBLE PREORDERSLet M be a nonempty subset of D.

DEFINITION 6.1. The set M is admissible with respect to (D£) if for each (T,£) 6 [a, 6) x Mthere exists at least one mild solution u : [T ,T] —» M, T < b, of (1)£) and (JC).

Applying either Theorem 3.1, or Theorem 3.2 to the differential equation

u'(t) + Au(t) 3 f M ( t , u ( t ) ) ,where JM stands for the restriction of / t o [a, b) x M, we get

THEOREM 6.1. Under the hypotheses of either Theorem 3.1, or Theorem 3.2, assume thatM is a closed subset of D. Then M is admissible with respect to (D£) if and only if thereexists a negligible subset Z in [a, 6] such that, for each ( t , £ ) € ([a, 6) \ Z) x M, we/(*,0eo£(0

We also have

COROLLARY 6.1. Under the hypotheses of Theorem 6.1, a necessary and sufficient conditionin order that for each £ G M there exists at least one saturated mild solution u : [T, T) ~ > M ,T < b, o/(I>£) satisfying (JC) is the tangency condition (7) with D replaced with M.

Now, let ^ be a preorder on M, i.e. a reflexive and transitive binary relation on M. It isconvenient to identity the preorder ^ on M with the multifunction P : M — > 2 defined by

for all £ 6 M.

DEFINITION 6.2. The preorder P : M — > 2M is admissible with respect to (D£) if for every(T, £) € [a, 6) x M there exists a mild solution u : [T, T] -> M to (D£) and (JC) such thatfor every s € [i",T] and for every t € [s ,T], u(t) £ P(u(s)).

Our main goal in the sequel is to characterize the admissibility of a given preorder Pwith respect to the differential inclusion (T>£).

THEOREM 6.2. Under the hypotheses either of Theorem 3.1, or of Theorem 3.2, assume thatM is closed in D and the graph of P is closed in D x D. ,Then a necessary and sufficientcondition in order that P be admissible with respect to (D£) is the tangency condition below.(S) There is a negligible subset Z o f [ a , b ) such that for each (i,£) € ([a,b) \Z) x M we

Proof. The proof of the necessity follows the same lines as that of the necessity of eitherTheorem 3.1, or of Theorem 3.2 and so we are not going to give details. We only note thatwe have to use the simple fact that the admissibility of P implies the admissibility of P(£)for each £ € M. The main point is that, the converse is also true, as Proposition 6.1 belowshows, and this is the essential clue to conclude the proof of the sufficiency. Indeed, sinceP(r)} C P(0 for all £ e M and for all r) e P(£), it follows that 7^(ri)(rj) C 7^(rj) for allf 6 M and for all 77 G P(f,)- Thus, assuming condition (S) we get that, for every £ 6 M,P(£) satisfies the tangency condition in Theorem 6.1. Therefore, for each £ G M, P(£) isadmissible with respect to (2)£) and Proposition 6.1 below comes into play to deduce thatP is admissible with respect to (33£). D

PROPOSITION 6.1. Under the hypotheses of Theorem 6.2 the preorder P is admissible withrespect to the differential equation (2)£) if and only if, for every £ € M, the set P(£) isadmissible with respect to the differential equation (D£).

Proof. Clearly, if P is admissible with respect to (D£), then, for all x € M, P(£) is admissiblewith respect to (!>£). To show the converse, assume that, for each £ € M, P(£) is admissiblewith respect to (D£). Let £ e M and r £ [a, b). We shall show that there exists at least onesolution u : [r ,T] -> M to (D£), with u(r) = £ and such that u ( [ s , T } ) C -P(w(s)) for eachs G [T, r]. To this aim we proceed in several steps.

In the first step we note that, reasoning as in Lemma 3.1 [13], one can show that thereexists a 6 (T, b) such that for every noncontinuable solution u : [T, T) — > M to (X>£) withU(T) = £ we have & <T. According to Corollary 6.1, there exists a solution u : [T, T) — > Mto (D£) with u(r) = £ and W ( [ T , < T ] ) C P((,).

In the second step we observe that, for every solution v : [T, a] —> M to (2)£), withV ( T ) = £ and U ( [ T , < T ] ) C P(£), and for every t> 6 [r, cr) , there exists a solution u> : [ r, a ] — > Mto (2)£) such that w equals v on [ r, ^ ] and iw([ ;/, cr ]) C -P(u;(c)).

In the third step we observe that, according to the first two steps, for every nonemptyand finite subset S of [r, a], with T £ 51, there exists a solution u : [T, a] — > M to (D£) and(JC) with u ( [ s , a - } ) C P(u(s)) for all s G S.

In the fourth step we consider a sequence (5n)ngi\i of nonempty finite subsets of [T, a}such that: T G Sn and 5n C Sn+i for each n G N; the set 5 = Un6N5n is dense in [T, <r] . Forexample we can take

Further we apply the third step to get a sequence of solutions (un : [T, a] — > M)n67v to (T>£)and (JC) such that u n ( [ s , c r ] ) C P(un(s)) and for each n G N and each 5 G 5n. Now, applyingeither Theorem 2.2, or Theorem 2.1, we can assume, taking a subsequence if necessary, thatthe sequence (un}n^ converges uniformly on [r, a] to a solution u G C ( [ r , a ] ; X ) of (X>£).Clearly U(T) = £.

In the fifth step we show that u([s ,cr]) C P(u(s)) for all s G S D [ T , < J ] . Indeed, given 5as above, there exists n G N such that 5 G Sn- Then s G Sm and w m ( [ s , c r ] ) C P(um(s)) forall m G N with n < m. At this point, the closedness of the graph of P in D x D implies thatu ( ( s , f f } ) c P ( u ( s ) ) .

In the sixth and final step, taking into account that S Pi [r, a] is dense in [T, a], u iscontinuous on [r,cr] and the graph of P is closed in D x .D, we conclude that the precedingrelation holds for every s G [ T, cr ] and this completes the proof . D

7. SOME PROBLEMS OF INVARIANCELet us consider the differential inclusion

u' G Au + F(t,u),

where A : D(A) C X —* 2 is an m-accretive operator, D C D(A) is locally closed, whileweF : [ a , b ) x D(A) —> 2 is a given multifunction. By a mild solution of (DJ) on [T,T]

mean a continuous function M : [T, T] —> D(A) for which there exists g G Ll(r,T',X) suchthat g(t) G -F(£, w(i)) a.e. for t G [T, T] and M is a mild solution on [ r , T ] of u' G Au + g inthe sense of Definition 1.7.5, p. 25 in Vrabie [31] .

We say that D is invariant with respect to (DJ) if for each (r, £) G [a, 6) x D each mildsolution M : [ r ,T] -> D(A) of (DJ) satisfying (3C), i.e. U(T) = f, satisfies also u(t) G £> foreach t G [r,T].

Let Y be a nonempty and closed subset in X. We recall that a multivalued mappingQ from y into 2* is called lower semicontinuous on Y if for each open subset U m X theinverse Q~(C/) = {t 6 [a, 6); F(<) H C/ ^ 0} is open in r.

Let (fi,M) be a measure space. A multivalued mapping F from ft into 2X is calledmeasurable is for each open subset U m X the inverse F~(U) = {w G f t ; F(UJ) D f/ 7^ 0} ismeasurable, i.e. F~(U] G M.

DEFINITION 7.1. A multivalued mapping _F from [a, 6) x V into 2X is called a Michael-mapping (M-mapping for short) if for each closed subset K in [a,b)xY for which F\K is lowersemicontinuous, each closed subset C in K and each continuous selection fc : C -^ X of F\c,there exists at least one continuous selection /K : K —> Y of F| - such that f c ( t , y ) = /W(£ ,y )for every ( < , y ) G A'.

We note that whenever X is separable and Q is nonempty, closed, convex valued thenQ is an M-mapping. This is consequence of the well-known Michael' Selection Theorem. SeeTheorem 9.1.2, p. 355 in Aubin-Frankowska [2].

DEFINITION 7.2. A multivalued mapping F : [a, 6) x Y — » 2X is called a Caratheodorymapping if it has nonempty values and satisfies :

the mapping F is jointly measurable on [a, b) x Y endowed with the LebesguexBorelmeasure space structure ;

(CM?) for almost every i G [a, 6), the mapping F(t, •) is lower semicontinuous on Y ;(C MS) for every r > 0 there exists a locally integrable function lr : [ a , b ) —» R such that

sup{||j/|| ; y € F(t,x)} < tr(t) for almost every t G [a, 6) and for every x G Fn 5(0, r).

In our setting, the next result is a slight extension of Theorem 3.2 in Artstein-Prikry [1].Its proof, which is not evident, is inspired from that of Theorem 2.3 in Carja-MonteiroMarques [11].

THEOREM 7.1. Let X be a separable real Banach space, Y a nonempty and dosed subsetin X and F : [a, b) x Y — > 2X a Caratheodory M-mapping. Then, there exists a negligiblesubset Z in [a, 6) such that, for each (T, £) G ([a, b) \ Z) x Y and each rj G F(T, £), thereexists at least one Caratheodory function f : [ a , b ) x Y — > X with f ( t , x ) G F(t,x) for each

= ri and

+ | | /M*))->7ll<fe = 0 (7-1)

for each continuous function u : [r, T) — > Y satisfying U(T) = £.

Proof. First let us observe that we may assume without loss of generality that b < +00. ByTheorem 2.1 in Artstein-Prikry [1] we have that, there exists a sequence (Kn)n^- of compactsubsets in [a, 6) such that, for each n G N* we have:

( i ) A ( [ « , 6 ) \ A ' n ) < i ; _(ii) F\KnxY is lower semicontinuous ;

(iii) Kn C Kn+1.Next, let Ln be the set of all density points of Kn which, at the same time, are Lebesgue

points of all the functions belonging to the family {i[>n ; n G N*}, where, for each n 6 N*,^n(t) — £n(t)x\[a,b)\Kn(t) for t 6 [a, 6) and in is given by (CM^). Since almost all points of ameasurable set are density points, it is easy to see that X(Ln) = X ( K n ) and, by the definitionof Ln, we have

n = 1 (7_2)MO h ' '

for each t G Ln. Since, for each n G N*, Ln contains only Lebesgue points of i(>n and if>n\Kn = 0,from (7.2) we deduce

il>n(s)ds = 0 (7.3)(t,t+h]\Kn

for every t G Ln. SetZ=[a,b)\ U Ln

and let us observe that Z is negligible. Let (T, £) G ([ a, b)\Z) x Y. By ( i i i ) and the definitionof Ln it follows that Ln C Ln+\ for each n G N*. Therefore, there exists nT£ G N* such thatT G Ln C A'n and ||£|| < n —1 for each n > nT^. Since F is an Af-mapping, for each £ G X andeach 77 G ^(T, £), there exists a sequence of continuous functions ( f n ) n > n r ^ , fn '• Kn x Y —> Xsuch that f n ( t , x ) G F(t,x) for each (i ,z) G A'n x y, f n ( r , £ ) ~ r/, f n ( t , x ) - fn+i(t,x) foreach n > nT^ and ( i ,z) G A'n x Y. Indeed, for n = nT£, in Definition 7.1 we consider the setK = Kn x y and C = {(T, £)} and /C(T, 0 = ??• We then obtain a continuous selection /#„ ofF\KnxY, denoted for simplicity by /„ : Kn x Y —> X, such that /n(r, £) = r/. From this pointwe proceed inductively. Namely, given the continuous selection /m of F^m^Y, m > ^T,^, weextend it to a continuous selection fm+i of F^m+lXy- By induction this clearly proves theexistence of the sequence (/7l)n>nr? with all the properties mentioned above.

Let us define / : [a, 6) x Y —> X, by

0 i f t e Z .Obviously / is a Caratheodory function, f ( t , x ) G F(t,x) for each ( t , x ) G ([a, 6) \ Z) x Yand /(T, £) = r/.

To prove (7.1), let u : [r, T) —> y be a continuous function. We may assume with noloss of generality that T is small enough so that ||u(t)|| < n for each t G [ r , T ) . Recall that||£|| = | U(T)|| < n — 1. Let £ > 0 be arbitrary and let us observe that inasmuch as T G in,by (7.2) and (7.3), there exists [J,(e,n) > 0 such that

A([r,r + fe]\/^ ( . if W , ) d a <£ (?.4)

" 3 / z 7[T,T+A]\7fn 3for each /z G [0 , / z (e ,n ) ) . Since the restriction of / to [a ,6) x Kn is continuous there exists<5(e,n) > 0 such that

for each s G [T.T + 6 ( e , n ) } n Kn. Then, by (7.3), (7.4) and (7.5) we have

"• J[r,T+h]\Kn

\ I \\fn(S,u(sD-r,\\ds<n J[T,r+h]nKn,r+h]nKn

\\f(S,u(S))\\dS+l-[T,T+h]\Kn

ft J[T,T+h]\Kn n J[r,T+h]nKn

(r,T+h}\Kn n « J[r,T+h]nK

- r,\\ ds <

for each n G N* and each h G (0,S(e,n)) n (0, /z (e ,n) ) and this completes the proof. D

DEFINITION 7.3. An element (r,£) G [a, 6) x D(A) is called an existence point for (TO) if,for each 77 G F(T,£] and each Caratheodory selection / : [T, T) x .D(/4) —> A" of F satisfying/(Ti£) = ^ there exists at least one mild solution u : [T, T) —>• D(A) of u' G Au + f ( t , u )satisfying U(T) — £.

REMARK 7.1. If —A generates a compact semigroup each element ( r ,£) G [a, 6) x -D(/4) isan existence point. See Theorem 3.8.1 of Vrabie in [31], p. 131. Therefore, whenever X isfinite dimensional, [a, 6) x jD(/4) contains only existence points.

The first result concerning the invariance of D with respect to (CD3) is :

THEOREM 7.2. Let X be a separable real Banach space, A : D(A) C X — »• 2* an m-accretiveoperator, D a locally closed subset in D(A) and F : [a, b) x D(A) -^ 2X a Caratheodory M-mapping. Then, a necessary condition in order that D be invariant for (D£) is the tangencycondition :(T3) there exists a negligible subset Z in [ah) such that, for each (r, £) G ([a, 6) \ Z) x D

which is an existence point o/(D3), we have

Proof. Let D be invariant with respect to (T>3). Since _F is a Caratheodory M-mapping,by Theorem 7.1, there exists a negligible subset Z in [a, 6) such that, for each (T, £) G([a, 6) \ Z) x .P(;4) and for each rj G F ( T , £ ) there exists at least one Caratheodory function/ : [a, 6) x D(A) -> X such that f ( t , x ) G f(t ,x) for each ( f , z ) G ([a, 6) \ Z) x/ r > £ = ' and

for each continuous function u : [r, T) —> -P(A) satisfying U(T) = {. Let (r, £) G [a,6) \ Z anexistence point of (DJ) and let u : [T,T) —> D(A) be a mild solution of the problem

where / : [a, b) x D(A) — >• X is a Caratheodory function satisfying all the conditions men-tioned above. Obviously u is a mild solution of (2)J), and since D is invariant with respectto (KJ), we necessarily have u(t) G D for each t G [T, 71). From this point the proof followsexactly the same lines as those in the proof of Theorem 5.1. D

REMARK 7.2. We do not know whether or not the condition (77) is sufficient for invariancein this general setting. More than this, we do not know whether or not the stronger tangencycondition below(S3) there exists a negligible subset Z in [a, b) such that, for each (T, £) G ([a, b) \ Z) x D,

we have

is sufficient for invariance even under the extra-hypothesis that, either — A generates a com-pact semigroup, or D is compactly embedded in X. The only thing we are able to prove inthis specific setting is that, whenever F belongs to a quite narrow but important class ofmultivalued mappings, the tangency condition (S3) is necessary and sufficient for invariance.We will present later on such a particular case. See Theorem 7.4. We emphasize however,that the condition (S3) is sufficient for the viability of the set D even if F is a CaratheodoryM-mapping as Theorem 7.3 below shows.


THEOREM 7.3. Let X be a real Banach space, A : D(A) C X —> 2X an m-accretive operator,D a closed subset in D(A) and F : [ a , b ) x D —> 2X a Caratheodory M-mapping. Assumethat either the semigroup generated by —A is compact, or that X is separable and D is locallycompact. Then, a sufficient condition in order that D be viable with respect to (D£) is thetangency condition (§3).

Proof. The necessity follows from Theorem 7.2 combined with Proposition 4.1. To prove thesufficiency, take (r,£) 6 ([a, 6) \ Z) x £>, 77 <E F(t,£) and let / : [a, b) x D -> X be theCaratheodory selection given by Theorem 7.1. Clearly / satisfies the tangency condition inboth Theorems 3.1 and 3.2. The conclusion follows from Theorem 3.1 and Remark 5.2. D

We finally consider a special class of multivalued mappings F, for which, under someextra general hypotheses, (S3) is even a sufficient condition for invariance of D with respectto (IXJ). More precisely, we introduce:

DEFINITION 7.4. Let Y a nonempty subset in X. We say that F : [a, b) x Y —> 2X is asuperposition mapping if there exist a complete, separable metric space V, a lower semicon-tinuous mapping G : [a ,6) —» 2V with nonempty, convex and closed values and a function/ : [ a, 6) x V xY -> X such that:

for each (v,u) £ V x Y the function f ( - , v , u ) is measurable;for almost each t G [a, 6), the function f ( t , -, •) is continuous;

(SP3) for each ( t , w ) € [a, 6) x Y, we have

F(t,u)= \J /(*,«,«).«€G(<)

DEFINITION 7.5. We say that the function / : [a, b) x V x Y —> X has the uniquenessproperty if there exists a continuous function w : R_|. —> R+ such that

(it! - u2,f(t,v,ui) - f(t,v,u2))+ < u(\\ui - u2 | |) | |wi - w2||

a.e. for t G [a, 6), for each v G V and each MI, w2 € V and such that the differential inequalityx' < ui(x), x(0) = 0 has only the solution x ~ 0.

THEOREM 7.4. Let X be a real Banach space, A : D(A) C X —> 2* an m-accretive operatorwith —A generating a compact semigroup, D a closed subset in D(A) and F : [a, 6) x D(A) —>2^ a superposition mapping satisfying condition (CM^) in Definition 7.2. Assume that thefunction f in Definition 7.4 has the uniqueness property. Then a necessary and sufficientcondition in order that D be viable and invariant for (1)8.) is the tangency condition (SJ).

Proof. Necessity. Let (T,£) £ [a, 6) x D, rj G .F(r,£) and let us observe that, by (SPz) inDefinition 7.4 there exists v^ G G(r) such that /(T, v,,,£) = 77. Since G is lower semicontinuouswith nonempty, convex and closed values, by Michael' Selection Theorem 9.1.2, p. 355 inAubin-Frankowska [2] it follows that there exists a continuous function v : [ T, 6) —» V suchthat V(T) = Vr, and v ( t ) € G(t) for each t £ [T, b). Clearly, g : [ r , b ) x D(A) -> A", definedby g(t,u) — f ( t , v ( t ) , u ) is a Caratheodory function. Since the semigroup generated by — A iscompact, we are in the hypotheses of Theorem 3.8.1 of Vrabie, [31], p. 131. Accordingly, theCauchy problem

has at least one mild local solution u : [ r, T) —> D(A). Obviously u is a mild solution ofand, inasmuch as D is invariant with respect to (DO), we necessarily have u ( t ) G D for eacht G [ r , T ) . From this point the proof follows exactly the same lines as that of Theorem 5.1and therefore we do not enter into details.

Sufficiency. We begin by showing that D is viable. To this aim let v : [a, 6) —> V be acontinuous selection of G. We define g : [r, 6) x D(A) —> X by

9(t,u) = f(t, v(t), u)

for each ( t , u ) G [a, 6) x D(A). Clearly g is a Caratheodory function. Let now Z be thenegligible subset in [a, b) which corresponds to go = g\[a,b)xD by means of Theorem 4.1.Since F satisfies (S3), we easily conclude that go satisfies (7). So, by Theorem 3.1, for each(T, £) G [a, b) x D, the Cauchy problem (CO3) with g replaced by go has at least one saturatedsolution u : [T, T) — > D which obviously is a mild solution of (TO).

To complete the proof, i.e. to prove thejn variance of D, we have merely to show thateach saturated mild solution u : [T, T) — > D(A) of (DO) satisfies u ( t ) G D for each t G [T, T).Thus, let u be such a solution and let <?„ G LI(T, T; A") be such that w is a mild solution of

u' G AM + #«

and <7u(t) G F(t, u ( t ) ) a.e. for £ G [ T, T). At this point, let us remark that, since G is obviouslymeasurable, by virtue of Fillipov's Theorem 8.2.10, p. 316 in Aubin-Frankowska [2], thereexists a measurable function v : [a,6) — > V such that gu(t) = f ( t , v ( t ) , u ( t ) ) for almost everyt G [a, b). Next, with v as above, let us consider the function g : [a, 6) x D — > X, definedby g ( t , x ] = f(t,v(t),x). By the sufficiency part of Theorem 3.1 we know that the Cauchyproblem (CO3) has at least one saturated solution w : [T, 7\) — > D. Inasmuch as / has theuniqueness property and D is closed, it follows that T\=T and w coincides with u on [T, T).This completes the proof. D

REFERENCES[1] Z. ARSTEIN AND K. PRIKRY, Caratheodory selections and the Scorza Dragoni property, /. Math. Anal.

Appl., 127(1987), 540-547.[2] J.-P. AUBIN AND H. FRANKOWSKA, Set-Valued Analysis, Birkhauser, Basel, 1990.[3] J. P. AUBIN AND A. CELLINA, Differential Inclusions, Springer Verlag, Berlin-Heidelberg-New York-

Tokyo, 1984.[4] V. BARBU, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden,

1976.[5] D. BOTHE, Flow invariance for perturbed nonlinear evolution equations, Abstr. Appl. Anal., 1(1996),

417-433.[6] D. BOTHE, Reaction diffusion systems with discontinuities. A viability approach, Nonlinear Anal.,

30(1997), 677-686.[7] H. BOULIGAND, Sur les surfaces depourvues de points hyperlimites, Ann. Soc. Polon. Math., 9(1930),

32-41.[8] A. BRESSAN AND V. STAICU, On nonconvex perturbations of maximal monotone differential inclusions,

Set-Valued Anal., 2(1994), 415-437.[9] H. BREZIS, On a characterization of flow-invariant sets, Comm. Pure Appl. Math., 23(1970), 261-263.

[10] H. BREZIS AND F. BROWDER, A general principle on ordered sets in nonlinear functional analysis,Advances in Math., 21(1976), 355-364.

[11] O. CARJA AND M. D. P. MONTEIRO M A R Q U E S , Viability for nonautonomous semilinear differentialequations, ./. Differential Equations, 165(2000), 000-000.

[12] O. CARJA AND C. URSESCU, The characteristics method for a first order partial differential equation,An. §tiinf. Univ. Al. I. Cuza lasi Sec/. / a Mat., 39(1993), 367-396.

[13] O. CARJA AND I. I. VRABIE, Some new viability results for semilinear differential inclusions, NoDEANonlinear Differential Equations Appi, 4(1997), 401-424.

[14] O. CARJA AND I. I. VRABIE, Viability results for nonlinear perturbed differential inclusions, Panamer.Math. .}., 9(1999), p. 63-74.

[15] I. CHI§-§TER, Monotone solutions for single-valued perturbed nonlinear evolution equations, Commun.Appl. Ana!., to appear.

[16] M. G. CRANDALL, A generalization of Peano's existence theorem and flow-invariance, Proc. Amer.Math. Soc., 36(1972), 151-155.

[17] S. FEFERMAN, Independence of the axiom of choice from the axiom of dependent choices, J. SymbolicLogic, 29(1964), 226.

[18] G. HADDAD, Monotone trajectories of differential inclusions and functional differential inclusions withmemory, Israel J. Math., 39(1981), 83-100.

[19] P. HARTMAN, On invariant sets and on a theorem of Wazewski, Proc. Amer. Math. Soc., 32(1972),511-520.

[20] L. MALAGUTI, Monotone trajectories of differential inclusions in Banach spaces, J. Convex Anal.,3(1996), 269-281.

[21] R. H. MARTIN JR., Differential equations on closed subsets of a Banach space, Trans. Amer. Math.Soc., 179(1973), 399-414.

[22] M. NAGUMO, Uber die Lage der Integralkurven gewonlicher Differentialgleichungen, Proc. Phys. Math.Soc. Japan, 24(1942), 551-559.

[23] N. H. PAVEL, Invariant sets for a class of semi-linear equations of evolution, Nonlinear Anal, 1(1977),187-196.

[24] N. H. PAVEL AND I. I. VRABIE, Equations devolution multivoques dans des espaces de Banach, C. R.Acad. Sci. Pans Ser. I Math., 287(1978), 315-317.

[25] N. H. PAVEL AND I. I. VRABIE, Semilinear evolution equations with multivalued right-hand side inBanach spaces, An. §tiin}. Univ. Al. I. Cuza Ias,i Snc\. I a Mat., 25(1979), 137-157.

[26] F. SEVERI, Su alcune questioni di topologia infinitesimale, Ann. Polon. Soc. Math., 9(1930), 97-108.[27] SHI SHUZHONG, Viability theorems for a class of differential operator inclusions, J. Differential Equations,

79(1989), 232-257.[28] C. URSESCU, Caratheodory solutions of ordinary differential equations on locally compact sets in Frechet

spaces, "Al. I. Cuza" University of Ia$i, Preprint Series in Mathematics of "A. Myller" MathematicalSeminar, 18(1982), p. 1-27.

[29] C. URSESCU, Caratheodory solutions of ordinary differential equations on locally closed sets in finitedimensional spaces, Math. Japan., 31 (1986), 483-491.

[30] I . I . VRABIE, Compactness methods and flow-invariance for perturbed nonlinear semigroups, An. §tiin}.Univ. Al. I. Cuza Ias,i Sec}. I a Mat., 27 (1981), 117-125.

[31] I. I. VRABIE, Compactness Methods for Nonlinear Evolutions, Second Edition, Pitman Monographs andSurveys in Pure and Applied Mathematics 75, Addison-Wesley and Longman, 1995.

FACULTY OF MATHEMATICS, "AL. I. CUZA" UNIVERSITY OF IASJ, IAS.I 6600, ROMANIAE-mail address: ocarjaSuaic.ro

FACULTY OF MATHEMATICS, "AL. I. CUZA" UNIVERSITY OF IAS.I, IAS.I 6600, ROMANIACurrent address: P. O. Box 180, Ro, Is. 1, la^i 6600, RomaniaE-mail address: ivrabieQuaic.ro


Almost Periodic Solutions to NeutralFunctional Equations

C. Corduneanu University of Texas at Arlington, Arlington, Texas

One problem oftenly encountered in the applications of functionalequations is the existence of periodic or almost periodic (in time) solu-tions.

For basic definitions and properties concerning almost periodicfunctions we shall refer the reader to our book [1].

We shall consider the functional equation

(Vx)(t) = (Wx)(t), t£R, (I)

where V and W stand for operators (not causal, in general) on thespace AP(R,Rn), consisting of Bohr almost periodic functions on R,with values in Rn.

Let us first assume that V in (1) is a linear operator on AP(R, Rn),and rewrite the equation (1) in the form

(Lx)(t) = (Nx)(t), ten, (2)with N standing, in general, for a nonliner operator on the spaceAP(R,Rn).

The special case of (2)

( V x ) ( t ) = f ( t ) , teR, (3)

with / 6 AP(R, Rn), is solvable in case, and only in case,

sup | ( L a ; ) ( t ) \ > m B u p \ x ( t ) \ , te R, (4)

for some positive m, and any x G AP(R^Rn). Condition (1) is thewell known condition for the invertibility (with bounded inverse) of thelinear continuous operator L, taking into account that supremum is thenorm in AP(R,Rn).

Based on the solvability of the equation (3), under condition (4),we can proceed to the discussion of the equation (2). It turns out that(2) is also uniquely solvable in AP(R, Rn], if N is Lipschitz continuous,with a sufficiently small Lipschitz constant K:

\Nx - Ny\Ap < K\x - y\Ap. (5)

Indeed, from (2) we obtain

m sup x(t) -y(t)\ 1, (6)

with x°(t) arbitrary in AP(R, Rn) is convergent in this space whenK < m. We can obviously write

xm+l(t) - xm(t) < Km~l sup xm(t) - xm-\t] , (7)sup

and since Km"1 < 1, the assertion is proved. The uniqueness of thesolution of (2) is obtained from

777,sup \x(i) — y(t)\ < K sup \x(t) — y(t}\,

which has been established above.The discussion conducted so far, leads to the following result:

Proposition 1. Consider the equation (2) in the space AP(R, Rn], andassume that the operators L and N satisfy (4), resp. (5). If K < m,then the iteration process defined by (6) is convergent in AP(R, E"1} tothe unique solution of (2).

Remark. The existence of the solution to equations (1) and (2), can beinterpreted as existence of coincidence points to the couple of operators(V, W), resp. (L,N).

Example. As an illustration to Proposition 1, we shall consider theequation

x(t)+ I k(t- s)x(s)ds = f(t,x(t),x(t + h ) ) , (8)JR

in which k = (fcjj)nxm is integrable on R, while / is almost periodicin the first argument and Lipschitz continuous with aspect to last twoarguments. One chooses h G R arbitrarily.

The condition equivalent to (4) can be written as

det[J + k(iw}}\ >p,>Q,weR, (9)

wherek(uj) = [ k(t)e-iultdt, u e R, (10)

JRis the Fourier transform of k.

According to Proposition 1, equation (8) has a unique solution inAP(R, Rn) if (9) is verified and the Lipschitz constant for / is suffi-ciently small.

We shall consider again equation (2), and notice that under con-dition (4) it can be rewritten in the form

x(t) = L-l((Nx)(t)), t E R. (11)

In this form, the application of fixed point method appears to be ap-propriate. We shall make such assumptions that will allow us to obtainexistence of a solution by means of Schauder fixed point theorem forcompact operators.

In order to secure the compactness of the operator L~1N it sufficesto assume that N is a compact operator on AP(R,Rn). Since (4)implies that L~l\ < m"1, one can write

< m^KNx^t)], t E R. (12)

Denotea(r) = sup|(Mc)(t)|, \x(t)\<r, (13)

assuming, of course, that the supremum in (13) is finite for each r > 0.From (12) and (13) we derive

L~l(Nx)(f) <m-la(r) for x(t}\ < r. (14)

Therefore, the operator L~1N will take the ball of radius r, centeredat the zero element of AP(R, Rn), into itself, if for this r one has

m~la(r] < r. (15)

Let us point out the fact that we need only one value of r > 0,such that (15) be valid. Such values for r do exist, for instance, if weassume

Qi(T*)limsup —^- < m. (16)

r-»oo r

In particular, when a(r) grows slower than r at infinity, condition (16)is verified.

Summarizing the discussion carried out above, the following exis-tence (only!) result can be stated:

Proposition 2. Consider equation (2), with L and N continuousoperators on AP(R, Rn). Moreover, assume that L is linear and in-vertible, while N is compact from AP(R, R71) into itself, and such that(15) or (16) is valid. Then equation (2) has a solution in AP(R,Rn).

Remark. The compactness of a set S C AP(R, Rn) is equivalent tothe following conditions: a) S is bounded, i.e., there exists M > 0 suchthat x(t)\ < M, t (E R, for each x G S\ b) S is equi-continuous, i.e., foreach e > 0, there exists 8(e) > 0, such that t, s E R, \t — s\ < 6 implies\x(t) — x(s)\ < £ for any x G S', c) 5" is equi-almost periodic, i.e., foreach e > 0, there exists £(e) > 0, such that \x(t + r) — x(t]\ < s, t 6 R,for al least one T in any interval (a, a + £) C -R, and any x E S.

Let us return now to the equation (1), and consider the case similarto (3), namely

(Vx)(t) = f ( t ) , t<=R, (17)

in which V is acting on AP(R, Rn) and / e AP(R, Rn). Since V is, ingeneral, a nonlinear operator, we do not have a condition of the form(4), to guarantee the existence of the inverse operator V~l.

Following E. Zeidler [2], we shall impose on the operator V in (17)a condition of monotonicity:

m\x(t) - y(t}f < < (Vx}(t) - ( V y ) ( t ) , x ( t ) - y(t] > . (18)

In (18), m > 0 is fixed, while x, y E AP are arbitrary.As we shall see, condition (18) assures the existence of the inverse

operator, which means that equation (17) is solvable in AP(R,Rn).

Actually, an iteration process can be applied in order to obtain theexistence of the solution for (17).

Let us consider the auxiliary operator

(Txx(t) = x(t) - X[(Vx)(t) - /(*)], t e R, (19)

where A is a positive number. It is obvious that any fixed point of T\in AP(R, Rn] is a solution to the equation (17). It will be shown nowthat we can find A > 0, such that T\ is a contraction on AP(R,Rn).One more condition will be required for V, namely

\Vx-Vy\AP<M\x-y\Ap, (20)

where M > 0 is fixed and x, y E AP(R, Rn) are arbitrary.The following equality follows easily from the properties of the

scalar product (in Rnl):

\Txx-Txy2AP = \x-y\\p-2A < Vx-Vy,x-y > +X2\Vx-Vy AP->

which leads to the inequality

\Txx ~ Txy\lP < (1 - 2mA + M2A2)|^ - y\2AP, (21)

if we take into account (18) and (20).^From (21) we derive that T\ is a contraction, if we can achieve

1 — 2mA+M2A2 < 1 for some positive A. This is obvious if we choose 0 <A < 2mM~2. Therefore, with such A, the operator T\ is a contraction.

Summarizing the above discussion on equation (17), we can statethe following result.

Proposition 3. Consider equation (17), with V acting on AP(R^Rn)and f G AP(R,Rn) arbitrary. IfV satisfies the monotonicity condition(18), then (17) has a unique solution in AP(R,Rn). This solutioncan be obtained by the iteration process xm+1(t) = (TAxm)(t), m > 0,0 < A < 2MM~2, starting with an arbitrary x°(t) € AP(R,Rn}.

We shall consider now the general equation (1), under the basicassumption that a solution of this equation does exist. Since this solu-tion is automatically almost periodic, it is interesting to establish some

connection between the almost periods of the solution and those of thedata.

Assume that the operator V satisfies the monotonicity condition(18). If x(t) is a solution of (1), then the following inequality holds:

m\x(t + T) - x(t)\2 << (Wx)(t+T)-(Wx)(t),x(t+T)-x(t) > . (22)

Equation (22) leads to

m x(t + r)- x(t)\2 < \(Wx)(t + r) - (Wx}(t}\ • \x(t + r) - x(t)\,

from which we derive, based on the inequality 2ab < ea2 + £~lb2, withs = m,

\x(t + r) - x(t)\ < m-l\(Wx)(t + T) - (Wx)(t)\, (23)

for t 6 .R and T a fixed real number.The inequality (23) can be easily dealt with to find out the connec-

tion between the almost periods of x(t) and those of the equi-almost pe-riodic set {Wy}, where y € AP(R, R™) is such that \y(t}\ < sup \x(t)\,while W is assumed compact on AP(R,Rn). Namely, one reads from(23) that any me-almost periodic for the functions in {Wy}, \y(t)\ <sup |ic(i)|, t € -R, is an e-almost period for x(t).

Let us point out the fact that the almost periods of the functionsin {Wy}, \y(f)\ < sup|x(i)|, t G R, depend only of the properties ofthe operator W.

These remarks are useful if we look for solutions of equation (1),in the form

Finally, in concluding this Appendix, let us consider an alternateapproach in regard to the almost periodicity of solutions of functionalequations, such that the case of functional differential equations can becovered.

Let us assume that in equation (2), the operator L is a differentialoperator of the form

(Cx)(t) = x(t) - (Lx)(t), t e R. (24)

Of course, it is necessary to choose another underlying space thanAP(R,Rn). It appears natural to consider the space AP^(R,Rn),consisting of all functions such that x(i),x(t) 6 AP(R,Rn), the nat-ural norm being sup(|or(i)| + |rc(i)|), t £ R. Endowed with this norm,AP^(R, Rn) becomes a Banach space.

The invertibility of £, given by (24), in the space AP^(R,Rn]can be discussed by means of the equation (Cx)(t) = f ( t ) . The generalcase, to the best of our knowledge, has not been investigated in theliterature. The case (Lx)(t) = A(i)x(t] is throroughly investigated in[3].

The case when L is a causal operator may be dealt with on thesame lines as the case mentioned above. This idea can be motivatedby the fact that the equation (£x)(t) = f ( t ) , with L causal, possessesan integral representation of the solutions (see Ch.3).

REFERENCES[1] C. Corduneanu (Almost Periodic Functions) (Second English

edition, Chelsea Publ. Co., New York, 1989; currently distributed byAmerican Math. Society).

[2] E. Zeidler (Nonlinear Functional Analysis and Its Applications),II. Springer, Berlin, 1983.

[3] M.A. Krasnoselskii et al. (Nonlinear Almost Periodic Oscilla-tions, John Wiley, New York, 1973).


The One Dimensional Wave Equation withWentzell Boundary Conditions

ANGELO FAVINI, GISELE Ruiz GOLDSTEIN,JEROME A. GOLDSTEIN, AND SILVIA ROMANELLI

DlPARTIMENTO DI MATEMATICA, UNIVEKSITA* DI B O L O G N A , P IAZZA DI PORTA S.DONATO 5,40127 BOLOGNA, ITALY

E-mail address: [email protected]

CERI, UNIVERSITY OF M E M P H I S , M E M P H I S , TENNESSEE 38152E-mail address: giseleOceri.rnoiTiphis.edu

DEPARTMENT OF M A T H E M A T I C A L SCIENCES, UNIVERSITY OF MEMPHIS, MEMPHIS, TEN-NESSEE 38152


DlPARTIMENTO INTERUNIVEKSITARIO DI MATEMATICA, UNIVERSITA' DI BARI, VIA E.ORABONA4, 70125 BARI, ITALY


ABSTRACT. We prove that in C[0,1] the mixed problem for the wave equationd2u/dt2 - c2d2u/fix2, u(x,0) = f ( x ) , du/9t(x,0) = g ( x ) , d2u/dx2(j,t) = 0, forx G [0,1], .7 = 0.1 and t G R is governed by a cosine function whose generatoris the operator Au — d2/dx2 with domain including Wentzell boundary conditions(Au(j) = 0, for j = 0, 1). Relations with squares of first order operators are alsoconsidered.

1. INTRODUCTION AND MOTIVATIONConsider the parabolic problem

r-\

~ = Au + h(x, t) (x € ft CC R™, t > 0)

u(i,0) = /(x) ( x e Q )u(x,t) = o (x 6 dft, t > o).

A standard approach to this problem is to write it as

duh ( t )

where u : [0,oo) — > X, X being a Banach space of functions on £1 and AQ is therealization on X of the Dirichlet Laplacian. Focussing on the maximum princi-ple leads to the supremum norm, and demanding that the problem is governedby a strongly continuous (or (C'0)) semigroup [13] requires that AQ is densely de-fined. Thus we are led to take X := C'o(fi), the space of all continuous func-

*Work supported by G.N.A.F.A. ( I .N.D.A.M.) and by University of Bari Research Funds.


tions on ft that vanish on the boundary 9ft. But then h(t) e A" requires thath(x,t) = 0 for all x 6 9ft, which is too restrictive. We prefer to assume h eC(ft x [0,oo)), or that h e C'([0, oo); C(ft)). Thus we want to extend the semi-group TO := (To(<) : t > 0} generated b}r the Dirichlet Laplacian AQ to T,a (C'0) contraction semigroup on C(ft). Its generator A will be an extension ofAO; which one is it? Now take A" := C'(ft). Let A be the Laplacian definedon D(A) := {u e C(ft) : the distributional Laplacian AM is in Co (ft)}. TheWentzell boundary condition [18] implicit in this definition is Ati = 0 on 9ft. Forthe problem

-^ = AM x £ ft, t > 0

M ( x , 0 ) = / ( x ) x e f tAw = 0 x e 9ft, t > 0,

the boundary condition AM = 0 on 9ft, coupled with ut = AM implies ut — 0 on9ft; hence w(x , t ) — f ( x ) for x e 9ft and t > 0 (assuming / e C(ft)). This is thehomogeneous Dirichlet boundary condition when / e Co(ft), but it is more generaland ties an inhomogeneous Dirichlet boundary condition to the initial conditionin a linear way. This leads to the definition of a linear operator A which is m-dissipative and densely defined on C(ft); the semigroup T that A generates is thedesired extension of TO .

We have recently made a systematic study of parabolic problems involvingWentzell boundary conditions and generalized Wentzell boundary conditions [1]-[12]. The case of Wentzell boundary conditions for the wave equation is muchtrickier. There are a few simple results and enticing calculations but lots of openquestions. These results and problems will be the focus of this paper.

We shall work in one dimensional space for the following reason. W.Littman[15] showed that the wave equation utt = AM is wellposed in an Lp(Rn) contextfor an n > 2 if and only if p = 2, whereas when n = 1, d'Alembert's formula givesthe solution in terms of translation operators and translation is continuous in manynorms, including the norms of LP(R), 1 < p < oo and BUC(R) (= the space ofall real-valued bounded uniformly continuous functions defined on R). So we shallwork in the context of ft := (0,1) and X := C(ft) = C[0,1].

2. THE WAVE EQUATION

Of concern is the mixed problem for the wave equation

(2.1) | (1C) M ( x , 0 ) = / ( x ) , f f ( x , 0 ) = f f ( x ) a : e [0 , l ]

We study the problem in the space C[0,1]. The function v := | f satisfies

{ 9\ _ C2 oi| ^

v ( j , t ) = 0 for .? =0,1.

The mixed (Dirichlet) problem is well known to have a a unique solution given byd'Alembert's formula, namely

1 r~ ~ 1 1 rx+ct

(2.3) v(x, t)=- \J'(x + ct) + J'(x - ct)} + ——' —•(-•. x-ct

where ft is the odd, 1-periodic extension of ft € C[0,1]. For our original problem, weassume /, g 6 C*2[0,1], so that f",g" £ C[0,1]. In order that /",<?" be continuouson R, we must further assume that f"(j) = g"(j) = 0 for j' = 0,1, so that

f",g" G C0(0,1) := {w G C[0,1] : u>(0) = u>( l ) - 0}.

When the initial value problem for u" = Au,u(0) = /, w'(0) = 0 is well posed forA a densely defined linear operator on a Banach space X, then the unique solutionis given by

«(*) = <?(*)/,where C := [C(i) : t G R} is the cosine function generated by A, (Cf. e.g. [13].)If w" = Aw,iu(Q) = 0,w'(Q) = g, then z = w' satisfies z" = Az, z(0) = g, z'(0) =w"(Q) = Aw(0) - 0, whence

z (t) = C(t)g

(2.4) w(t) = S(t)g := C(s)g ds.Jo

Thus the unique solution of

u" = Au, u(Q) = f , u'(0) = g

is given by

where S is given by (2.4). Hence the solution (2.3) of (2.2) can be rewritten as

where Co is the cosine function generated by -^ on Co(0, 1) and So is the corre-sponding sine function, denned as in (2.4) with the subscript zero added.

Now, let P be the canonical projection from C[0, 1] onto C'o(0, 1). Thus for

is the unique function in Co(0, 1) satisfying

(Pf)" = f", mease /eC 2 [0 , l ] ;

equivalently Pf differs from / by a linear function and P/(0) = P/(l) = 0.

Theorem 2.1. The mixed problem (2.1) is well posed in C[0, 1]. It is governed bya cosme function C and a corresponding sine function S given by (2.4); the unique

solution of (2.1) is

= C0(t)Pf + S0(t)Pg + (/(I) - /(0))x + /(O)+ t[(g(l)-g(Q))x

The proof is now merely a straightforward computation; we omit the details.

Corollary 2.1. Let A be the identity function \(x) := x. The cosine and sinefunctions associated with (2.1) are given by (see (2.4))

C(t)f = C0(t)Pf + (/(I) - /(0))A + /(O),

S(t)g = S0(t)Pg + t [ ( g ( l ) - g(0))\

Corollary 2.2. The operator A = -r-j with Wentzell boundary conditions onC[0, 1] (with

D(A) ~ {u 6 C2[0, 1] : u"(0) = u"(l) = 0})generates a strongly continuous uniformly bounded cosine function on C*[0,l]. Also,A generates a semigroup analytic in the right half plane.

Proof. The first sentence is an immediate consequence of Theorem 2.1. Since— 1> we S6*- ^e estimate

sup||C(i)| |<3,

which gives the uniform boundedness. The last assertion follows from Romanov'sformula [17], [13, Theorem 8.7, p. 120], and the semigroup generated by A is givenby

ofor t > 0, / 6 X. Clearly this is a well defined analytic function for Re t > 0.

3. THE FIRST ORDER SYSTEM

Let us write (2.1) as a system

(3.1) U' = AU, U(0) = F,where

The Wentzell boundary condition for this system is

(3.2) AU(t) = 0 on dtt = {0,1}.

This becomes

u t ( j , t ) = Uxx(jit) = 0 for all t 6 R and x = 0, 1.

This introduces an extra boundary condition, namely

This requires g € Co(0,1), which is an inessential requirement, according to theresults of the previous Section. Thus problem (2.1) is not equivalent to problem(3.1), (3.2). This observation complements a classical result of B. Nagy [16] and J.Kisynski [14] concerning cosine functions on C[0,1]. In many cases, the generatorA of a cosine function is of the form A = G2 + H where G generates a (Co) groupand D(H] D D(G) (or perhaps H is bounded or even a multiple of the identity).We show this to be the case for an example related to problem (2.1), working at aheuristic level.

4. SQUARES OF FIRST ORDER OPERATORSDefine B by

(4.1) £/(*):= c|-/(l - z) = c/'(l-*)

(where c £ R\{0}) with boundary condition

(4.2) a/(0) + /?/'(0) = 0.

ThusD(B) := {/ € C'lO, 1] : a/(0) + /3/'(0) = 0}.

Clearly, for / smooth enough,

B 2 f ( x ) = c2/"(*),

B2nf(x) = c2nf(2n\x),

thus for all polynomials (and functions analytic on \z < 1), g, we can calculateg(B). In particular,

00 jn r>n f °° 2k(r)2kf\( \ °°» t B f f ^ _ _ ^ r < t B J _ ^ c (u n(x)+^c^"> ^ nl ^ (2fc)! ' (2fc + l)!

n=0 k=0 V ' fc=0 V y

= [cosfe(cZ>)/](x) + [sm/i(cD)/](l - x);

thusetB = cosh(cD) + Rsinh(cD)

where D := £ and (x) := g(l - x). Recall that e±(cD/(x) = /(x ± c<), andthese calculations make sense when we determine how to extend functions / inG[0,1] to all of R (say to F 6 BUC(R)) in a consistent manner with the boundaryconditions. In particular we should have H-FHoo = ||/||oo-

CASE I. a = 0,/3 = 1. Then A = B2 = c2L>2 has boundary conditions /'(O) =0, /"(I) = 0 for /(<E G2[0,l]) in D(A). Thus we have a Neumann condition at 0and a Wentzell condition at 1. According to the Neumann condition, / in D(A)

should be an even function. Thus we are led to the space X :— Ce[— 1, 1] of evencontinuous functions on — 1, 1]. On this space, A — c2D2 with Wentzell boundaryconditions. Except for inessential changes, this is the case treated in Section 1.

CASE II. a = 1, 0 = 0. The boundary conditions for A = B2 = c2D2 are

Thus functions in C[0, 1] should be extended to be odd about 0, even about 1. andperiodic of period 2. This is a well understood case which has nothing to do withthe Wentzell boundary condition.

5. OPEN PROBLEMS

The main case of a ^ 0, /? ^ 0 leads us to (for A = B2)

a/(0) + /3/'(0) = 0, «/'(!) + /?/"(!) = 0.

This is very difficult to analyze and represents an open problem.If we again define B by (4.1) but replace (4.2) by

a/(0) + /?/(!) = 0,

then the second boundary condition for B2 becomes

Taking a = — /? leads to 1— periodic boundary conditions. The most general gameto play in this context is to replace (4.2) by

a/(0) + 0/'(0) + 7/(l) +*/'(!) = 0,

where ( a , / 3 , ~ f , § ) is a nonzero vector in R.4.Another open problem is as follows. Consider the wave equation with generalized

Wentzell boundary conditions on [0, 1], namely

d2u 2 d2u—— = c2 —— , 0 < x < l , t e R ,at2- ox1

r\

u(x, 0) = f ( x ) , -j£(x, 0) = g ( x ) , 0 < x < 1,*~l2 *~i

} + /3j 7 0 ' ) + 7j«(J) = 0, at j = 0, 1

where 70,71 > 0, /?i > 0 > /9o- Is this problem well posed in C[0, 1]? The corre-sponding heat problem is known to be well posed [7].

REFERENCES1. V.Barbu arid A.Favini, The analytic semigroup generated by a second order degenerate dif-

ferential operator in C[0, 1], Suppl. Rend. Circolo Matem. Palermo 52 (1998), 23-42.2. A.Favini, G.R.Goldstein, J.A.Goldstein and S.Romanelh, Co-semigroups generated by second

order differential operators with general Wentzell boundary conditions, Proc. Am. Math. Soc.128 (2000), 1981-1989.

3. ____, Nonlinear boujidary conditions for nonlinear second order operators on C[G, 1], Arch,der Mathern. (to appear).

4. ____, On some classes of differential operators generating analytic semigroups, EvolutionEquations and their Applications in Physical arid Life Sciences (G.Lumer and L.Weis eds.),M.Dekker, 2000, pp. 99-114.

5. ____, Generalized Wcnlzcll boundary conditions and analytic semigroups in C*[0,1], Pro-ceedings of the 1st International Conference on Semigroups of Operators: Theory and Appli-cations, Newport Beach, California, December 14-18, 1998, Birkhauser (to appear).

6. ____, Degenerate second order differential operators generating analytic semigroups in Lp

and W^'P, (submitted).7. ____, The heat equation with generalized Wentzell boundary conditions, (in preparation).8. A.Favini, J.A.Goldstein and S.RomanelH, An analytic semigroup associated to a degenerate

evolution equation, Stochastic Processes arid Functional Analysis (J.A. Goldstein, N.A.Gretskyand J.Uhl, eds.) M.Dekker, New York, 1997, pp. 85-100.

9. ____, Analytic semigroups on L^(0, 1) and on Lp(0, 1) generated by some classes of secondorder differential operators, Taiwanese J. Math. 3, No.2 (1999), 181-210.

10. A.Favini and S.Romanelli, Analytic semigroups on C'[0, 1] generated by some classes of secondorder differential operators, Semigroup Forum 56 (1998), 367-372.

11. ____, Degenerate second order operators as generators of analytic semigroups on C[Q, -j-oo]or on Lp , (0,-f-oo), Approximation and Optimization, Proceedings of the International Con-

a~ 2ferenceon Approximation and Optimization, Cluj-Napoca, July 29-August 1, 1996 (D.Stancu,G.Cornan, W.W.Brecknor and P.Blaga eds.), Volume II, Transilvania Press, 1997, pp. 93-100.

12. A.Favini and A.Yagi, Degenerate Differential Equations in Banach Spaces, Pure and AppliedMathematics : A Series of Monographs and Textbooks 215, M.Dekker, New York, 1998.

13. J.A.Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press,Oxford, New York, 1985.

14. J.Kisynski, On cosine operator functions and one-parameter groups of operators, StudiaMath. 44 (1972), 93-105.

15. W.Littman, The wave operator and Lp-norms, J. Math. Mech. 12 (1963), 55-68.16. B.Nagy, Cosine operator functions and the abstract Cauchy problem, Periodica Math. Hung.

7 (1976).17. N.P. Romanoff, On one parameter operator groups of linear transformations I, Ann. Math.

48 (1947), 216-233.18. A.D. Wentzell, On, boundary conditions for multi-dimensional diffusion processes, Theory

Prob. Appl. 4 (1959), 164-177.


On the Longterm Behaviour of a ParabolicPhase-Field Model with Memory

M A U R I Z I O GRASSBLU AND VITTORINO FATA

DlPARTIMENTO DI MATEMATICA "F. BRIOSCI l f '

POLITECNICO DI MlLANO

VIA E. B O N A R D I , 9 - MlLANO - ITALY

maugraOmate.polimi.it, pataOmate.polimi.it

Abstract. We consider a non-conserved phase-field model with memory effects in the internal energyand in the heat flux according to the Coleman-Gurtin law. Thus, the temperature and the orderparameter evolve according to a nonlinear parabolic integrodiff'erential coupled system. In particular,the equation derived form the energy balance contains two time-dependent convolution terms whichare characterized by two relaxation kernels ft and k, respectively. In a previous paper, the phase-fieldsystem was analyzed by the present authors jointly with C. Giorgi. We first proved that the systemwas indeed a dynamical system in a suitable phase space depending on the temperature history. Thenwe showed the existence of a uniform absorbing set and the existence of a uniform attractor of finitefractal dimension. All these results were obtained by assuming that the energy relaxation kernel (Iwas smooth, bounded, and concave. This assumption is thermodynamically compatible, but in thel i tera ture a more common assumption is that Q. must be smooth, summable, and convex. In this case,the dissipativity of the system, i.e., the existence of an absorbing set, is more delicate to prove, sincethe memory term in the internal energy is somehow antidissipative. Here we provide conditions whichensure the existence of a uniform absorbing set when ft is smooth, summable, and convex. Existenceof an attractor of finite fractal dimension is also discussed.

1. INTRODUCTION

In [12] (cf. also [3, 4]), we proposed and analyzed a nonlinear integrodiffereiitial coupledsystem describing, in absence of mechanical stresses, phase transitions in a material withmemory effects, like, e.g., certain high-viscosity liquids (see. e.g.. [17] and referencestherein). More precisely, we considered a material occupying a bounded domain fl CIR3 with smooth boundary 3ft, whose state at a point x G Q. at time t € El, wascharacterized by three variables; namely, the temperature variation field -O(x.t), its pasthistory i ) l ( x , s ) ~ i>(x,t. — s ) , s > 0, and the order parameter \(x,t). Then, referring to


[12] for the details, we derived the following evolution system

(1.1)

(1.2)

in fl, for any f € 1R. Here a and A: are (smooth) relaxation kernels which fulfill suitablethermodynamic restrictions (see below), / basically represents the heat supply, and Aand -y are smooth given functions. The instantaneous diffusion coefficient e is supposedto be positive (see [8]); for the case e = 0 the reader is referred to [11, 13, 14] andreferences therein.

In [12] we investigated the longterm dynamics described by system (1. !)-(!. 2) byassuming that A: and A/ were summable on (0, +00) and A: was convex; while « wasrequired to be bounded and concave with a(0) > 0 and <z% a" summable on (0, +00).These assumptions are thermodynamically compatible (see [11]); however, a more com-mon assumption on a (see, e.g., [10] and its references) is that a must be convex andsummable along with a' and a" on (0, +00). In this case, as we noted in [11], the dissi-pativity of the system, i.e., according to [19], the existence of a uniform absorbing set, isharder to prove. This is basically due to the fact that the integral term characterized bya' behaves in an antidissipative way. Here we prove that the existence of an absorbingset can be obtained even in this case, provided that « is suitably dominated by A; andthe Poincare inequality for )) can be used; in other words, ?) must, for instance, vanishon a portion of 90 with positive Lebcsgue measure (compare with [11]). This result,combined with the techniques developed in [12], allows us to prove the existence of auniform compact attractor of finite fractal dimension in the present case as well. Weshall assume a(0) > 0 even though the results still holds when « = 0. as it can be easilyrealized. It is worth pointing out that a similar analysis can also be performed in theconserved case, where equation (1.2) is replaced by a fourth-order equation (see [15]).

To introduce the dynamical system, we first specify the initial conditions at a giventime r G ]R for all the state variables. Thus, due to the presence of memory dependentterms, besides the values of %? and % at r, the whole past history of t) up to r must begiven, namely,

^(a) = T)o(a) in 0, Vj >0

where ??o(a) is the ;7i:!:aJ ^aj^ Azjfory of i).Concerning boundary conditions, we assume

c?n% = 0 on c*0 x (-r, +00)i) = 0 on <90 x 1R

3n being the usual outward normal derivative. Here we assume homogeneous Dirichletboundary condition for J just for the sake of simplicity. Indeed, it is not difficult to

check that mixed Neumarm-Dirichlet boundary conditions for 0 ca.n be trea.ted providedthat the Dirichlet's hold on a, portion of 9ft of positive Lebesgue measure so that thePoincare inequality can be applied.

In order to operate in a history space setting, along the lines of [11. 12]. we introducethe additional variable ;/' (cf. [9]), which is defined by

7/'(.t ,5) = I i)'(x.T)dr s >0 .Jo

This variable ?/ is easily seen to satisfy the equation

dtr)'(s) + dsifis) = d ( f ) in ft, (t, s) 6 (-, +00) x (0, +00)

along with the initial and boundary conditions

if = ?/o in ft x (0, +00),?(0) = 0 in ft x (T, +00)

where

T)O(X,S) = I • & 0 ( x , y ) d yJo

is the initial summed past history of $.Assuming a suitable and physically reasonable asymptotic behavior of kernels k

and a'(see below), making a formal integration by parts, and setting

/ i ( s ) = —k (s) and "(•s) = a (•?)

for any s > 0, the above choice of variables leads to the following initial and boundaryvalue problem, where we have set a(0) = e = 1.

Problem P. Find ( - i 9 , X ' , ? / ) solution to the system

/"OC

v(a)r,\a) da - / ^(a)A?/'(a) da = f ( t )

dtx(f) - Ax(i) + x3(t) = 7(

in fi, {or any t > r and any s > 0, which satisfies the initial and boundary conditions

•d = 0 on <9fi x (T,+OO)dn\ = 0 on 30 x (r,+oo)r y f O ) = 0 on Q x (T, +oc)

t9(-) = ?9o in ftX ( r ) = X0 in ft7/r = 7/0 in ft x (0, +00).

In the next section we shall give a rigorous formulation of problem P and we shallstate the result winch ensures P to be a dynamical system on a, suitable phase space.The main result, i.e., the existence of a uniform absorbing set, is proved in Section 3.The existence of a compact uniform attractor of finite fractal dimension is discussed inthe final section.

2. THE SOLUTION PROCESSBefore introducing the variational formulation of P and the related well-posedncss re-sult. some notation and assumptions are needed.

We recall that fi C IR3 is a bounded domain with smooth boundary d£l. We setH = L2(Q), V = Hl(ty, V0 = H£(Sl), W = f f 2 ( f i ) , and V0 = H2 n H^Sl), with theidentification H = H* (dual space). We denote the norm and the product oil a spaceX by (•, -}.\' and | • | y, respectively. In particular, due to the Poincare inequality

H (2.1)

we can take |t'||y0 = l lVu l^ -a . The symbol {-, •} will stand for the duality pairing betweenV0*

Given a positive function a defined on 1R+ — (0,+oo), and a real Hilbert spaceA , let L^(IR , X ) be the Hilbert space of A"- valued functions whose norms belong toI2(IR+) with respect to the measure a(s)ds.

The assumptions on the memory kernels are the following (see [11-13])

i / , / / e C 1 ( I R + ) n L 1 ( ] R + ) (Kl )v ( s ) > 0, fj.(s) > 0 V s e IR+ (K2)v ' ( s ) < 0, /-i'(s) < 0 V s e I R + . (K3)

In view of (K1)-(K2), we introduce the space M = I2(IR+, H) n £2,(IR+, V0). Further-more, we assume

7 e C f l(IR) and 7' e I°°(IR) (HI)A e C'2(IR) and A" 6 L°°(IR) (H2)f ( = L } o c ( T R , H ) (H3)i90 e H (H4)X0 £ V (H5)7/0 6 M. (H6)

Definition 2.1. Let (K1)-(K2) and (H1)-(H6) hold. Pick r, T e IR such that T > rand set /= [r,T]. A triplet ( t f , A ' , > ? ) which fulfills

•d e C ' ° ( / , f f )n i 2 ( / ,Vo)

is a solution to problem P in the time interval I provided that,

(dfd,v) + (X'(\)d,X,v)H + ( V t f , V i ' ) / / 3 + ( d , v ) H - IJo

+ / fj,(<r)(¥t](a), Vt '}tf3 du - ( f . v ) n V u G I-o- a.e. in / (2 .2)Jo

dfX - A,\ + A3 = -/(A') + X'(X)i) a.e. in Q x I(<9,7/ + d.7/, i>>}M = (d, i/')M V-0 e y\/f, a.e. in /dnX = 0 a.e. on dfl x /i9(r) = $0 a.e. in $7A'(r) = XQ a.e. in S7rfT = 770 a.e. in Q x IR+.

Here we point out that —d3 is the infinitesimal generator of the right-translation semi-group on M.

Assuming (K1)-(K3) and (H1)-(H6), along the lines of [11, 12], it is possible toshow that problem P admits a unique solution in every time interval /. In particular, Pgenerates a strongly continuous process of continuous operators on the product Hilbertspace "H = H x Y0 x j\4. Indeed, denoting by U/(t, T)ZQ the solution (i>, X, ??) to problemP at time / with source term / and initial data ~o — ($o, A'o , ' /o) G T~i given at time r,the two-parameter family of operators U f ( t , r ) , with t > r, r € IR. satisfies the usualproperties of a process (see, e.g., [16], Chapter 6). In particular, the crucial continuity-property follows from

Theorem 2.2. Assuming (K1)-(K3) and (Hl)-(HG), there holdsrT

\\Ufl(t,-)z0i - Uh(t:T)z02ln + I \ X i ( y ) - X2(y)\\2vdy

01 — -02!^ -

for some A > 0 depending only on I and on the size of zgt and /,, where \i(t) denotesthe second component of the vector Uft (t, r)zoi-

Remark 2.3. Here, for sake of simplicity, we assumed / € L*OC(TR,H). Nonetheless,the same results hold for the more general situation / € L^OC(]R., H) + L^OC(IR, V*).

3. U N I F O R M ABSORBING SETSIn this section we consider a family of processes { U f ( t , r), / G J7}, where T is a certainfunctional space, and we prove the existence of a bounded, connected, invariant uniformabsorbing set (as / G T} for U/(t. r) in "H. That is, we show that there exists a boundedconnected set BO C T~i such that U/(t,r)Bo C BO, for every / G J~ and every t > r;moreover, given any bounded set B C Ti. there is t* — t * ( B ] > 0 such that

\JUf(t,r)BcB0,

To this aim, we introduce the Banach space T of L\oc-translation bomided functionswith values in H, namely,

r , r+1 iT = / e Lloc(JR, H) • \\f\\T = sup / lf(y)lH dy <oo\.(. r£TR J r J

We also need to make some additional hypotheses on the memory kernels. We set/•CO /"DO

O.Q = / v(a) da > 0 and kg — I i.i(a)da > 0Jo ,/o

and we assume

n'(s) + 8fj,(s) < 0 for some 8 > 0, V s £ IR+' (K4)9

aoi'(.s) < ——^—^-80^(3) for some 50 6 (0,^) , V 5 6 1R+. (K5)

Notice that (K4) entails the exponential decay of the kernel /.t. This should be comparedwith the results in the literature concerning the decay of linear homogeneous systemswith memory (cf. [18] and references therein). We also remark that condition (K5) canbe rewritten in terms of the original kernels o. and k as follows

a'(0)a"(s) > ——^—T-Sok^s ) for some <S0 € ( 0 , 8 ) , V s € IR+.1 ~t~ Cp

In light of (K5), it is immediate to check that the spaces L^(1R+, H) Cl L^(IR+, Vb) andL?(IR , V Q ) coincide, and have equivalent norms. Thus in the sequel we agree to denote

W'e have the following uniform energy estimates.

Theorem 3.1. Let (K1)-(K5) and (H1)-(H6) hold, and let f 6 T. Then there exists > 0 and two continuous increasing functions Cj : IR+ —> IR+. j = 1,2, such thai

for every t > r, T E IR..

Proof. We perform some a priori estimates, which clearly hold in a proper approximationscheme (see [12]). Take v — i9 in equation (2.1); multiply equation (2.3) by Xt and thenintegrate over fL Adding the resulting equations, we have

n

(3.1)o

Then, multiply equation (2.3) by K/\', for AC G (0, 1) to be fixed later, and integrate overQ. so obtaining

where - y ( r ) = r + "/(>'). Finally, take ;/' = ;/ in equation (2 .4) to get

Using (K4) and performing an integration by parts, we have that

_ 12 J0

Set now

*2(<) = \m\\lt +Addition of (3.1)-(3.3), on account of (3.4), leads to

3 + 2||<C + 2\\dtX\\l! + 2K\\X\\2H + 2«|V,Y|2Ha + 2« |X'|4L,(

) ,^)H^- (3-5)o

Denoting

00

with <5o given by (K5), using the Young inequality and (2.1), we have the estimate

< ————Pri~ " l l ^ l l / / + T^-M2

M

2 / /Op

Therefore, exploiting once again (2.1),

(3.6)

Hence, setting

(recall that K,C < 1) substituting (3.6) into (3.5) we arc led to

dt v 'C}H +2{/,%/. (3.7;

Moreover, from (H1)-(H2), (2.1), and Young inequality, it is fairly easy to conclude that

< c + c(l + ||/||H)$ + «c|Wj|^3 + |<9Al/-/ + ~-^- (3.8)

for some c > 0 depending only on fi, /z, 7, and A. At this point, we choose K — if c, andwe put £ — e(/c)/2. Then (3.7)-(3.8) yield the differential inequality

~$2 + £$2 + £|j W||2H3 + IcV\fH < c + c(l + ||/|H)$. (3.9)

Making use of a Gronwall-type lemma (see, for instance, Lemma 2.5 in [12]), we end upwith

•>c C2e2s

$2(<) <2$2(-)e- s ( (- r ) + y+ ( 1 _ g _ . 2 ( l + ||/|r)-, V t 6 [r,+oo)

which implies the thesis at once. DExploiting Theorem 3.1, it is now fairly easy to find the required set BQ for the

family of processes (i7/(t,r), / 6 J-}, when J- is a bounded subset of T. Indeed,denoting M? = sup^g^- \ f \ - j - , let -B be the ball of ~H of radius ICi(Mf). It is thenimmediate to check that the set BQ = U/g^ U/> r U rgiR5 will do.

Integrating (3.9) over [t, t + 1], for t > T, we get another estimate, which is of someimportance for further asymptotic analysis (cf. [12]).

Lemma 3.2. Let (K1)-(K6) and (H1)-(H6) hold, and let f e T. Then there exists acontinuous function C% : IR x IR —> IR , increasing in both variables, such that

rt+\I I1V7„<] / •« .Ml 2 i H v C . . . M l 2 \ ,7,, ^ n (\\~ .. „ , ..\\-HI \\J \\T>sup supy (||Vt?(y)||2v3 + \\X(y)\\lv) dy < C3(\\z0\

where fl(y) and X(y) are the first and the second component, respectively, o f U f ( y , r ) z o .

Remark 3.3. The above results hold as well if we consider the same model withoutmemory in the internal energy (that is, a = 0). In this case, observe that the firstequation of P reads (compare with [3])

4. EXISTENCE OF A U N I F O R M ATTRACTOR

In this section, we assume (K1)-(K5) and (H1)-(H6), and we study the longterm be-havior of the family of processes U/(t, r) as / 6 H ( < y ) . for a given translation compactfunction g in LÔC(TR.,H).

Recall that g G i^ÎR, H) is said to be translation compact (cf. [7] and referencestherein) in L^ÎR, H) if the hull of g, denned as

is compact in i11oc(]R, H), where </' '(•) = < / ( • + r) is the translate of g by ?-.

Repeating with no substantial changes the argument developed in [12], we have

Theorem 4.1. There exists a compact set AC C l~i such that, for every ZQ G BQ andevery f G H(</), the solution U f ( t , r ) z o to P; for every T G IR. and every t > T, admitsa decomposition

Vf(t,r}z0 = z D ( t - T ) + zc(tT)

such that

for some M > 1 and £Q > 0, independent of ZQ and f , and

z c t ; r ) e 1C.

In particular, Theorem 4.1 says that AC is a uniformly attracting compact set forthe family {£//(<, r), f G H(</)}; that is, for any r G IR and any bounded set B C 7i,

f , 1lim sup 6 - f i ( U f ( t , T ) B , f C ) \ = 0

where ^(^1,62) = sup,ieS[ iiif22ge2 |si — 22 l-^ denotes the Hausdorff semidistance oftwo sets Si and ^2 in 7i.

Referring to [2, 16, 19] for a detailed presentation of the theory of attractors ofdynamical systems, we recall the following

Definition 4.2. A closed set A C Ti. is said to be a uniform attractor for the family{ U f ( t . r ) . f G H(<7)} if it is at the same time uniformly attracting and contained inevery closed uniformly attracting set.

Due to well-known results from [5, 6] (see also the monograph [20]). the existenceof a uniformly attracting compact set AC, together with the continuity of U , ( t , T ) asa map from T~i x H(</) to T~i. for every r G IR and t > T (which is a consequence ofTheorem 2.2), entail

Theorem 4.3. The. family { U f ( t . r ) , f G H ( < y ) } is uniformly asymptotically compactpossesses a compact and connected uniform attractor given by

f r(0) such that z ( t ) is any bounded complete. 1

trajectory of U/(t, r) for some f G H ( < / ) J

If we replace conditions (H1)-(H3) with

7 e C'2(IR) and 7' G L°°(IR) (HI')A(r ) = X0r A0 G El (H2')g is quasiperiodic in If (H3')

then there holds (cf. [12])

Theorem 4.4. The attractor A of the family { U f ( t , r ) , f G H(fir)} /KM /imie fractaland Hausdorff dimensions.

Recall that a function g : fi x IR — > IR is quasiperiodic (see. e.g., [1]) if

where G( - , a / ) € C'^T", ff) is a 27r-periodic function of w on the n-climensional torusT" and K = ( « ! , . . . , « ; „ ) are rationally independent numbers. Such g is translationcompact in L1

1OC(IR, H), and / G H(</) if and only if f ( x , t ) = G(x.K.t + ro0), for some

OTQ G T".

ACKNOWLEDGMENTSThis work has been partially supported by the Italian MURST Research Project "Sim-metrie, Strutture Geometriche, Evoluzione e Memoria in Equazioni a Derivate Parziali".

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On the Kato Classes of Distributions andthe #M0-Classes

A. GULISASHVILI Department of Mathematics, Ohio University,Athens, Ohio 45701, USA

1 IntroductionThe Kato class Kn on n-dimensional Euclidean space Rn was introducedand studied by Aizenman and Simon (see [1, 20]). The definition of Kn

is based on a condition considered by Kato in [11]. Similar functionclasses were defined by Schechter [18] and Stummel [23]. We refer thereader to [4, 10, 20] for more information concerning the Kato class andits applications. The Kato classes of order s were studied by Daviesand Hinz [3]. For the generalizations of the Kato class to the case oftime-dependent functions see [7, 13, 14, 15, 19].

It is known that the following conditions are equivalent for a func-tion V G L}oc: Condition A:

V G Kn.

Condition B: J~2\V\ G L°° and

oHma2 | |J-2(|V|)a | |00 = 0. (1)

Condition C:j-2|y| G BUC (2)

(see [8, 9], see also [6]). In (1) and (2), the symbol J~2 stands forthe Bessel potential of order —2, BUC denotes the space of boundeduniformly continuous functions on Rn, and (\V ) a ( x ) = \V(ax]\.

Since


condition B is equivalent to the following condition from [19]:Condition D: (/ - A)"1^] € L°° and

lim ||(JB-A)-1 |y||U = 0.E— >oo

It was shown in [19] that Condition D characterizes the Kato class.In the present paper we introduce and study two scales of classes of

tempered distributions on Rn (see definitions 2 and 3). We will call thescales in these definitions the scale of Kato classes of distributions andthe scale of .BMO-classes, respectively. Our main goal in this paperis to study whether conditions similar to conditions B and C abovecharacterize the Kato classes of distributions and the 5M0-classes.Our main results are contained in sections 2 and 3. In the proofs ofthese results we use the theory of Fourier multipliers and some ideas ofStein (see [21]), concerning the connections between the Bessel and theRiesz potentials.

2 The Kato classes of order s and the classesof distributions

We define the scale of the Kato classes of order s > 0 on Rn as follows:

Definition 1 Let V G L}oc and s > 0. Then we say that V G Ks,n iff

lim sup / \V(y}\gs(x- y)dy = 0a-"0+ x Jx-y<ay\<

wheref \X\-

9s(X)~\\x\s-

We also put K0in = BUG.

1 / 5 - 7 1 ^ 0 , 2 , 4 , .

Remark 1 There exists a positive constant rs such that the functionhs — rsgs coincides with the fundamental function for the fractionalpower (— A)s of the Laplace operator. In other words,

in the sense of distributions (see [16]). The class K^<n coincides withthe classical Kato class on Rn. The classes Ks^n with 0 < s < n werestudied by Davies and Hinz in [3] where the notation Ka was used. Forthe explanation why we put Koin = BUG see Remark 2 below.

Let S' denote the space of tempered distributions on Rn. For everyV G S' and a > 0 we define the dilation (V% of the distribution V asfollows:

If V is a function, then (V)a(x) = V(ax) for almost all x G Rn.For any r > 0 and x G Rn we will denote by B(x, r) the closed ball of

radius r in Rn centered at x. Throughout the paper we will denote by Athe class of functions A G C^ such that 0 < A < 1, supp(X) C B(0, 1),and A is equal to 1 in a neighborhood of 0. For s — n — 0, 2,4, • • • and0 < a < 1, we define a function gs<a by

an (~\ _ \T,\s~n In __y s , a ( x ) — \x\ m ] r -

\x\

The next definition introduces the Kato classes of tempered distri-butions on Rn.

Definition 2 Let V G S' and let s > 0 be a number such that s — n ^0,2,4, ••• Then we say that V G Ks,n iff for every a with 0 < a < 1and every A 6 A the distribution V*((A)i .</ s ) coincides with a functionfrom L°° , and moreover,

Urn | |V*((A)^.) l loo = 0.a—fO+ a

If s — n = 0,2,4, • • • , then we say that V G Ks,n iff for every a with0 < a < 1 and every A G A the distribution V * ((A) Lgs,a) coincideswith a function from L00 , and moreover,

lim | |y*((A) l f lf.,a)| |00 = 0.a— >-0+ a

It is not difficult to show that if V G L}oc, then \V\ G Ks,n & V GK,,n.

Let — oo < s < oo. The Bessel potential Js of order 5 is defined by

e e R\ 7 € s1.Here F : S' —> S' denotes the Fourier transform on S'. For / G L1 wehave

If s > 0, then the Bessel potential J s is a convolution type operator,

J— S Ft \ I £( \ X^f (j(x) = \ j(y)^s(x~JR"

where the Bessel potential kernel Gs is defined by

2/ — - s S r - J - T ^ / s x c ' V /

(27TJ2 2zl (|) JO 0

In (3), F denotes the gamma function. Formula (3) can be found in[16, 21]. For more information concerning the Bessel potentials see[12, 16, 21]. Note that the definition of the Fourier transform in thepresent paper differs from that in [21].

Theorem 1 Let V G L\oc and s > 0. Then the following are equiva-lent:(i)V&Ka,n.(H) J~S\V\ G£°° and

lim ct'\\J-3(\V\)a\\00 = 0.or— >0+

(Hi) J~S\V\

Remark 2 In the case where 5 = 2, Theorem 1 gives the equivalenceof conditions A — D for the classical Kato class. Theorem 1 also clarifieswhy we put KQJH

= BUC. Indeed, for 3 = 0 condition (Hi) in Theorem1 is \V\ G BUC which is equivalent to V G BUC.

Our next result generalizes the equivalence of parts ( i i ) and (Hi) inTheorem 1 to the case of tempered distributions.

Theorem 2 Let V G S' and s > 0. Then the following are equivalent:(i) V G L°°'-s and

lim a'\\J-(V)a\\00 = Q.a—>-0+

(ii) J~SV G BUC.

We will prove Theorem 2 first.Proof of Theorem 2. It is not immediately clear why the conditionJ~SV G L°° in part (i) of Theorem 2 implies that J~s(V)a G L°°. Thenext lemma shows that this implication is valid.

Lemma 1 Let V G S', s > 0, and J~SV G L°°. Then for every a with0 < a < 1 we have J~s(V)a G L°° and moreover,

sup as||J"

Proof of Lemma 1. It is easy to see that

as||J-5(V)«|U = as-n\\V*(Gs)i_\\oo. (4)a

We have(G,)L = Ga*Ya (5)

a

where

n^)(0 = «n(I^F)'- (6)

Consider the expansion

where \t\ < 1. The explicit formula for the coefficients Am>s is as follows:

m's ml

for all TO > 1. It follows that for all m > 1

14 I < r m~(1 + 2) C7\|^ls,m| 2^ Cstll •* . ^ ( J

Inequality (7) can be obtained, using the Gauss formula for the gammafunction (this formula can be found in, e.g., [2]). Since

1 + iei2 _ 2 f 1 1-a2

1 i 9 1 ^ ! ? V 1 i 9 I ^ 19 / '

we get from (6) that

oo

Am<s(-l)m=l

Therefore,

Ya = an~s8 + a~s ) Am,s(-l}m(l - a2)m(G2m) i (8)m=l

where 8 denotes the delta-measure at 0. Next, using (7), (8), and theequality |(j2m||i = 1, we get that

as~nYa = 8 + Xa (9)

where Xa G L1 for all 0 < a < 1 and

sup H^H^oo. (10)0<a<l

Now (4), (5), (9), and (10) show that Lemma 1 holds.Let us continue the proof of Theorem 2.

(0 => («')•Let V G S' and suppose that part (i) in Theorem 1 holds. The followingequality can be easily checked:

1 1 l + a 2 £ 2 ' / 2 _ a - £ '

.

It is known that there exists a function $s G L1 such that

for all ^ e ^R" (see [21], p. 134). It follows from (11) and (12) that

J~SV -J-sV*a-n$sa-1- = asV*a-nGsa-1-

Since $s G i1, the I/^-norm of the expression on the right-hand sideof (13) can be majorized by a constant multiple of the expression

aa\\V*(a-nG,(a-1-moo = a'\\J-'(V)a\ 00.

Therefore, the left-hand side of (13) tends to 0 in L°° as a — > 0. Hence,the function J~SV belongs to the space BUC, since it can be approxi-mated in the L°°-norm by an approximation of the identity.(») => (0-Let V G S' and J~SV G BUC. It is easy to check that

2|£ |2W2_ S | t |» S

" ' S I J l € l + ________________________].(14)

There exists a function \PS G L1 such that

for all £ € Rn (see [21], p. 134). Now we get from (14) and (15) that

a'V*(a-nGs(cT1-) = tf * (J~SV - J~SV * (cT^cr1-)]

+ {a'[J-'Vir(a-nG,(a-1-)]}. (16)

Since J~SV e L03, Gs G Ll, and vps € I1, the last two terms on theright-hand side of (16) tend to 0 in Z/°° as a — > 0+. The first twoterms also tend to 0, since J~SV G BUG , and any function from thespace BUG can be approximated in L°° by the approximations of theidentity. It follows from (16) that condition (i) in Theorem 2 holds.

This completes the proof of Theorem 2.Proof of Theorem 1. The equivalence of parts (n) and (in) in Theorem1 follows from Theorem 2. Our next goal is to prove the equivalence ofparts (i) and (n).

It is known that the local behavior of the Bessel potential kernel andthe corresponding Riesz potential kernel is the same for 0 < a < n. Itis also known that the Bessel potential kernels decay exponentially atinfinity. More exactly, the following estimates hold: If 0 < s < n, thenthere exist cs > 0 and cs > 0 such that

c,x\'~n <Gs(x)<cs\x\s-\ (17)

for all x with 0 < \x\ < 1. Moreover, there exist cn > 0 and cn > 0such that

cn In — < Gs(x) < cn In — (18)X X

for all 0 < \x < 1/2. If s > n, then there exist ds > 0 and ds > 0 suchthat

d, < Gs(x) < ds (19)for all 0 < x < 1. On the other hand, for every s > 0 we have

Ga(x) < bse'c^ (20)

for all x 6 Rn with \x\ > 1. In (20), bs > 0 is a constant, dependingonly on s, and c > 0 is an absolute constant (see [21], p. 132-133).

Remark 3 It is not difficult to prove, using (19), that for s > n weget Ks>n = L]oc<u. Here L}o^u stands for the space of functions / £ L}ocsuch that

sup / \f(y) dy < oo.x JB(x,l)

Using (17)-(19), we get the following estimate:

XB(o,a)9, < asas-n(Gs)±, 0 < a < 1, (21)

for 5 — n 7^ 0, 2, 4, • • • . Moreover, for s — n = 0, 2, 4, • • • we have

«*)flr, < as(Gs)i, 0 < a < l/\/2. (22)

Now the implication (n) => (z) in Theorem 1 follows from (21) and(22).

Let 0 < s < n and V G A'S)n. In the remaining part of the proof ofTheorem 1 we will denote by cs positive constants, depending only on5, which may vary from line to line. We have from (17), (18), and (20)that

G, < cs(gsXB(o,i) + r) (23)

where r(x) = e~'x ' . If 0 < s < n, then for every 0 < a < 1 we have

«s( V\)a *GS< cs(as(\V\)a * (xB(o,i)9,) + s(\V\)a * r).

It follows from the definition of gs that

as\$\V$a*Gs\\^<cs(\\\V\*(gsXB(0,a)}\\™ + 11(^1)**^. (24)

Let C denote the cube in Rn given by

C = {x = ( x i j - . - j X n ) e Rn : 0 < xt < 1, 1 < i < n},

and let Ck = C + k for all k £ Zn. Then

a ' l K I ^ D a ^ r H o o < assup{x

< csas-nsup ! \ V ( y } \ d y ^ sup r ( y ) ( 2 5 }x J\x-y\<a keZ"yeCk

Since V G Ks,rn the last expression in (25) tends to 0 as a. — > 0+. Now(25) implies the validity of condition ( i i ) in Theorem 1 for 0 < s < n.

If s = n, then (23) gives

(26)

Reasoning as in the proof of (25), we see that the last term in (26) canbe estimated by

Msup / \ V ( y ) \ d y . (27)x J\x — y\<ct

Since V 6 Kn,n, the expression in (27) tends to 0 as a — > 0+. Now (26)implies condition ( i i ) in Theorem 1 in the case s = n.

If s > n, then KS:H = L]ocu, and we have

Now reasoning as in (25), we get condition ( i i ) in Theorem 1 for s > n.This completes the proof of Theorem 1 .The next theorem is one of the main results in the present paper. It

gives a complete description of the classes K3>n in terms of the Besselpotentials in the case s = 2m where ra is a natural number.

Theorem 3 Let V G S' and let s = 2m where m is a natural number.Then condition

V € KZm,nis equivalent to conditions (i) and ( i i ) in Theorem 2 for s = 2m.

Remark 4 Theorem 3 is a generalization of Theorem 1 for the Katoclasses of distributions. However, we do not know whether Theorem 3holds for s ^ 2m.

Proof of Theorem 3. We will show that condition V G K^m^ impliescondition (ii) in Theorem 2, and that condition (i) in Theorem 2 impliescondition V e K2m,n.

Let V G K-im,n- Then for every A (E A we have

V*(Xg2m)£L™. (28)

Recall that we denoted by r2m the constant for which (— A)m(r2m(?2m) =8. We have

V* (Xgim) = V* G2m * (/ - A)m(Xg2m)m-\

= E bjV*((-&yG2n)*(Xg2m) + V*G2m*(-&r(\g2m) (29)j=o

where bj are the binomial coefficients. Equality (29) holds in the senseof distributions. Using (12), it is not difficult to prove that for every0 < j < m — I the function (— A)-JG<2m is in Ll. Hence, we have

m — 1

h= XX-AX'G^eL1. (30)3=0

Since the function A is equal to 1 in a neighborhood of 0, we have

) = r^6 + r (31)

where r G C£° is a function which is equal to zero near 0. It followsfrom (29), (30), and (31) that

r£j~*mV = V*(\g2m} - V*(Xg2m)*h- V*G2m*r. (32)

For any function r as above there exist functions AI and A2 suchthat

| A 2 | U G A

andr = A 1 - A 2 . (33)

Indeed, we may put AI = r + p + a and A2 = p + a, where p G C™is a nonnegative function which is equal to zero in a neighborhood of0 and for which T + p > 0. The function a G C* ° is nonnegative andsupported in a neighborhood of 0 where the functions p and r are equalto 0.

It follows from (33) and from the definition of the class K2m^n thatfor every function T G C^° which is equal to zero in a neighborhood of0, we have V * r G L00. Hence,

V * G 2 m * r € £ ~ (34)

and we get from (30), (32), (34), and from the definition of the classTS f Vi a t_£X2771 fl UIlCll

j-2my £oo_

Let a be such that 0 < a < 1 and let 2m — n ^ 0, 2,4, • • •. Then(32) implies that for every A G A there exists a function ra G C£°,depending on A, equal to zero near 0, and such that

r^J-2mV + (J-2mV)*Ta = V*((X)Lg2m)-V*((X)Lg2m)*h. (36)Q a

Since h G Ll, it follows from the definition of the class K2m,n that theexpression on the right-hand side of (36) tends to 0 in L°° as a —> 0+.Now we get from (35) that J~2mV * ra G BUC for every 0 < a < I.Since the space BUC is a closed subspace of L°°, we get from (36) that

This proves that for 2m — n ^ 0, 2, 4, • • • condition V G Kim,n impliescondition (zz) in Theorem 2.

Let 2m — n = 0, 2, 4, • • • . Then

m,a} -V*

where /i G L1 is given by (30), and pa G C£° is a function, dependingon A and satisfying pa(x) = 0 for all x in a neighborhood of 0. Next,reasoning as in the previous part of the proof, we see that conditionV £ Kim,n implies condition (zz) in Theorem 2 for 2m — n = 0, 2, 4, • • •

Now suppose that condition (i) in Theorem 2 holds. Let 2m — n ^0, 2, 4, • • • Then we have

(37)

where Y = (I - A)m(\g2m)- It follows that

m-l

Y = r^8 + Y: b,(-&y(\g,m} + r (38)3=0

where T G C£°. Since

m-l

Y: 6,-(-A)J'(A^m) G L\j=0

we have from (37), (38), and from condition (z) in Theorem 2 that

Therefore, we have V G K-2m,n-If 2m -n = 0,2,4, • • - , then

and the proof proceeds exactly as in the case 2m — n ^ 0 , 2 , 4 , • • •This completes the proof of Theorem 3.

3 The BMO-classesIn this section we introduce the BMO-scale of distributions on Rn.First we give well-known definitions of the spaces BMO and VMO.


Let V G L}oc. Then it is said that V belongs to the space BMO ifffor all balls B(x,r) in Rn

\f(y) - fB(*,r)\dy < M, (39),m(B(x,r)) JB(X,T)

where the constant M > 0 does not depend on x £ Rn and r > 0 andwhere

,T) = ,p, —— 77 / f(u)du' m(B(x,r)) JB(x,r)denotes the mean value of the function V over the ball B(x,r}. Thefunctions in BMO are defined up to an additive constant. The normof a function V G BMO is given by

where the infimum is taken with respect to all constants M satisfying(39). It is clear that L°° C BMO. The opposite inclusion does nothold. Various examples of unbounded functions in BMO can be foundin [22], p. 141. It is also true that BMO C S'.

A function V G BMO belongs to the space VMO iff1 r

lim sup ——r- / \f(y) - fB(x,r)\dy = 0.r^°+xeRn m(B(x,r)) JB(x,r)The space VMO is a closed subspace of the space BMO. It is knownthat

V eVMO<=>lim\\V-V(--h)\\BMO = 0 (40)h—>0

(see [17], Theorem 1). It is also known that BUC C VMO. We referthe reader to [22] for more information concerning the spaces BMOand VMO.

In the next definition we introduce the BMO-analogues of theclasses Ks<n.

Definition 3 Let Let s > 0 be such that s — n / 0,2,4, • • •. Let V G S'.Then we will say that V G BStH iff for every a with 0 < a < 1 and A G Athe distribution y*((A)i_<7s) coincides with a function from BMO, andwe have

lim ||y * (^S(^)J-)||BMO = 0.a—>0+ a

In the case s — n = 0, 2,4, • • •, we say that V G Bs,n iff for every a with0 < a < 1 and every A G A the distribution V * ( (A) i_g S j a ) coincideswith a function from BMO, and moreover,

lima-+0+

The next theorem is a _E?M(9-version of Theorem 2.

Theorem 4 Let V 6 S' and s > 0. Then the following are equivalent:(i) J~SV e BMO and

lima—>0+

(ii) J~SV e VMO.

Proof of Theorem 4- The next lemma is similar to Lemma 2. The onlydifference in the proof is that we should use the jBMO-norm instead ofthe L°°-norm.

Lemma 2 Let V € S', s > 0, and J~SV € BMO. Then for every awith 0 < a < 1 we have J~s(V)a € BMO. Moreover,

sup as\\J-s(V}a\\BMo<cs\\J-sV BMO.a:Q<ct<l

Let us continue the proof of Theorem 4. Suppose / (E BMO andg G Ll. In general, the convolution f -kg is not defined as an absolutelyconvergent integral. However, the operator / —> f*g is defined on thespace BMO as the adjoint to the corresponding convolution operatoron Hl. Moreover, we have

\\f*9\\BMO< \\f\\BMO\\9\\l. (41)

Let V G 5" be such that J~3V G 5M0 and condition (i) in Theo-rem 4 holds. Using the previous remark, concerning the convolution ofBMO and Ll functions, we see that equality (13) can be considered asan equality for the bounded linear functionals on Hl. Next, using (13)and reasoning as in the proof of the implication («') => ( i i ) in Theorem2 with BMO instead of L°°, we get

lim ||J-y-J-4y*(a-n$,(a-1-)||BMO = 0 (42)a—>0+

where $s is defined by (12). By (40), the function ha = J~SV *(a~n$s(a^1-)) belongs to the space VMO for every 0 < a < 1. Sincethe space VMO is a closed subspace of the space BMO, (42) impliesJ~SV € VMO. This proves the implication (i) =$> (ii) in Theorem 4.

Let V e 5" and J~SV e VMO. Using (41), we see that the last twoterms on the right-hand side of (16) tend to zero in BMO as a —> 0+.Next we will prove that the first two terms also tend to zero. The proofof this fact is more complicated than the proof of the corresponding

fact in Theorem 2. Here we need to show that the convolution of thefunction J~SV with the function a~n$s(a~1-) is defined as an absolutelyconvergent integral.

We have\J-V(x)\

X n+8 dx < oo. (43)

Inequality (43) with 8 = I follows from (2) on p. 141 in [22]. Thecase 8 > 0 can be obtained similarly. Now we get from (43) that thefunction J~SV * g exists as an absolutely convergent integral for anyfunction g € i-1, satisfying the estimate

<

for all \x\ > 1 and some 8 > 0.Next we will show that the function $s satisfies condition (44). The

following formula holds (see [21]):CO

$ (x) = — 5^ AmaG2m(x). (45)^ \ / £__J / / t j O 6111, \ / \ /

m=l

Using formula (3) for the Bessel potential kernel G^mi we obtain thatfor \x\ > I and 0 < 8 < 1,

t

2 e 2 —>- j it>o

Now using Stirling's formula (see [5]) for the gamma function, we getp/2m+5\

V 2 / ^ - / ^ T \T——^—— < cm2 (47)(m- 1)! ~ v ;

for all m > m0. In (47), the constant c does not depend on 8 and m,and the constant mo does not depend on 8. It follows from (46) and(47) that there exists a constant cn>s such that

" C|-(n+5' (48)

for all m > 1 and x > 1. Next (7), (45), and (48) show that thefunction $s satisfies condition (44). Hence, the function given by

Rn a

is locally integrable. Therefore,

BMO<

Since

we have

/JRn

J-sV(x) - J-sV*(a-n$s(a-l-))(x)

= I [J-sV(x) - J~sV(x - y}]a-n<$>s(y-}dy. (49)JRn a

Using the definition of the space BMO and (49), we get

\\J-V -J-'V*(a-n<S>3(a-l-))\\BMoV(-) - J-SV(- - y)\\BMody. (50)

Since J~SV €. VMO and

l ima" / |*,(a^0+ J\y\>e

for every e > 0, we get from (40) and (50) that

lim \\J-'V-J-°V*(a-n$a(a-1-)) BMO = 0.a— >0+

It follows that the first two terms on the right-hand side of (16) tendto zero in BMO as a — s- 0+. This shows that the left-hand side of(16) tends to zero in BMO as a — > 0+. This proves the implication( i i ) =/> (i) in Theorem 4.

The proof of Theorem 4 is thus completed.The next theorem is the 5MO-version of Theorem 3. We do not

know if Theorem 5 holds in the case s ^ 1m.

Theorem 5 Let V € S' and let m be a natural number. Then condition

V e B2m,nis equivalent to conditions (i) and (ii) in Theorem 3 for s = 2m.

Proof of Theorem 5. We will show that condition V G B-2m,n impliescondition (n) in Theorem 3, and that condition (i) in Theorem 3 impliescondition V G -f^m.n- The proof will be similar to that of Theorem 3.

Let V G B2m,n- Then for every A G A we have

Using (32) and the definition of the class -B2m,jra, we get that for everyfunction T G C^ which is equal to zero near 0, we have V *T G BMO.Hence, V*G2m*T G BMO, and (32) gives

Next we will prove that the expression on the right-hand side of(36) tends to 0 in BMO as a -> 0+. Since (51) implies that J"m-kra GVMO, and we know that V MO is a closed subspace of BMO, weobtain from (36) that J~2mV G VMO.

Now suppose that condition (i) in Theorem 3 holds. Let 2m — n 7^0, 2, 4, • • •. Then we have

| |V*((A)i02m)| |BMO = a n | (V) a *((0 2 m )

\(V)a*Y\\BMO= a2m

where Y = (I—A)m(Ag2m)- Then, reasoning as in the proof of Theorem3, we get that

For 2m — n = 0, 2, 4, • • • the proof is similar.

References[1] M. Aizenman and B. Simon, Brownian motion and Harnack's in-

equality for Schrodinger operators, Comm. Pure Appl. Math. 35(1982), 209-271.

[2] J. B. Conway, Functions of One Complex Variable, Springer, NewYork, 1978.

[3] E. B. Davies and A. Hinz, Kato class potentials for higher orderelliptic operators, J. London. Math. Soc. (2) 58 (1998), 669-678.

[4] M. Demuth and J. A. van Casteren, Stochastic Spectral Theory forSelfadjoint Feller Operators: A functional integration approach,Birkhauser Verlag, Basel, 2000.


[5] M. D. Greenberg, Foundations of Applied Mathematics, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1978.

[6] A. Gulisashvili, Sharp estimates in smoothing theorems forSchrodinger semigroups, J. Functional Analysis 170 (2000), 161-187.

[7] A. Gulisashvili, On the heat equation with a time-dependent sin-gular potential, submitted for publication.

[8] A. Gulisashvili and M. A. Kon, Smoothness of Schrodinger semi-groups and eigenfunctions, Int. Math. Res. Notices 5 (1994), 193-199.

[9] A. Gulisashvili and M. A. Kon, Exact smoothing properties ofSchrodinger semigroups, Amer. J. Math. 118 (1996), 1215-1248.

[10] J. W. Johnson and M. L. Lapidus, The Feynman Integral andFeynman's Operational Calculus, Oxford University Press, Oxford,2000.

[11] T. Kato, Schrodinger operators with singular potentials, Israel J.Math. 13 (1973), 135-148.

[12] V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin, 1985.

[13] Qi Zhang, On a parabolic equation with a singular lower orderterm, Transactions Amer. Math. Soc. 348 (1996), 2811-2844.

[14] Qi Zhang, On a parabolic equation with a singular lower orderterm, Part 2: The Gaussian bounds, Indiana Univ. Math. J. 46(1997), 989-1020.

[15] F. Rabiger, A. Rhandi, R. Schnaubelt, and J. Voigt, Non-autonomous Miyadera perturbation, Differential Integral Equa-tions 13 (2000), 341-368.

[16] S. G. Samko, A. A. Kilbas, and 0.1. Marichev, Fractional Integralsand Derivatives: Theorey and Applications, Gordon and BreachScience Publishers, Amsterdam, 1993.

[17] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer.Math. Soc. 207 (1975), 391-405.

[18] M. Schechter, Spectra of Partial Differential Equations, North Hol-land, Amsterdam, 1986.


[19] R. Schnaubelt and J. Voigt, The non-autonomous Kato class,Arch. Math. 72 (1999), 454-460.

[20] B. Simon, Schrodinger semigroups, Bull. Amer. Math. Soc. 7(1982), 445-526.

[21] E. M. Stein, Singular Integrals and Differentiability Properties ofFunctions, Princeton University Press, Princeton, NJ, 1970.

[22] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Or-thogonality, and Oscillatory Integrals, Princeton University Press,Princeton, 1993.

[23] F. Stummel, Singulare elliptische differentialoperatoren inHilbertschen Raumen, Math. Ann. 132 (1956), 150-176.


The Global Solution Set for a Class ofSemilinear Problems

Philip KormanInstitute for Dynamics and

Department of Mathematical SciencesUniversity of Cincinnati

Cincinnati Ohio 45221-0025

AbstractFor a class of semilinear Dirichlet problems we present an exact

multiplicity result. Our proof simplifies the previous one in T. Ouyangand J. Shi [11]. By an indirect argument we sidestep the necessity ofproving positivity for linearized equation, which was the most difficultstep in [11], as well as in the earlier paper of P. Korman, Y. Li and T.Ouyang [6].

1 Introduction

We consider a class of semilinear Dirichlet problems

(1.1) Au + A/O) = 0 for |z < .R, u = Q for \x = R,

on a ball of radius R in Rn. Here A is a positive parameter, and the non-linearity f ( u ) is assumed to generalize a model case f ( u ) == u(u — b)(c — u),with positive constants b and c, and c > 26 (in case c < 26 the problem (1.1)has no nontrivial solutions, see e.g., [6]).

We now list our assumptions on the nonlinearity /(«). We assume thatf ( u ) e C2(R+), and it has the following properties

(1.2) /(O) = f(b) = f ( c ) = 0 for some constants 0 < b < c,

(1.3) f ( u ) < 0 for u G (0, 6) U (c, oo),f ( u ] > 0 for u e (-oo,0) U (6,c),

(1.4)

(1.5) There exists an a € (0. c), such thatf"(u) > 0 for u € (0, a) and f"(u) < 0 for u € (a, c).

We define 0 to be the smallest positive number, such that J0 /(s) ds = 0.Clearly, 9 € (b, c). After T. Ouyang and J. Shi [11], we set p = a - j^ (i.e.p is the first Newton iterate when solving f ( u ) = 0 with the initial guess a).

uf'(u}We define K(u) = Our final assumption is the following. If 9 K(0) on (M)K(u) is noriincreasing on (8, p)

K(u) < K(p) on ( p , a ) .

(If 0 > p this assumption is empty.)We are now ready to state the main result.

Theorem 1.1 Assume that f ( u ) satisfies the conditions listed above. Forthe problem (1.1) there is a critical XQ > 0 such that the problem (1.1) hasexactly 0, 1 or 2 nontrivial solutions, depending on whether A < XQ, A = AQor A > AQ. Moreover, all solutions lie on a single smooth solution curve,which for A > AQ has two branches denoted by 0 < u~(r,X) < u+(r,X),with u+ (r, A) strictly monotone increasing in A and limA— *oo u+ ( r> A) = cfor r € [0, 1). For the lower branch lirn\-»oo w~ (r, A) — 0 for r ^ 0, whileu~ (0, A) > b for all A > A0.

In present generality this theorem was proved first by T. Ouyang andJ. Shi [11]. In two dimensions (with some extra assumptions on /(n)) thistheorem was proved in P. Korman, Y. Li and T. Ouyang [6], where thegeneral scheme for proving such results was developed. One of the crucialthings in that approach was proving positivity of any non-trivial solution ofthe linearized problem

(1.7) &w + \f'(u)w = 0 for x < R, w = 0 for x\ = R.

This turned out to be a difficult task, and it was the only reason the paper[6] was restricted to two dimensions. Later, T. Ouyang and J. Shi [11] wereable to prove that w(r) > 0 by using Pohozhaev type identity. Their proof israther involved. We also mention that one-dimensional version of this resultwas proved in P. Korrnan, Y. Li and T. Ouyang [7], where more generalnonlinearities of the type f ( u ) = (u — a)(u — b)(c — u) were considered,and where references to earlier work in case n = 1 by J. Smoller and A.Wasserman, and S.-H. Wang can be found.

In this work by using an indirect argument, we are able to avoid havingto prove that w(r) > 0, which considerably simplifies the proof, and makes itmore transparent. We show that it suffices to prove that w(r) cannot vanishexactly once. We show that our assumptions on f ( u ) make the function

t(yt behave almost the same way as in the important paper of M.K. Kwongand L. Zhang [8], and then our proof that w(r] cannot vanish exactly onceis similar to Lemma 8 of [8].

We outline our arguments next. It is known that for large A our problem(1.1) has a positive solution. When continued for increasing A this solution,after possibly some turns, has to tend to c as A —> oo. When continued fordecreasing A this solution has to turn, since no positive solutions exist for A >0 small. The lower end of our solution curve, after possibly some turns, hasto tend to 0 as A —> oo. If one assumes that w(r) > 0 at any one of the turns,we show that the result follows. It is important on this step that w(r~) cannotvanish exactly once. It then remains to consider the case when conditionw(r) > 0 is violated at all turning points. Assume for simplicity there isonly one turning point on the solution curve. Since condition w(r) > 0 isviolated, it follows by the Crandall-Rabinowitz bifurcation theorem (whichis recalled below) that the lower and upper solution branches intersect nearthe turning point. By uniqueness for initial-value problem these brancheswould have to intersect for all A. But the upper branch tends to c, whilethe lower one tends to zero, and hence they have to separate eventually, acontradiction. In case of more than one turning point, the argument is moreinvolved, although the idea is similar.

Next we state a bifurcation theorem of Crandall-Rabinowitz [1].

Theorem 1.2 [1] Let X andY be Banach spaces. Let(\,~x) e RxX and letF be a continuously differentiate mapping of an open neighborhood of (A, x)into Y. Let the null-space N ( F x ( X , x ) ) — span {XQ} be one-dimensional andcodim R(Fx(\,x)) = 1. Let F\(X,x) £ R(Fx(\,x)). If Z is a complement of


span {.TO} in X, then the solutions of F(\,x) = F ( X , x ) near (A, x) form acurve (A(s),x(s)) — (\ + T(S),~X + SXQ + Z(S)), where s —> (T(S),Z(S}) 6 Rx Zis a continuously differentiate function near s = 0 and r(0) — r'(0) — 0,z(0) = z'(0) = 0.

Throughout the paper we consider only the classical solutions of (1.1).Without loss of generality we set R = 1. Also notice that by the maximumprinciple all non-trivial solutions of (1.1) are positive.

2 Preliminary results

We list some consequences of our conditions on f(u). We define /3 > 0 to

be the unique number where /'(/?) = ——. Clearly, f3 € (0,7), where 7 is

the larger root of f'(u) — 0. The following lemma was proved in [6].

Lemma 2.1 We have

(2.D «n«)- /(«)(>° ' fr uej°/]v ; J v ; ^ M < 0 for u € (/?,c).

Lemma 2.2 #(0) = 1, K(u] < 1 on (0,6).

Proof: The first statement follows by L'Hospital rule. Notice next thatfor u f ( u ) .It follows that K(u) < 1.

Lemma 2.3 K'(u) < 0 on ( a , / 3 ) .

Proof: Compute

K\U) = H£ /2The first term in the numerator is negative for u > a, and the second oneis negative by Lemma 2.1 (notice that f'(u) > 0 on (a, f3}}.

Lemma 2.4 K(u) < I on ((3,c).

Proof: On (/?, c) we have /(«) > 0, and f'(u}u < f ( u ) by Lemma 2.1,and the proof follows.

Lemma 2.5 If 0 1.

Proof: By the definition of p

Using this and our last condition in (1.6),

proving the lemma.

The lemmas above imply the following result.

Theorem 2.1 Assume 0 < p. For any UQ € ( 0 , p ) we define 7 = K(UQ).Then 7 > 1, and

(2.2) « / / ( « ) - nv ' ^ v ' IJ v ; | < 0 for u o , .

Proof: The above lemmas imply that the horizontal line y = 7 intersectsthe graph of y = K(u) exactly once, and the graph of K(u) lies above theline y = 7 in the region where /(«) > 0. This proves the first inequality in(2.2), arid second one follows similarly.

We study multiplicity of positive solutions of the Dirichlet problem, de-pending on a positive parameter A

(2.3) An + \f(u) = 0 for \x\ < 1, u = 0 on \x\ = 1,

with nonlinearity f(u) satisfying all of our assumptions. By the classicaltheorem of B. Gidas, W.-M. Ni and L. Nirenberg [3] positive solutions of(2.3) are radially symmetric, which reduces (2.3) to

(2.4) u" + ^— -u' + \f(u) = 0 for 0 < r < 1, u'(0) = u(l) = 0.r

We shall also need the corresponding linearized equation

(2.5) w" + —— -w' + Xf'(u)w = 0 for 0 < r < I , w'(0) = w(l) - 0.

The following lemma was proved in [4].

Lemma 2.6 Assume that the function f(u) 6 C2(R+), and the problem(2.5) has a nontrivial solution w at some A. Then

(2.6)

We recall that solution of (2.4) is called singular provided the corre-sponding linearized problem (2.5) has a nontrivial solution. The followinglemma follows immediately from the equations (2.4) and (2.5).

Lemma 2.7 Let (\,u) be a singular solution of (2.4)- Then

/-i(2.7) / (/(«) - /») wrn-1 dr = 0.

Jo

The following lemma is a consequence of the previous two.

Lemma 2.8 Let (A, u) be a singular solution of (2.4)- Then for any real 7

(2.8) / (7/(«) - /'(«)«) wrn~l dr = i/(l)w/(l).Jo 2A

Proof. Multiplying (2.6) by 7 - 1, and adding (2.7), we obtain (2.8).

The following lemma is known, see e.g. E.N. Dancer [2]. We present itsproof for completeness.

Lemma 2.9 Positive solutions of the problem (2-4) are globally parameter-ized by their maximum values n(0, A). I.e., for every p > 0 there is at mostone A > 0, for which u(Q, A) = p.

Proof. If u(r, A) is a solution of (2.4) with u(0, A) = p, then v = u(-k-r)v A

solves(2.9) v" + ——-v' + f ( v ) = 0, u(0) = p, t/(0) = 0.rIf u(0,fj.) = p for some fj, ^ A, then w,(-4=r) is another solution of the sameproblem. This is a contradiction, in view of the uniqueness of solutions forinitial value problems of the type (2.21), see [12].

The following lemma restricts the region where w(r), solution of thelinearized problem (2.5), may vanish. Its first part is due to T. Ouyang andJ. Shi [11], see also J. Wei [13], and its second part is due to M.K. Kwongand L. Zhang [8].


Lemma 2.10 Any nontrivial solution of (2.5) cannot vanish in either in-terval where 0 p it follows that any nontrivial solution of (2.5) is positive, andthe main result of the present paper then follows similarly to [6].

The following lemma follows the idea of Lemma 8 of M.K. Kwong andL. Zhang [8].

Lemma 2.11 Under our conditions on f ( u ) any non-trivial solution of(2.5) w(r] cannot have exactly one zero on (0,1).

Proof. Since w(0) ^ 0, see [12] for the appropriate uniqueness result (ifw(0) = u/(0) = 0 then w = 0), we may assume that u>(0) > 0. Assume thaton the contrary w(r) has exactly one root at some r = ro, i.e.

(2.10) w(r] > 0 on (0,r0), w(r] < 0 on (r0,l).

By Lemma 2.10 U(TQ) 6 ( 8 , p ) . Setting 7 = K(u(ro}}, we see by Theorem2.1 that/ o n \ ff \ t'i \ / < ° for all u<u(ro)(2.11) 'jf(u)-uf(u){ n c u ) \v ' u v ; J v ; 1 > 0 for all u > u(r0).

Since 7 > 1, we obtain by Lemma 2.8 (notice that w/(l) > 0 by (2.10))

r1

(2.12) / J7/(u) - «/'(«)] w(r}rn~l dr < 0.

In view of (2.10) and (2.11) the quantity on the left is positive, and we havea contradiction in (2.12).

Lemma 2.12 Let u(r, A) and v(r, A) be two solution curves of (2.4), whichare continuous in A, when the parameter A varies in some interval I. Assumethat for some AQ € / solutions u(r, AQ) and v(r,\o) intersect. Then u(r, A)and v(r, A) intersect for all A € I.

Proof: In order for the solution curves to separate, there must ex-ist AI (the last A at which they intersect) and a point r\ € [0,1] at whichu(r\, AI) = v(ri, AI) and ur(ri, AI) = vr(ri, AI). But this contradicts unique-ness for initial value problems.

Next we study the linearized eigenvalue problem corresponding to anysolution of (2.4):

(2.13V + —— -V + */'(«)¥> + W> = 0 on (0, 1), <p'(0) = <p(l) = 0.

Comparing this to (2.5), we see that at any singular solution of (2.4) fj, = 0is an eigenvalue, corresponding to an eigenfunction ip = w.

We shall need the following generalization of Lemma 2.6.

Lemma 2.13 Let if > 0 be a solution of (2.13) with /j, < 0. (I.e. tp is aprincipal eigenfunction of (2.13).) Then

(2.14) /V^V"-1^ > iu'(iy (i).Jo 2A

Proof. The function v = rur — ur(l) satisfies

(2.15) Au + A/» + p.v = f j . v - 2A/(u) - X f ' ( u ) u ' ( l ) for•u = 0 on a; = 1.

x

Comparing (2.15) with (2.13) we conclude by the Fredholm alternative

(2.16) fj, JQ v(prn~ldr - 2A J,,1 f(v)<prn-ldr -

Integrating (2.13)

rl rl-A / f'(u)(prn-ldr = </(!) + fj, / yrn~ldr.

JQ Jo

Using this in (2.16), we have

/•I /-I2A / f(u}(prn-ldr = n r

Jo Joand the proof follows.

We now define Morse index of any solution of (2.4) to be the number ofnegative eigenvalues of (2.13). The following lemma is based on K. Nagasakiand T. Suzuki [10].

Lemma 2.14 Assume that (A,?/) is a singular solution of (2.4) such thatK/(!) < 0 and

(2.17) / f"(u)w\n-ldr < 0.Jo

Then at (X,u) a turn to "the right" in (X,u) "plane" occurs, and as wefollow the solution curve in the direction of decreasing u(Q,X), the Morseindex is increased by one.

Proof. To see that the turn is to the right, we observe that the functionr(s), denned in Crandall-Rabinowitz theorem, satisfies r(0) = r'(0) = 0 and

X f1 f(2.18) r"(0) = -V^ ' -ftf(u)wrn~ldr

see [6] for more details. By our assumption the numerator in (2.18) isnegative, while by Lemma 2.2 the denominator is positive. It follows thatr"(0) > 0, and hence r(s) is positive for s close to 0, which means that theturn is to the right.

At a turning point one of the eigenvalues of (2.13) is zero. Assume it isthe f.-th one, and denote /j, = /ie. Here /i = fj,(s), and /z(0) = 0. We nowwrite (2.13) in the corresponding PDE form and differentiate this equationin s

(2.19)for x < 1, (ps = 0 on x\ = 1.

At (A, u) the Crandall-Rabinowitz theorem applies, and hence we have:|i(0) = 0, </?(0) = w, A'(0) = 0, and us(Q) — —w (considering the chosenparameterization). Here w is a solution of the linearized equation (2.5).The equation (2.19) becomes

(2.20) A</?s - A/'»2 + \f'(u)(ps + fjf(Q)w = 0.

Multiplying (2.5) by </?s, (2.20) by w, subtracting and integrating, we have

, AJin^VV -

It follows that across the turning point one of the positive eigenvalues crossesinto the negative region, increasing the Morse index by one.

Lemma 2.15 Assume that (\Q,UQ) is a singular solution of (2.4), i.e. theproblem (2.5) has a nontrivial solution w(r). Then

(2.21) w(r] > 0 for all r € [0,1)

if and only if for A close to AQ any two solutions on the solution curve passingthrough (Ao,?io) d° n°t intersect.

Proof: In view of Lemma 2.6 the Crandall-Rabinowitz theorem applies at(XQ,UQ) (see [6] for more details). According to that theorem near the point(Ac, «o) solutions differ asymptotically by a factor of w(r), which implies thelemma.

The following lemma was proved in [6], see also [11] and [13].

Lemma 2.16 Assume that (Ao, UQ] is a singular solution of (2.4), such that(2.21) holds. Then the inequality (2.17) holds, and the conclusions of theLemma 2.14 apply.

Next we study eigenvalues and eigenfunctions of radial solutions of Laplaceequation on a ball. Since singularity at r = 0 is introduced by the polarcoordinates, and is not present in the original equation, it is natural to ex-pect spectral properties similar to that of regular Sturm-Liouville problems.Surprisingly, we were not able to find any references.

Lemma 2.17 Consider an eigenvalue problem

(2.22) / + V + b(r)y + \y = 0, for 0 < r 1, and b(r} € C2[0,1]. Assume that A = 0 is aneigenvalue of (2.22), and let yo(r) be the corresponding eigenfunction. Thenthe problem (2.22) has an infinite sequence of eigenvalues \\ < A 2 < . . . ,with \n —> oo as n —> oo, and the n-th eigenfunction has precisely n — 1roots on (0,1) for all n > I . (One of \k 's is equal to zero.)

Proof: We convert the problem (2.22) into an integral equation, usingthe modified Green's function. We claim that any solution of the equation

(2.23) y" + V + b(r}y = 0,

that is bounded at r = 0 must be a multiple of the first eigenfunction yo(r).Indeed, writing the first two terms of the Taylor's series of the solutionwith a remainder term, we easily conclude that y'(0) = 0 for any boundedsolution. If we now fix a constant a so that y(0) = c«/o(0), then we shall havey(r] — ayo(r) for all r > 0, in view of uniqueness for initial value problemsof the type (2.23), see [12]. Let y2(r) be a solution of (2.23) with y2(l) = 1.Since y^ is not a multiple of yoi it follows that yz(r) —> oo as r —>• +0. Aformal use of Frobenius method at r — 0 shows that y2 ~ /3r~a+l as r —»• 0,with some constant j3. Setting y(r) = r~a+iz(r), we see that the resultingequation for z(r} has all solutions bounded near r = 0, which justifies theasymptotic formula for y2(r} near r — 0.

Notice that the problem (2.23) can be put into an equivalent self-adjointform(2.24) (ray'}' + rab(r}y = 0.

The modified Green's function for (2.24) subject to the boundary conditionsy'(0) = y(l) = 0 has the form

for r < £for r > £,

where K is a constant. By the above remarks we have, with some constantc> 0,

(2.26) |G(r,OI <<£l~a f o r r < £<crl~a fo r r > f.

We now multiply the equation (2.22) by ra, and convert it into an integralequation for the function z ( r ) = r^y(r]

(2.27)o

with the kernel G(r, £) = G(r,£)rz£*. Using (2.26) it is a standard exercise/•! /•!

to show that / / G2 drd^ < oo, see pages 178 and 421 in [14]. This meansJo Jo

that (2.27) is an integral equation with a compact and symmetric kernel.It follows that its spectrum is discrete, and eigenvalues tend to infinity.Moreover, we conclude that the minimum characterization of eigenvaluesapplies, from which it follows that the fc-th eigenfunction cannot have more

than k — 1 interior roots, see p. 173 in [14]. On the other hand, the sameminimum characterization implies that y\ is of one sign, and 7/2 must vanishat least once. Also, by Sturm's comparison theorem yk+i must have at leastone more interior root than y^. We then conclude that y^, then 2/3, and soon have the desired number of interior roots.

3 Proof of the main result

We are now ready to prove the Theorem 1.1. We begin by noticing thatexistence of positive solutions under our conditions follows by the Theorem1.5 in P.L. Lions [9], see also [11]. Indeed the result in [9] implies existence ofa critical A, so that for A > A there exists a maximal positive solution of (2.3),while for A > A there exists at least two positive solutions. Since positivesolutions of our problem (2.3) are radial, we consider its ODE version (2.4).We now continue the curve of maximal solutions for decreasing A. It wasshown in [6] that this curve cannot be continued for all A > 0, and hence acritical point (Ao,«o) must be reached, at which the curve will turn. By thedefinition of a critical point, the linearized equation (2.5) has a nontrivialsolution w(r). We claim that the theorem follows provided that

(3.1) w(r)>0 for all re [0,1).

By the Crandall-Rabinowitz Theorem near the turning point (Ag, UQ) thesolution set has two branches n~(r, A) < u+(r, A), for r 6 [0, 1), A > AQ. Bythe Crandall-Rabinowitz Theorem we also conclude

(3.2) 7i|(r> A) > ° for A close to AO (for a11 r e [°. !))•Arguing like in P. Korman, Y. Li and T. Ouyang [6], we show that the sameinequality holds for all A > AQ (until a possible turn), see also T. Ouyangand J. Shi [11] and J. Wei [13]. We claim next that solutions u+(r, A) arestable, i.e. all eigenvalues of (2.13) are positive. Indeed, let on the contraryH < 0 be the principal eigenvalue of (2.13), and </? > 0 the correspondingeigenvector. The equation for u\ is

(3.3) u'( + u'x + A/»A + /(«) = 0 for r 6 (0, 1],»/A (0)=7, ,A ( l )=0.

From the equations (2.13) and (3.3) we obtain

(3.4)o o

The right hand side in (3.4) is positive by our assumptions, inequality (3.2),and Lemma 2.13, while the quantity on the left is zero, a contradiction.

We show next that for A > AQ both branches u+(r, A) and ii~(r, A) haveno critical points. Indeed, if we had a critical point on the upper branchu+(r, A) at some A > AQ, then by the Crandall-Rabinowitz Theorem solutionof the linearized equation would be positive at A = A (since u\ > 0 as weenter the critical point). But then by Lemma 2.14 we know precisely thestructure of solution set near (A, u+(r, A)), namely it is a parabola-like curvewith a turn to the right. This is impossible, since the solution curve hasarrived at this point from the left. Turning to the lower branch u~(r, A), weknow by Lemma 2.14 that each solution on this branch has Morse index ofone, until a possible critical point. At the next possible turning point oneof the eigenvalues becomes zero, which means that the Morse index of theturning point is either zero or one. If Morse index is zero, it means that zerois a principal eigenvalue, and so solutions of the corresponding linearizedequation are of one sign, and then we obtain a contradiction the same wayas on the upper branch. If Morse index = 1, it means that zero is a secondeigenvalue, i.e. by Lemma 2.17 w(r] changes sign exactly once, but thatis impossible by Lemma 2.11. It follows that if condition (3.1) is satisfiedat the first turning point, then our Theorem 1.1. follows. But exactly thesame arguments show that having w(r) > 0 at any turning point will implyTheorem 1.1.

It remains to rule out the possibility that condition (3.1) fails at allturning points. By Lemma 2.15 this means that the branches u~(r,X) aridu+(r, A) intersect near any turning point (\Q,UQ). When we continue theupper branch u+(r, A) for increasing A, then, after possibly some more turns,u+(r, A) —> c as A —>• oo for all r 6 [0,1), see [6] for more details. Similarly,for the lower branch we have u~(r,X) —» 0 as A —»• oo for all r £ (0,1),after possibly some additional turns, see [6]. (Notice that u~(Q, A) > 9.) Itfollows that for A sufficiently large

(3.5) u~(r, A) < u+(r, A) for all r € [0,1).

We now pick the leftmost turning point on our curve (i.e. the turning-point with smallest A; if there is more than one such point, take any one ofthem). In Figure 1 this is the point A. By above, condition (3.1) is violatedat this point, and hence solution branches contain intersecting solutions nearA. As we increase A solutions on both branches continue to intersect byLemma 2.12, until a possible turning point. If both branches have no more

uCO)

A

c

Figure. 1. Solution, curve with several turning points.


turning points, solutions will intersect for all A, contradicting (3.5). Assumethat as we continue both branches u~ (r, A) and u~ (r, A) for increasing A thefirst turning point happens, say, at the upper branch at a point B. By B'we denote the point on the lower branch, which has the same A coordinateas B. By Lemma 2.12 solutions at B and B' intersect. We now continue theupper branch for decreasing A until the next turning point, which we callC. By C' we denote the point on the lower branch, which has the same Acoordinate as C. Moving leftwards on both branches, we conclude by Lemma2.12 that solutions at C and C' intersect. We denote by E the next turningpoint on the upper branch (if it exists), and by E1 the corresponding pointunder it on the lower branch. By moving to the right on both branches andusing Lemma 2.12 , we conclude that solutions at E and E1 intersect. Wecontinue the process until the upper branch passes over B for the last timeat a point B. We conclude that solutions at B' and B intersect. We nowresume moving forward in A on both lower and upper branches. If anotherturning point is encountered, we repeat the above procedure. We concludethat solutions on upper and lower branches corresponding to the same Aintersect for all A. This contradicts (3.5). We conclude that w(r] > 0 at anyturning point, and the theorem follows.

Remark. After completing the proof, we conclude that the solution curvehas exactly one turn, and w(r) > 0 there. This is simpler than the previousstrategy (of [6], [11] and [13]) of directly proving that w(r) > 0.

Acknowledgement. It is a pleasure to thank S. Aizicovici and N. Pavel fora very well organized and stimulating workshop, and Y. Li and T. Ouyangfor useful comments.

References

[1] M.G. Crandall and P.H. Rabinowitz, Bifurcation, perturbation of sim-ple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52,161-180 (1973).

[2] E.N. Dancer, On the structure of solutions of an equation in catalysistheory when a parameter is large, J. Differential Equations 37, 404-437(1980).

[3] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related propertiesvia the maximum principle, Commun. Math. Phys. 68, 209-243 (1979).


[4] P. Korman, Solution curves for semilinear equations on a ball, Proc.Amer. Math. Soc. 125(7), 1997-2006 (1997).

[5] P. Korman, Exact multiplicity of positive solutions for a class of semi-linear equations on a ball, Preprint.

[6] P. Korman, Y. Li and T. Ouyang, An exact multiplicity result for a classof semilinear equations, Commun. PDE. 22 (3&4), 661-684 (1997).

[7] P. Korman, Y. Li and T. Ouyang, Exact multiplicity results for bound-ary value problems with nonlinearities generalising cubic, Proc. RoyalSoc. Edinburgh 126A, 599-616 (1996).

[8] M.K. Kwong and L. Zhang, Uniqueness of the positive solution of/(u) = 0 in an annulus, Differential and Integral Equations 4, 582-599(1991).

[9] P.L. Lions, On the existence of positive solutions of semilinear ellipticequations, SIAM Review 24, 441-467 (1982).

[10] K. Nagasaki and T. Suzuki, Spectral and related properties about theEmden-Fowler equation —An = \eu on circular domains, Math. Ann299,1-15 (1994).

[11] T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a classof semilinear problems, J. Differential Equations 146, 121-156 (1998).

[12] L.A. Peletier and J. Serrin, Uniqueness of positive solutions of semilin-ear equations in Rn, Arch. Rat. Mech. Anal. 81, 181-197 (1983).

[13] J. Wei, Exact multiplicity for some nonlinear elliptic equations in balls,Proc. Amer. Math. Soc. 125, 3235-3242 (1997).

[14] H.F. Weinberger, A First Course in Partial Differential Equations, JohnWiley & Sons (1965).


Optimal Control and Algebraic Riccati Equations underSingular Estimates for eAtB in the Absence of

Analyticity. Parti: The Stable Case

Irena Lasiecka and Roberto Triggiani*Department of Mathematics

Kerchof HallUniversity of Virginia

Chai-lottesville, VA 22904

Abstract

We study the quadratic optimal control problem over an infinite time horizon in thecase where the free dynamics operator A and the control operator B yield a singularestimate for eAiB. Here, eAt is the corresponding s.c. semigroup which, by assumption,is not analytic. In this Part I, eAt is assumed (exponentially) stable. The resultingabstract model covers systems of coupled Partial Differential Equations, which possessan analytic component, but which are not themselves analytic. Two applications aregiven to hyperbolic/parabolic structural acoustic problems. Here a hyperbolic PDE(a wave equation within an acoustic chamber) is coupled with a parabolic PDE (theflexible wall which is either modeled by an elastic equation with structural damping[A-L.l] or else by a thermoelastic equation with no rotational inertia [L-T.4-6]).

1 Mathematical Setting and Formulation of the Con-trol Problem

Dynamical Model. Let U (control), Y (state) be separable Hilbert spaces. In this paper,we consider the following abstract state equation

y ( t ) = A y ( t ) + Bu(t) + w(t) on, say, [D(A')]'; j /(0) = y0 £ V, (1 .1)

subject to the following assumptions, to be maintained throughout the paper:

"Research partially supported by the Na t iona l Science Foundation under Grant DMS-9804056.


(H.I) A : Y D U(A) —> 1'' is the infinitesimal generator of a strongly continuous (s.c.)semigroup e''" on Y. Moreover, e.Ai is (exponentially) uniformly stable: that is, thereexist constants M > 1, u} > 0, such that

lk'U|ko-) < Me""', t>0. (1.2)

(H.2) B is a linear operator U = V(B) -> [D(A*)\', the dual space of the domain !>(/!"),with respect to the pivot space Y. Here A* is the adjoint of A in Y. Thus, eAt can beextended as a s.c. semigroup on [V(A*}\' as well.

(H.3) There exist constants 0 < 7 < 1 and T > 0, such that the following singular estimateholds true:

where (Bu,v)Y = (u,B*v)u, u£(J,v& V(B*) D V(A").

(H.4) The function w is a deterministic disturbance, satisfying

w € L 2 ( 0 , c o ; r ) (1.4)

to be kept fixed throughout.

Optimal Control Problem. With the dynamics (1.1), we associate the followingquadratic cost functional over an infinite time horizon:

(1.5)o

where y ( t ) = y ( t ; y o ) is the solution of (1.1) due to u(t), for fixed w, and, moreover,

(H.5)) , (1.6)

where Z is another Hilbert space. The corresponding Optimal Control Problem is:

For fixed lu as in (1.4), minimize Jw(u,y] over all u GL2(0, oo; f/), where y is the solution of (1.1) due to u (1-")(and w).

Remark 1.1. What makes the above optimal control problem different from those stud-ied in the literature [B-D-D-M], [L-T.l], [L-T.2], is, of course, the new set of assumptionsimposed on model (1.1); in particular, the presence of the singular estimate of hypothesis(H.3) = (1.3), while, however, the semigroup eAt is only assumed to be strongly continuous.Explicitly, eAt is not assumed to be analytic. In [L-T.l, Chapters 1, 2. 6], the singular es-timate (H.3) = (1-3) was also available in the treatment of those chapters; however, it was

so a-posteriori, as a consequence of the original assumption that the s.c. semigroup eAt be,moreover, analytic. Thus, wri t ing eAtB — ( — A)'1eAt( — /l)~7£? would at once yield estimate(1.3) under the two assumptions in those aforementioned chapters: (i) analyticity of thesemigroup of eAt on Y, and ( i i ) the property ( — A}~~*'B 6 L(U;Y). In the present paper,by contrast, the s.c. semigroup eAt is not assumed to be analytic. The above set of assump-tions on (1.1) are motivated by the structural acoustic problem, with hyperbolic/paraboliccoupling, where these assumptions are, in fact, properties of the coupled dynamics: see theexamples in Section 4 below, with both 7 < | (the easier case: see (3.1.16) below) and| < 7 < 1 (the more challenging case). Similarly, in line with the structural acoustic prob-lem, we are taken a pure distributed L2-disturbance w (that is G = Identity in the notationof [L-T.2, Eqn. (6.1.1.1) of Chapter 6]. D

Lemma 1.1. Assume (H.I) , (H.2), (H.3). Then, for any 0 < WQ < w, there exists aconstant k > 0 (depending on M, cj, w, w0, T, 7, and computed below), such that

y £ / < A: , V t > 0; 0 < w0 < w. (1.8)

Proof. Let t > T, the latter being the constant defined in (1.3). By using (1.3) and(1.2), we obtain the following estimate in the operator norm:

* , V t > T, (1.9)v, n

since f i t ) = (^) e-(w-^o)' jlas {^s maximum at t = t = f/(w — WQ) given byujo^e"1 over [0,oo]. For all t > T, we then have: f ( t ) < f ( t ) if i> T, and /(*) < /(T) iff < T. This identifies the constant k. n

Preliminaries. The solution to problem (1.1) is given by

y ( t ) = eAty0 + (Lu)(t) + ( W w ) ( t ) ; (1.10)

/

teA(i~T} Bu(r)dT ( l . l l a )

: continuous L2(0,oo;[7) -^ L2(0,oo;Y) ( l . l lb )

: continuous C([0, oo]; (/) -> C'([0, oo]; F); ( l . l lc)

/

(e-4( '-T)w;(r)£/r (L12a)

.: continuous L2(0,oo;Y) -> L 2 (0 ,oo ; y)n C([0, cxs' ; V). ( l -12b)

The regula.rity properties noted in ( l . i lb-c) , (1.12b). due to the Young's inequality [S.I]and (H.3), will be formalized in Proposition 3.1.2 below. The L2-adjoints of L and W, are,respectively,

(L*f)(t) = I B * e A ' ( T - t ] f ( T ) d T (L13a)

: continuous L2(0,oo;Y) -> L2(0,oc;U) (1.13b)

: continuous (7([0, oo]; Y) -> C([0, oo]; U); (1.13c)

(W'v)(i] = j eA'(T-l)v(T)dT (l.lia.)J t: continuous L2(0,oo ;y) -* L2(Q,oo:Y) (1.14b)

: continuous C([0,oo};Y) -> C([0, oo]; K). (1.14c)

2 Statement of Main ResultsThe main result of this paper is the following theorem.

Theorem 2.1. Assume (H.I) , (H.2), (H.3) = (1.3), (H.4) = (1.4), (H.5) = (1.6). Then:(al) For each y0 g Y and w g £2(0, oo; K) fixed, there exists a unique optimal pair

{ M ™(*; yo),y^(t', y o ) } of the optimal control problem (1.5), (1.7) for the dynamics (1.1), or(1.10), which satisfies the following properties:

« ° ( - ; y o ) 6 £2(0,00; £7) n <?([(), oo];tO; (2.1)

y°w( • ; yo) e £2(o, oo; r) n C([o, oo]; F). (2.2)(a2) The operator $(i) e £(V) defined by

= y°w=0(t; y0) e C*([0, oo]; V) n £2(0, oo; F), yo e K, (2.3)

describes a s.c. semigroup on K, which, moreover, is (exponentially) uniformly stable on Y.(a3) The bounded operator P £ £(V) defined on Y by

, ,T e Y (2.4)

is non-negative, self-adjoint on Y : P = P* > 0,

^x^u^^x^u}^. (2.5)o

(a4) The gain operator B'P is bounded Y —* U:

U}. (2.6)

(a,5) The infinitesimal generator of the s.c. semigroup $( /) in (a'2) is the operator Ap =,4 — BB'P, with maximal domain T>(Ap) on Y,

AP = A - BB'P, -D(Ap) = {x € Y : [I - A"1 B(B~ P)}x e T>(A)}, (2.7a)

so that the s.c. uniformly stable semigroup $(/) may be also denoted as

$(*) = eApt = e(A-BB'p)t, t > 0; \eApt c(y) < MPe^"\ t > 0 (2.7b)

for some constants Mp > 1, wp > 0.(a6) The following singular estimate holds true: for any 0 < Wi < up (where LOP is defined

in (2.7b)), there exists a constant kp (depending on Mp, wp, 7, T, u>\) such that

C(U,Y) t/;r) < kP - , V f > 0.

(a?) The operator P possesses the following additional regularity properties:

P e C(V(AP); T>(A')), or A'P € C(D(AP); Y); (2.9)

P 6 £(V(A);T>(A"P)), or XpP € £(P(/l); K). (2.10)

(a8) The operator P in (2.4) satisfies the following Algebraic Riccati Equation,

(A'Px, z)Y + (PAx, z)Y + (Rx, Rz)z = (B'Px, B-PZ)V

either V x , ? y e P(/l); or V;c , j / € I>(/lp). (2.11)

(a9) Moreover, the operator P in (2.4) is the unique operator satisfying the ARE (2.11)within the class of non-negative, self-adjoint operators P = P" E C(Y) such that B'P G£(}'"; U) [a property enjoyed by P by (a4)].

(alO) The optimal pair {u^(<; j/o),?/2/(i; yo)} satisfies the following feedback relation:

t ; y 0 ) + r w ( t ) ] (2.12)

= -B*Py%,(t;y0)-B'rw(t)£L,(0,o0;U)nC([Q,oo];U). (2.13)

B"Py°w( • :,y0) and ^-^(O & 7,2(0, oo; f / ) n G'([0, oo]; f / ) . (2.14)

Here, r w ( f ) is defined by

rw(0 = Pw(t- 2/o = 0)- Pj/° (f; y0 = 0) e L2(0, oo; Y) n C'([0, oo]; Y), (2.15)

with

/^(< ; l / 0 )= / e- 4 ' ( T -" / r J Ry°(r ; ? ; / 0 )^ -6L 2 (0 .oo :V)n6 < ( [0 ,oo] ;K) , j,0 € K (2.16)

( a l l ) Moreover, r w ( t ) is likewise given by

/•CO /-XJ

r w ( t ) = I eA'r(T-t]Pw(r)dT, so that B'r^t) = I B'eA'r(T~t]PW(T)(IT. (2.17)Jt J t

Thus pw(t'. fjo) is the unique solution of the problem

P w ( t ; y o ) = -A'pw(t-y0) - R*Ry^(t;y0), t> 0, y0 G Y; (2.1Sa)

lim pw(T;y0) = 0, (2.1Sb)T—oo v y

while /"„(<) is the unique solution of the problem

rw(t) = -A'Prw(t) - Pw(t), t > 0, in [D(A)}'; (2.19a)

T1™ j

r^(T)=:0 ' (2.19b)

(a!2) Finally, the optimal dynamics may be rewritten as

>Jw(t;yo) =

o Jo

eL2(Q,oo;Y)nC([Q,oo];Y). O

Additional results are given in the sections below.

3 Proof of Theorem 2.1

3.1 Existence of a Unique Optimal Pair, Characterization, andRegularity Properties: Proof of (al)

Proof of (al). First, as already noted in ( l . l lb) and (1.12b), assumptions (H.I) , (H.2),(H.3) guarantee the regularity properties Lu € £3(0,00;}') and Ww € £2(0, oo; Y), thusfulfilling the notion of well-posedness of the dynamics (1.1) , needed in the cost functional(1.5). Next, the stability assumption (1.2) implies that both the Finite Cost Conditionand the Detectability Condition (see Equations (2.1.12) and (2.1.13) of [L-T.2, Chapter 2])are automatically satisfied. Then, the usual argument of, say, [L-T.2, Theorem 1.2.1.1 ofChapter 1, p. 14; Theorem 6.2.1.1 of Chapter 6, p. 563, etc.] yields that: there exists aunique optimal pair {M° (t; y0), y^(t; y0)} in L2(0,oo;t/) x I2(0, oo; Y) of the optimal controlproblem, satisfying the optimality condition. We obtain

Proposition 3.1.1. Assume (H. I ) through (H.5). Then:

(i) for any w G L2(0,oo:l' ') fixed, the optimal control problem (1.5), (1.7) for the dy-namics (1.1), or (1.10), admits a unique optimal pair {«°,( • : ? /o ) , ? / ° ( • ;2 /o )} satisfying theoptimality condition

«°( • ; j / 0 ) = -L'R-RyK • ; y 0 ) G L2(0,oo;U). (3.1.1)

( i i ) the optimal pair is given explicitly by the following formulas:

u°w( . ;j/0) = -[/ + L- R' RL}-1 L' R- R (eA ' ya + Ww] G L 2 ( Q , o o ; U ) (3.1.2a)

= « ° = o ( - l ! t o ) + « ° ( - ; ! / o = 0); (3.1.2b)

Vl ( ' i yo} = (1 + LL*R*R}~1 [eA ' y0 + Ww] G L2(0, co; F) (3.1.3a)

= {/ - L[I + L*R*RL}-1L' R'R] [eA ' y0 + Ww] (3.1.3b)

= y ° = 0 ( - ; j / o ) + y ° ( - ; y o = o), (3.i.3c)*]-lR[eA- y0 + Ww] eL 2 (0 ,oo ;Z) (3.1.4a)

= ^=o( - ;yo) + ^ ° ( - ; y o = 0), (3.1.4b)where the inverse operators in the above formulas are well-defined as bounded operators onall of L2(0,oo; • ) by (l . l lb), (1.12b), (1.13b), (1.14b) (for the inverse occurring in (3.1.3a),see [L-T.2, Chapter 2, Appendix 2A]). Moreover, with J/Q G Y, the corresponding optimaldynamics is

y°w(t;y0) = eMy0 + {Lu°w( • ; y 0 } } ( t ) + (Ww}(t) € L 2(0,oo ;y). (3.1.5)

( i i i ) For yo G V, the optimal cost satisfies the following relations

J°(y0) = JM( • ;yo),y°w( • ;yo}) = J°=0(yo) + J°(y0 = o) + xw(y0); (3.1.6)

J°=0(y0) - (RcA-y0,[I + RLL*RrlReA-y0}L2(0^z); (3.1.7)

^(2/o = 0) = (w^WR^I+RLL'R'^RWw)^^^; (3.1.8)

Xw(y0) = 2 (RcA ' j/o, [/ + /ML-flT1^™)^^

= linear in w. n (3.1.9)

We next prove the additional regularity C'([0,oo]; • ) for u°w and ?y°. To this end, andgenerally to build further a theory, as described by Theorem 2.1, we make the preliminaryobservation that the abstract model of the present chapter differs c r i t ica l ly from the two main


settings of the literature [B-D-D-M], [L-T.l], [L-T.2]. Indeed, neither is the s.c. semigroup eAt

analytic — as in [L-T.2, Chapters 1, 2, and 6] — nor does the pair {A. B] satisfy the 'abstracttrace condition' as in [L-T.2, Chapters 7, 9]. The key feature of the present new setting isassumption (H.3) = (1.3) on 0 < < < T — hence, its consequence (1.8) for all < > 0 under thestability hypothesis (1.2). As noted in Remark 1.1, this makes the present setting a shadowyresemblance akin to the 'analytic case' of [L-T.2, Chapters 1, 2, or 6]. This observationprovides the key guide to the treatment that follows.

First, all the critical smoothing properties, in the analytic or parabolic case, of theoperators Z,, Z/* — beginning with the preliminary regularity Z,2 — * Z/% in (l.llb), (1.13b) —confif&ue (o /:oM (rue under assumptions (H.I), (H.2), (H.3). This is so, since it is the singularestimate (1.8) on the kernels of these operators, which plays the key role in these results inthe parabolic or analytic chapters of [L-T.2]. Thus, we still have available, under the presentsetting (H.I), (H.2), (H.3), the following properties:

(i) The regularizing properties of the operators Z/ and A* on Z,p-spaces ([L-T.2, Theorem1.4.4.3 of Chapter 1, p. 38 for T finite; Theorem 2.3.5.1 of Chapter 2, p. 143 for T = oo, aswell as Theorem 6.9.1 and Theorem 6.23.1 of Chapter 6, p. 590 and p. 620, the latter forT = oo]). This is recorded as Proposition 3.1.2 below.

(ii) The regularizing properties of the operators Z/ and Z,* on the space -,(?([ , ]'; • )with singularity on the left ([L-T.2, Proposition 1.4.5.4 of Chapter 1, p. 49]). This is recordedin Proposition 3.3.1 below.

We collect the conclusions reached in (i) above in the next statement.

Proposition 3.1.2. Assume (H.I), (H.2), (H.3), (H.5). Then, with reference to theoperators L and Z/" in (1.11 a), (1.13a). we have:

0)

Z, : continuous Z,2(0,oo;[/)-»Z,2(0,oc;y) (3.1.10)

continuous C([0,oo];C/) -* C([0,oo];}'); (3.1.11)

Z," : continuous Z/2(0,oo;y)->Z.2(0,oo;[/) (3.1.12)

continuous C([0,oo];y) -> C([0,oo];[/); (3.1.13)

(iii)[/+ L'R'RL}-1 €£ ( I 2 (0 ,oo ; f / ) )n / : (C( [0 ,oc ] ;C / ) ) ; (3.1.14)

( iv )[/ + LL'R'R}-1 e £(C([0, oo]; y)) n £(L2(0, oo; V)) : (3.1.15)

(v)Z, : continuous L 2 (0 ,oo ;C/ ) -> C([0, oo]; V). if 7 < r; (3.1.16)

i

L : continuous L p (0 ,oo;£ / ) -> C([0, oo ; F), if p > 1/(1 -7); (8.1.17)

L : continuous Lrt(Q,oo;U) -> LT2(0, oo; Y), (3.1.18)

where r\ is any positive number satisfying r j < 2/(2j ~ 1), where 2/(2~{ — 1) > 2, forI < 7 < 1; for 0 < 7 < | we may take r\ = oo; and r^ is any positive number satisfying?-2 < 2/(47 - 3), where 2/(47 - 3) > 7^ for | < 7 < 1; for 0 < 7 < |, we may take r2 = oo.

(vi i i ) there exists a positive integer 710(7) depending on 7, such that for all positiveintegers n > "0(7), we have

(L'R'RL}n : L2(0,oo;[7) -> C*([0, oo]; F). D (3.1.19)

Regarding properties (3.1.14) and (3.1.15) in the space C — that is, [/ + L*R*RL}~1 GC([0, oo]: U) and [I + LL*R*R]~l G C'([0, oo]; F)— these are achieved by the usual boot-strapargument as in, say, [L-T.2, Corollary 2.3.5.2 in Chapter 2, p. 145, and Corollary 6.23.2 ofChapter 6, p. 621]. This plays alternatively between the optimality condition (3.1.1) andthe optimal dynamics (3.1.5), starting with the £2(0,00; • )- regularity for w^ and y^ as in(3.1.2a), (3.1.3b), and using the smoothing properties of L and L*, while Ww e C([0, oo]; Y)by (1.12b).

Proposition 3.1.3. Assume (H. I ) through (H.5). Then, for any y0 G F and any to G(0, oo: F) (fixed), the optimal pair established in Proposition 3.1.1 satisfies the additionaljularity properties

•<( • ; y 0 ) G C([0,oo];£/) ; y°w( • ; y 0 ) G C([0,co];F). (3.1.20)

Proof. Apply (3.1.14) and (3.1.15) in the space C to the explicit formulas (3.1.2a) foru°w and (3.1.3a) for y°w, where (eA ' y0 + Wiu] G C*([0, oo]; V) by (1.2) and (1.12b). Of course,(1.13c) for L* is also used. This way, the regularity properties in (3.1.20) are achieved. D

The proof of property (al) of Theorem 2.1 is complete.

3.2 The Operator $(t); The Functions pw, the Operator P: Proofof (a2), (a3), (a4), (a5)

The Operator $(<). This was defined in (2.3) by (recall (3.1.3a)):

= !/°= 0(<;yo) - {[/ + LI' R' R}"1 [eA ' y 0 } } (t)

£ L 2 (0 ,oo;V)nC'( [0 ,oo] ; y) . (3.2.1)

The regularity noted in (3.2.1) follows from (3.1.2a) and (3.1.3a) for L2 , as well as (3.1.20)for C.

Proposition 3.2.1. (property (a'2)) Assume (H. I ) through (H.o). Then the operator<J?(i) defined in (3.2.1) is a s.c. semigroup on V, t > 0, which, moreover, is (exponentially)uniformly stable.

Proof. Because of the regularity $(i)j/0 € C([0, oo]; }'") in (3.2.1) already established,it is the semigroup property that needs to be proved next. But this can be done preciselyas in past chapters, several times, as a consequence of the optimality condition: see [L-T.2, Proposition 1.4.3.1(ii), Eqn. (1.4.3.3), p. 32 (evolution property in the case T < oo)of Chapter 1; Lemma 2.3.2.1, p. 132, or Theorem 2.3.6.1, p. 146, of Chapter 2; Theorem6.10.1, p. 593, and Theorem 6.12.1, p. 596 of Chapter 6; Lemma 6.24.2, p. 622 for T = ooin the stable case, Chapter 6]. See also [L-T.2, Theorem 6.25.1 of Chapter 6]. Uniform(exponential) stability follows, as usual, by Datko's Theorem, since $ ( t ) t j o € £-2(0, oo; Y) forall y0 6 Y. n

The Functions pw(t;y0), r w ( t ) , and the Operator P 6 £(Y). For y0 e Y, we define(as in [L-T.2, (6.2.2.1) of Section 6.2.2, p. 564, of Chapter 6]) the function

J t

which is the unique solution of the problem

Pw(t; yo) = -A'pw(t; 3/0) - R'Ry°w(t\ y0); (3.2.3a)

lim pw(T; y0) = 0, (3.2.3b)

with zero initial condition at T = oo. Moreover, we introduce the operator P £ C,(Y) (see[L-T.l], [L-T.2]),

/

CO /-CO

eA'tR*R$(t)y0 = e.A'(T~i]R*R$(T - t)y0dr. (3.2.4)Jt

Lemma 3.2.2. (property (a4)) Assume (H.I) through (H.o). Then, wi th reference to(3.2.2) and (3.2.4), we have for ya e Y:

(i)

0e L2(0,oo;y)nG'([0,oo];K). (3.2.7)

(3.2.8)/

OO

B'eA^

= B'pw(t-y0] G L 2 (0 ,oo ; [ / )nC([0 ,oo] ; fy) . (3.2.9)

( i i i ) P £ £(Y) and moreover B*P: continuous Y —> f/:

= 5*1^=0(0; ,r), ,r £ F. (3.2.11)v Jo

(iv)

-«S,=o( f;yo) = B'Pw=0(t;y0) = B'P$(t)y0£ L2(0, oo; (7) n C([0, oo]; £/). (3.2.12)

Proof, (i) The decomposition of pw in (3.2.5) is a consequence of using the decompositionof t/,° in (3.1.3c) in the definition of (3.2.2). Moreover, the regularity of pw noted in (3.2.5)follows from using the stability assumption (1.2) and the regularity (3.1.3a) [no need of(3.1.20)] for y° in the denning integral (3.2.2).

(ii) The steps in (3.2.8), (3.2.9) are self-explanatory, once one recalls the optimalitycondition (3.1.1), and then (1.13a) for L* and (3.2.2) for pw. The regularity noted in (3.2.9)is the one established in (3.1.1) and (3.1.20) for u^.

( i i i ) The operator P defined in (3.2.4) is plainly in £(Y), by the stability hypothesis (1.2)and the regularity (3.1.3a) of j/JJ,. It is, however, at the level of establishing B"P G £(Y; U)from its definition (3.2.11) that we critically use the assumed singular estimate (1.3) on0 < t < T, as propagated to all / > 0 in (1.8): since 0 < 7 < 1 by hypothesis, then (3.2.11)yields via (1.8) and (3.2.1) in the C'-space:

\\B"Px\\Y (t)x\\dt (3.2.13)Jo

( /•OO -LJQt \

/ -^~dt ||$( • ^llc([o,oc];n (3.2.14)Jo ' • /

(3.2.15)

and (3.2.15) proves (3.2.10), as desired.( iv ) We return to pw=o(t',yo) = P3>(t)yo in (3.2.7). apply B~ on both sides and obtain

(3.2.12), where the noted regularity follows from (3.2.10) on B'P and (3.2.1) on 3>(t)yQ.n

Property (a3). The non-negative, self-adjoint property of P is contained in identity(2.5): this is then proved in the same way as, say, [L-T.2, Proposition 1.4.4.8 of Chapter 1,p. 44; Corollary 6.26.1.2, Eqn. (6.26.1.7) of Chapter 6, p. 628].

Property (a5). One only needs to show that the infinitesimal generator of the s.c. (ex-ponential) uniformly stable semigroup $(i) is, in fact, Ap = A — BB"P. But this follows,as usual (see the proof of [L-T.2, Theorem 2.3.8.1, p. 150 of Chapter 2]), as a consequenceof the optimal dynamics (3.1.5) for w = 0, via (3.2.1) and (3.2.12) for u°=0. D

3.3 Singular Estimate for eApiB: Proof of (a6)Orientation. As in the proof of, say, [L-T.2, Theorem 6.26.3.1], in order to derive thatthe operator P defined in (3.2.4) satisfies the Algebraic Riccati Equation (2.17) on T>(A),we need to differentiate strongly eApt on T>(A). This, in turn, is accomplished if we canestablish the same singular estimate for eAptB that holds true under (H.3) for eAiB. Theresulting Theorem 3.3.2 below is a delicate new point of the present development, which wasnot explicitly needed in our treatment of the abstract analytic or parabolic case of [L-T.2,Chapters 1, 2, and 6].

We begin by collecting the conclusions reached in point ( l i i ) of Section 3.1.

Proposition 3.3.1. Assume (H.I) through (H.5). With reference to the operators Land L* in (1.11) and (1.13), and recalling that 0 < 7 < 1 as postulated in assumption (H.3),we have:

(i) Let 0 < r < 1. Then for any 0 < T < oo,

L : continuous ,.(7([0, T}; U) -^(r+^_i) C([0,T];Y). (3.3.1)

(ii) Let ?• > 0, and e > 0 arbitrary,

L' : continuous rC([0, T]; Y) -^(r+7_1+£) C([0, T}: U). (3.3.2)

( i i i ) Let 0 < r < 1. Then, there exists a positive integer m = m(r) such that

(L*R*RL)m : continuous TC([Q, T}: U) -> C ( [ 0 , T } ; U ) . (3.3.3)

( iv)[i + L'R'RL]-1 e £(7C([o,r] ;r)). (3.3.4)

(v)

[/ + LL'R'R}-1 = I - L[I + IS R-RL]-1 L'R'R 6 £(7C'([0,T] ; V)) , (3.3.5)

where we recall that , if X is a Banach space, then

(3.3.6)o<(<?see [L-T.2, p. 3, p. 46 of Chapter 1].

Proof, (i), (ii). (iii). Properties (3.3.1), (3.3.2), (3.3.3) are precisely the propertiesestablished in [L-T.2, Proposition 1.4.5.4, p. 49, of Chapter 1], in the analytic case. Asargued in point ( I i i ) of Section 3.1, the proofs given there continue to hold true on any[0,T], 0 < T < oo, under assumption (H.3): this is so since the key role in these proofs inprecisely the singular estimate (1.3) for the kernels of the operators Z/ and .L*. Moreover,in view of the stability assumption (1.2), we may also take the case T = oo in these threeresults (i), (ii), (iii).

(iv) The proof of property (3.3.4) on the space ^C"([0,T]; • ) with singularity on the leftis nothing but the exact counterpart of the property

[/ -I- L'R'RL}-1 <E £(C7([0, T]; U), (3.3.7)

which is [L-T.2, Theorem 1.4.4.4, p. 40, of Chapter 1] on the space C'7([0,T]; U) with singu-larity on the right. Indeed, property (3.3.7) was proved there as a consequence of two mainingredients: that given 0 < 7 < 1, there is a corresponding positive integer n (depending on7) such that

(L'R'RL)n : continuous C7([0,T]; U) -> C([Q,T];U); (3.3.8)

: continuous L2(0,T;U) -> C([0,T];U) (3.3.9)

(see [L-T.2, Corollary 1.4.4.2, p. 38, and Theorem 1.4.4.3(v), Eqn. (1.4.4.19), p. 39]). Then,(3.3.8), (3.3.9) => (3.3.7) in the proof of [L-T.2, Theorem 1.4.4.4].

But the counterpart of property (3.3.8) this time on -yC([0, T}; U) with singularity on theleft continues to hold true in the present setting under assumption (H.3), by (3.3.3). Thesame holds true for property (3.3.9), which was noted in (3.1.19). Then, the same argumentgiven in the proof of [L-T.2, Theorem 1.4.4.4] yields (3.3.4).

(v) This follows from property ( iv) . D

Theorem 3.3.2. Assume (H.I) through (H.5). Then, with reference to the semigroup= eApt (feedback semigroup when w = 0), we have:

( i i ) for any 0 < MI < cjp, there exists a constant A:t > 0 (depending on Mp. u>p, wi , T,such that

Y.U} < h ~, Vt>0,Q<^<ujP. (3.3.11)

Proof, (i) We return to the explicit formula (3.2.1) for iu = 0, which we apply with Bx,x € Y, in place of y0, thus obtaining

®(t)Bx = eAptBx = [I + LL-R'R}-l[eA'Bx], x £ Y. (3.3.12)

By assumption (H.3) = (1-3), we have CA ' Bx £7 C ( [ 0 , T } ; Y ) continuously on x <E Y.We then invoke Proposition 3.3.1(v), Eqn. (3.3.5) on (3.3.12), and then conclude that

continuously on x £ Y. But this, in view of (3.3.6), means precisely estimate (3.3.10).( i i ) Part (i), Eqn. (3.3.10), implies Part ( i i) , Eqn. (3.3.11) as in the proof of Lemma 1.1.

D

As a consequence of Theorem 3.3.2, we obtain

Proposition 3.3.3. Assume (H.I) through (H.5). Then, the s.c. semigroup $(<) = eApt

in (3.2.1) is strongly differentiable on T>(A): that is, more precisely, if x £ T>(A), then fort > 0:

jeAptx = cAptAPx = eApt(A - BB'P)x (3.3.13)

= eAptAx - eAptB(B'Px) £ Y, x £ V(A). t > 0, (3.3.14)

and in fact,

d , .7te' < MPe-»"'||Ax\\ + k, —— \\B*P\\c(Y.,u)\\x\\Y, t > 0. (3.3.15)

Y

Proof. The steps in (3.3.13), (3.3.14) are self-explanatory. Of course. (3.3.13) makessense, at least, in [D(Ap)]'. The point is that, under present assumptions, (3.3.14) makessense actually in Y. Estimate (3.3.15) follows from (3.3.14) by invoking (2.7b) (alreadyproved in Section 3.2) and (3.3.11) of Theorem 3.3.2 on eApt and eApiB. respectively, as wellas (3.2.10) of Lemma 3.2.2(iii) on B'P. D

3.4 Additional Regularity Properties of P: Proof of (a7)We shall recall [from [F-L-T.l], [L-T.3], [L-T.2, Chapter 11, vol. 3] more precise versions ofthe regularity properties (2.9) and (2.10) of Theorem 2.1. (a,7). For these, assumption (H.3)is not really needed.

Proposition 3.4.1. Assume ( H . I ) . ( I I . 2 ) , (H.5) . Then the following identities hold true:

( i )A'Px = -R'Rx - PApx G Y, V.r G V(AP],

and soA"P : continuous 'D(Ap) —>• V.

(»)A'PPx = -R'Rx - PAx G F, V.T 6 £>(/!),

and soApP : continuous 'D(A) —> 1'.

Proof. This result was established already in [F-L-T.l] without assuming the stabilityhypothesis (H.3). Here we provide the counterpart proof under (H.3).

(i) Let x G T>(Ap). Then, recalling (3.2.4), we integrate by parts arid obtain

A'eA>tR*ReAptxdt (3.4.3)

A>tR~Re^ APx dt (3.4.4)

o

(by (3.2.4)) = -R'Rx - PAPx G Y,

recalling the exponential stability in (H.3) = (1.2) and in (2.7) in (a4) (already proved inSection 3.2) at t = oo. Then (3.4.5) establishes (3.4.la) from which (3.4.1b) follows by theclosed graph theorem: indeed, the bounded operator P is acted upon by the closed operatorA' which is boundedly invertible. so that A"P is closed [K.I, p. 167].

(ii) From (3.2.4) we obtain for x.y € Y:

/ fix \ / /-OO \

(Px,y)y=( eA"tR'ReAptxdt,y} =(x, e'^ R'ReAty dt } , (3.4.6)\Jo / Y \ JO ) Y

and so /•oo

P'y=l eA'r'R~ReAty dt, y £ Y. (3.4.7)Jo

But P = P' by part (a3) (already proved in Section 3.2). Hence

Px = / eArtR'ReAtxdt, x € Y, (3.4,8)Jo

which provides an alternative expression for P over (3.2.4). Next, we apply to (3.4.8) thecounterpart argument to that employed in ( i ) . For x G T>(A), we integrate by parts and


obtain

A'PPx = / A*PeA'rlR~ReAtxdt (3.4.9)Jo

°° e^r'R'Re^Axdt (3.4.10)

(by (3.4.8)) = -R-Rx-PAxeY. (3.4.11)

Thus, (3.4.11) establishes (3.4.2a) from which (3.4.2B) follows by the closed graph theorem.n

3.5 The Operator P Satisfies the ARE on T>(A): Proof of (a8)We proceed as, say, in the proof of [L-T.2, Theorem 2.3.9.1 of Chapter 2, Theorem 6.15.1 andTheorem 6.26.3.1 of Chapter 6] in the present circumstances, by critically using Proposition3.3.3.

Theorem 3.4.1. Assume (H.I) through (H.5). Then, the operator P £ £(Y) defined in(3.2.4)—which was noted in property (a3) to be non-negative, self-adjoint: P — P* > 0 —satisfies the following Algebraic Riccati Equation on ~D(A}: or on T>(Ap): that is

(A'Px,z)Y + (PAx,z)Y + (Rx,Rz)z = (B*Px,B*Pz)v

); or else V . T , Z e V(AP}. (3.5.1)

Proof. Let, at first . x,z e Y. From the definition (3.2.4) with $(*) = eApt, we have

(Px,z)y= f" (ReAptxJReAtz)zdt = f (ReA^T~^x, ReA^z)z dr, x,y(=Y. (3.5.2)/o Jt

We next specialize to x,z 6 T>(A) and differentiate (3.5.2) in i. We obtain, recalling Propo-sition 3.3.3,

Az)zdt-(Rx,Rz}z, x,z&V(A). (3.5.3)

At first, we may consider the first integral in (3.5.3) as an improper integral, since (3.3.14)applies for r > t. However, invoking (3.5.2) and Ap — A — BB"P ', we rewrite (3.5.3) first as

(PApx, z)Y + (Px. Az}Y + (Rx, Rz)z = 0, ,r, z e V(A). (3.5.4)


which is well-defined, by recalling (3.4.2a) of Proposition 3.4.1, so that A'PPz £ V. and then.since AP = A - BB'P, as

( P ( A - BB'P}x, z)Y + (P.r, Az)Y + (Rx, Rz}z = 0, x, z 6 V(A). (3.5.5)

Finally, recalling that B' P 6 C(Y:U] by (3.2.10), we obtain from (3.5.4),

(PAx, z)Y + (A'Px, z)Y + (Rx, Rz}z = (B'Px, B~Pz}v, (3.5.6)

where each term is well-defined and Theorem 3.4.1 is proved for x,z 6Let now x,z G T>(Ap). Then, the steps through (3.5.4) hold plainly true: finally (3.5.4)

again yields (3.5.5), this time by virtue of the property A* P 6 C(T>(Ap)\ Y] claimed in(3.4.1) of Proposition 3.4.1. O

3.6 The Function rw. Feedback Synthesis of Optimal Control:Proof of (alO), (all)

Proposition 3.6.1. (property (alO)) Assume (H.I) through (H.5). Then, with reference tothe function pw(t',yo) defined in (3.2.2), we have:

(0pu,(t;yo = Q) = Py°v,=0(t;y0) + pw(t;y0 = Q) (3.6.1)

- P7/°( f ;yo) + r u , ( f ) € L 2 ( 0 , o o ; y ) n C ( [ 0 , o o ] ; K ) , (3.6.2)

where the function rw(t) is defined by

rw(t) = Pw(t- y0 = 0) - Pyl(t- y0 = 0) € L2(0, oo; Y) n C([0, oo]; Y); (3.6.3)

( i i ) for 3/0 G ^ i the optimal control is written in feedback synthesis as

-u°w(t;y0) = B*pw(t-y0) = B'{Pyl(i-y0) + r w ( t ) } (3.6.4)

= B*Py°w(t;y0) + B * r w ( t ) e L 2 ( Q , o o ; U ) r > C ( [ Q , o o } - U ) , (3.6.5)

whereB'Py°w( • ;yo) and 5-^(0 £ L 2 (0 ,oo ; ( / )nC( [0 ,oo ] ; f / ) ; (3.6.6)

( i i i ) the optimal dynamics may thus be rewritten as

y°w (t; yo) = APy°w(t; y0) - BB'rw(t) + w in [D(A*]}' , (3.6.7)

i.e., in the sense that

(3.6.8)


A^^ BB'rlu(r}dr

G Z,2(0, oo; Y) n C([0, oo]; Y). (3.6.9)

Proof, (i) Eqn. (3.6.1) results from substituting (3.2.7) into the right side of (3.2.5), andrecalling the definition (3.2.1) of <£(i)?/o- The passage from (3.6.1) to (3.6.2) uses identity(3.1.3c) on ?/° via the definition ot rw in (3.6.3).

(ii) The first identity in (3.6.4) was proved in (3.2.9) of Lemma 3.2.2(ii). Then, thesecond identity in (3.6.4) follows by substituting (3.6.2). The final breaking up as in (3.6.5)is legal, once we establish the regularity properties in (3.6.6). The regularity (3.6.6) forB*Py°w follows at once from B*P e C(Y;U) in (3.2.10), and from the regularity of y° in(3.1.3c) (in L2) and in (3.1.20) (in C). Similarly, from (3.6.3),

B*rw(t) = B'p,L,(t-y0 = 0)~B'Py°w(t-y0 = 0) (3.6.10)

= -«° (i; j/o = 0)- B'Py°w(t; y0 = 0) e L2(0, oo; [/) n C([0, oo]; £/)(3.6.11)

we establish the regularity of B*rw in (3.6.6), since u°u( • ; y0 = 0) and y^( • ;y0 = 0) havethe required regularity, see (3.1.2a), (3.1.3a) (in L2) and (3.1.20) (in C), and B*P is boundedas in (3.2.10).

( i i i ) Eqn. (3.6.7) is obtained by substituting (3.6.5) into the optimal dynamics, andrecalling the definition of Ap. Then, (3.6.9) is the unique solution of (3.6.7).

We note explicitly, that the first integral term in (3.6.9) is well-defined in £2(0,00;}'") nC([0. oo]; Y'}, by virtue of the singular estimate (3.3.11) on eAptB, and the regularity in(3.6.6) for B*rw(t). D

Property (all). Equation satisfied by pw. It remains to prove property (al l) .That pw defined by (3.2.2) satisfies the backward initial value problem (2.18) follows bydirect differentiation using the regularity (3.1.20) for y^. The latter, plus the stability (1.2),justify that pw(oo;yo) = 0, as stated in (2.18b), again by (3.2.2).

Equation satisfied by rw. We finally establish that the function rw(t) satisfies thebackward initial value problem (2.19).

Proposition 3.6.2. Assume ( H . I ) through (H.5). Then, the function rw defined by(3.6.3) satisfies

(rw(t),x)Y = -(A*Prw(t),x)Y - (Pw(t),x)Y, V x 6 £>(/!). (3.6.12)

Proof. Let x € *D(A). We start from the defining formula (3.6.3) for rw and differentiatein t. thus obtaining

( r w ( t ) , x)Y = (pw(t- ijo = 0) - Py°w(t; y0 = 0),x)Y, x e T>(A). (3.6.13)

Next, we substi tute Eqn. (2.18) for pw [already proved above and Eqn. (3.6.7) for y°a. Weobtain from (3.6.13):

( i ' w ( t ) , x ) y = -(A'Pw(t;y0 = 0 ) , x ) y - (R'Ry^(t;y0 = 0) , . r )y

- ( P [ A p y t ( t ; y o = 0)- BB'rw(t] + w ] , x ) Y (3.6.14)

= - (Pw(t; 2/o = 0), Ax}Y - (y°u(t; y0 = 0), [R'Rx + A'PPx\}Y

+ (B*rw(t},B'Px)u - (Pw, x ) Y , x 6 V(A). (3.6.15)

Notice that each term in (3.6.15) is well-defined (by the regularity of pw and B*rw in(3.6.2) and (3.6.6), respectively: by B* P e jC(Y- U) in (2.6) or (3.2.10)) and, moreover, thatR'Rx + A'pPx = -PAx, x £ V(A) by (3.4.2a). Using this identity in (3.6.14), we rewriteit as

(rw(t),x)Y = -(Pv(t; y0 = 0), Ax)Y + (y° (*; 2/0 = 0),PAx)Y

+ (rw(t),B(B'P)x}Y - (Pw,x)Y (3.6.16)

= - (\Pw(t\ yo - 0) - Pyl (t- y0 = 0)],Ax)Y

+ (rv,(t),B(B'P)x)Y-(Pw,x)Y, xeV(A), (3.6.17)

where each term in (3.6.17) is well-defined. Recalling the definition of rw(t) given by (3.6.3)in the first term of the right side of (3.6.17), we rewrite it as

(rw(t),x)Y = -(rw(t),Ax)y + (rw(t),B(B'P)x)y-(Pw..x)y (3.6.18)

= -(rw(t), [A - BB'P}x)Y - (Pw,x)Y, x € V(A) (3.6.19)

= -(rw(t),APx)Y-(Pw,x)Y, x£V(A). (3.6.20)

Finally, from (3.6.20) we obtain

( f w ( t ) , x ) y = -(A'Prw(t),x)Y-(Pw,x), , rel>(,4), (3.6.21)

which is precisely (3.6.12). D

4 Illustrations. Structural acoustic problems satisfy-ing (H.I), (H.2), (H.3) with 7 = | + e or 7 = f + e

In this section we provide two structural acoustic models which, once written abstractly asin (1.1). satisfy all the required assumptions (H. I ) , (H.2), ( I I . 3) = (1.3). the latter one wi theither 0 < 7 < \, typically 7 = | + e (Example 4.1), or else 7 = | + e > \ (Example 4.2).

Example 4.1: A class of structural acoustic problems with constant 0 < 7 < |

The model. We consider the following class of structural acoustic problems, where ft isan acoustic chamber with flexible (elastic) wall F0, assumed flat and rigid wall IV Thus, let$1 C K", n = 2,3, be an open bounded domain with boundary F — FQ U FI, where FQ andFI are open, connected, and disjoint parts, F0 D FI — 4>, in R.™"1, and FQ is flat. We alloweither F to be sufficiently smooth (say, C2), or else 0 to be convex: this assumption wil l thenguarantee that solutions to classical elliptic equations with L2(fi)-non-homogeneous termsbe in H2(Q) [Gr.2]. Let z denote the velocity potential of the acoustic medium within thechamber. For simplicity of notation, we tcike equal to 1 both the density of the fluid and thespeed of sound in the fluid. Then zt is the acoustic pressure. Let v denote the displacementof the flat flexible wall F0, modeled by an elastic beam or plate equation (n = 2, or n = 3).The structural acoustic model here considered is as follows:

acousticchamber: dv

dz,2 = 0

in (0 ,Tj x f ] ,

i n ( 0 , T ] x F 0 ,

(4. la)

elastic wall Mkvtt + Avt + Av ± zt = Bu in (0,T] x [D(A^)]' (4 ' ld )

I z(0, • ) = ~o, ^(0, • ) = -!. "(0. ' ) = «o, vt(Q, -)=v0 in 0. (4.1e)

Assumptions. In (4.1a-c), / £ -^2(0, T; (H)) denotes the deterministic externalnoise within the chamber; and the non-negative constant (/,-, when positive, introduce inte-rior/boundary damping in the model. Equation (4. Id) is an abstract version encompassingseveral 'concrete' elastic models, as documented below. At the abstract level, we make thefollowing assumptions:

(al)

Mk : I2(F0) D

are two positive, self-adjoint operators

D T>(A)

(the stiffness operator, and the elastic operator, respectively, the first depending on a non-negative parameter k > 0; in concrete situations, if k > 0. the elastic model on F0 accountsfor rotational forces);

(a2)

' ' CT>(Ml); (4.3)

(a3) there is a positive constant p < | such that

A~"t3 € C(U; I2(Fo)), equivalently B : continuous U -> [D(A")}\ (4,4)

where [ ]' denotes duality with respect to /^(Fo) as a pivot space, and U is the controlHilbert space.

(a4) Moreover, the parameter p in (4.4) satisfies either one of the following additionalassumptions:

(a4i) either

p < —, if ft is a smooth domain,l (4.5a)

p < —, if ft is a parallelepiped,

(a4ii) or else

Remark 4.1.1 (on the control operator B). In concrete PDE examples of the struc-tural acoustic problems, such as they arise in smart material technology, all of the aboveassumptions are satisfied. First, in this case, the control operator B is given by

JBu = T aiUiS', u = [ U l , . . . , uj] e EJ = U, (4.6)

where: (i) if dim F0 = 1 (dim ft = 2), then £,• are points on FQ, a} are constants, and 6'^ arederivatives of the Dirac distribution supported at £y; ( i i ) if dim FQ = 2 (dim ft = 3), then£i denote closed regular curves on F0, a3 are smooth functions and 6'^ denotes the normalderivative supported at <£,•:

-(£'(£,•), dim r0 = 1 , . .3 ^ (4.7a)

- / V<? • //(/£,, dim F0 = 2 (4.7b)

where i/ is the unit outer normal vector to the closed curve £,-. and e > 0 is arbitrary.Thus, from (4.7) we have that, a-fortiori, (8'^ , A~l-i + ~ltl1)L2(r0) is well defined V -0 6 L2(Fo),since A~^» + *ty G I)(>t8 + 4 ) c //2+ c(F0) . Hence the operator B defined by (4-6) satisfiesassumption (a3) = (4.4) with / 9 = | + e, V e > 0 small [Las.3], [L-T.2, vol.2, p. 907]. Then,such / > = ! + £ satisfies both conditions (<i-{i) = (4-5a) as well.

Remark 4.1.2 (on the stiffness operator M.k}. In concrete PDE examples where theparameter k > 0 (and so the elastic beam/plate-model on F0 accounts for rotational forces).

,-Vfji is the translation by the identity of the realization oi ( — A ) on TO subject to appropriateL~ i

boundary conditions. Thus, P(y\/f|) C / / ' (To) C / / s (Fo) and assumption (a.4ii) = (4.5b) issatisfied as well. O

The above structural acoustic model ( 4 - 1 ) , subject to assumptions (al) through (0,4),satisfies assumption (H.3) — (1.3) with ~) = p < |. The following claims are shown in[Las.3], [A-L.l]. The structural acoustic problem (4.1) can be rewritten in the abstractform (1.1), with operators A and B explicitly identified, and w — [O,/, 0,0]. Moreover,the operator A is the generator of a s.c. contraction semigroup eAi on an appropriate finiteenergy space Yk given by

Yk = H l ( t t ] x L2(f t) x V(A*) x V(M\), (4.8)

for the variables [ z ( t ) , Z t ( i } , v ( i } , v t ( t } } = eAt[z0, r1: t?0, Ui] .Finally, such operators A and B do satisfy assumption (H.3) = (1-3) for 7 = p < |, the

constant in assumptions (a3) and (a4). In particular, if the control operator B is defined by(4.6), then we have -y = = | + e, V £ > 0.

Remark 4.1.3 (on the uniform stability of problem (4.1)). There are several configura-tions—that is choices of the damping constants c/,- in (4.1a-c) and corresponding geometricalconditions—which ultimately yield uniform stability on Yk of the associated s.c. semigroupcAt [L.2], as required by assumption (H. I ) = (1.2). They include the following cases: (1)</2 = c/s = 0, c/i > 0 (viscous damping); (2) d\ = d^ = 0, d? > 0 (boundary damping on rigidwall FI) , with no geometrical conditions; (3) f/i = d? = 0, d3 > 0 (boundary clamping onflexible wall FQ) under the geometrical condition that fi is convex and there exists a point.,r0 € K.71 such that (x — XQ) • v ( x ) < 0, V x G FI [L-T-Z.l, Appendices]. Additional cases arealso possible [L.3].

Remark 4.1.4. At the price introducing heavy notation, it would be possible to includeinto problem (4.1), also the case where the elastic wall FQ is curved and. accordingly, modeledby a shell equation to be written abstractly as in (4.Id), see [Las.3]. n

Concrete illustrations of the abstract elastic equation (4.Id). As canonicalillustrations of the abstract elastic equation (4.Id)—say with no coupling term zt and withno control: u = 0—we may take the classical Euler-Bernoulli equation on FQ (k = 0) or thecorresponding Kirchhoff equation on F0 (A- > 0):

vtt - k/\vtt + A2i> + A2u, = 0 on (0, T] x F0, (4.9)

under a variety of B.C. on (0,T] x <9F0: hinged, clamped, free B.C.. etc. [A-L.l], [Las.3],[L-T.3]. Then, A is the realization of A2 on ^(Fo) subject to the appropriate B.C. Finally,Mk = I + kA\ where A\ is the realization of ( — A) on L2(ro) under suitable B.C.

Conclusion. Under the above assumptions, including those of Remark 4.1.3, model(4.1) is covered by the abstract theory of Sections 1-3, with 7 < |.

Example 4.2: A class of structural acoustic problems with constant ~ < 7 < 1

The model. We use, when possible, the same notation as in Example 4.1. We consideragain an acoustic chamber fJ endowed with a rigid wall Tl and a flexible wall F0, where,now, however, we introduce two main changes over Example 4.1: (i) the wave equation in zdisplays a 'strong' damping on the wall F0 (much stronger than damping on zt in (4.1c) ofExample 4.1: see operator D in (4.10c) below); ( i i ) the flexible wall F0 accounts now alsofor thermal effects, and is therefore modeled by a thermoelastic beam or plate, where w and9 denote displacement and temperature. Accordingly, the new model is now given by

acousticchamber:

thermo-elasticwall

zu-Az-diZ + J i n ( 0 , r ] x f t ,

a~ + z = Q i n ^ T l x T t ,

a = 0 : Dirichlet B.C.; a > 0: Robin B.C.

^ + Dzt = vtdv in (0,7'] xF 0 ,

vtt - kAvtt + A2v + M + zt = Bu in (0, T] x F0,

9t-M- Af ( = 0 in (0, T] x F0,

plus Boundary Conditions

ZtIQ •) = =, v ( Q . ) = Un, V l ( Q , - ) = vi in ft.

(4.10a)

(4.10b)

(4.10c)

(4.10d)

(4.10e)

(4.10f)

where

L2(r0) if k = o. '// ( I n ) u A ' > 0

where, for k > 0, Hl(ro;k) is topologized by

In (4.10b-c), // is the unit outward vector to the boundary F = d^l. By contrast, here below,when dim fi = 3 and so dim FQ = 2, we shall let v be the unit outward normal to 6T0 as theboundary of F0; and, f be the unit tangent vector to 9Fo, oriented counterclockwise. System(4.10) is supplemented with Boundary Conditions (B.C.). We shall consider explicitly threesets of B.C. for the thermoelastic component:

Hinged B.C.:v = A-i> = 0 = 0 on (0, T] x #r0; (4.12)

Clamped B.C.:

v = '— = 0 = 0 on (0,T] x 9r0. (4.13)ov

Free B.C. when dim f2 = 3, dim FQ = 2:

A« + £^ + # = 0(4.14a)

— &v + B2v - "i -^- vtt + — = 0 on (0, T] x <9F0 (4.14b)v '

\9 = 0, A > 0, V

where, with constant 0 < /.i < 1, we have

(4.14d)

C > 0 (4.14e)

Remark 4.2.1. The parameter A; > 0 in (4.10d) is critical in describing the character ofthe dynamics of the uncoupled free thermoelastic system (4.10d-e) [that is with no couplingterm zt and with u = 0]: for k = 0, such thermoelastic problem generates a s.c. analyticsemigroup ('parabolic' case) [L-T.2, Chapter 3, Appendices 3E-3I], [L-T.4-6], [L-L.l], whilefor k > 0 the corresponding s.c. semigroup is 'hyperbolic-dominated' in a technical sense[L-T.3]. n

Throughout this example, we let

A = realization in L ^ ( T o ) of A2 subject to hinged, or (4-15)clamped, or free homogeneous B.C.

Regarding the control operator B in (4.10d), we shall assume the same hypothesis (A3)= (4.4), here restated as

(hi) there exists a positive constant p < |, such that

A~"B e C(U\ i2(r)), equivalently B : continuous U -x [D(A"}}' (4.16)

[an assumption satisfied with p = | + c, if B is the operator defined in (4.6)]. In addition, wemake the following assumption on the tangential positive self-adjoint operator D occurringin (4.10c):

(h2) With p as in (4.16), there exist positive constants f>i, <52, such that V(Di) = V(AP°)and

V z e V ( D ^ } (4.17)

whereif - < p < -, then: p - - < p0 < -; ( l . lSa )

4 2 4 4i f / ? < -, then: p0 = 0. (4.1Sb)

A typical example of such tangential operator D is a realization of the Laplace-Beltramioperator on £2(To).

Remark 4.2.2 (on assumption (h2)). (a) If the constant p in (4.16) satisfies p < \. thenthe damping operator D may be taken to be the identity operator on L2(Fo).

(b) If, however, \ < p < | [as in the case of the control operator B in (4.6)], then astronger, unbounded damping operator D is needed. More precisely:

(bl) Let j (A*) is topologically equivalent to ^f ^(To) subject to appropriate B.C. [e.g., in, ithe case of hinged or clamped B.C., then V(A») = //0

20(r0)].

(b'2) Let | < p < | so that | = | — | < P — \ < \, and we can take po satisfying| < Po < \- Then, (*): 0 < 4/?0 - f < |- Then, the following two subcases need to beconsidered.

(b2i) Assume either hinged or clamped B.C. for the operator A in (4.15), see (4.12) or(4.13). Then (*) above implies [Gr.l]

V(A"°} = H'^(Y0) C H 1 ( T 0 ) . (4.19)Thus, in this case, Dzt well-defined requires, by assumptions (4.17) and (4.19), that z gr0 = 0.To ensure this, we then take a = 0 in (4.10b), so that the z-problem is endowed with Dirichlet,rather than Robin, B.C.

(b2ii) Assume now free B.C. for the operator A in (4.15), see (4.15). Then, (*) aboveimplies

V(Apa) = H4pa(T0) c //'(Fo), (4.20)and then we can allow a > 0 in (4.10b): that is either Robin or Dirichlet B.C.

The above structural acoustic mode/ (4-10), subject to assumptions (hi), (h2), and k = 0(no rotational forces accounted for) satisfies assumption (H.3) = (1.3) with 7 = 2/7 [Las.3].Thus, if B is the control operator denned by (4.6), then i < 7 = 2(| + e) = | + 2 e < l .

When k = 0 in (4.10d) [Euler-Bernoulli rather than Kirchhoff equation] and assumptions(hi), (h2) above are in force, then the structural acoustic model (4.10) can be rewritten inthe abstract form (1.1), with operators A and B explicitly identified. Moreover, the operatorA is the generator of a s.c. contraction semigroup eAt on an appropriate finite energy spaceY given by

Y = //^(ft) x L2(.Q) x V(A^) x L2(r0) x L 2 ( Y 0 ) (4.21)for the variables { z ( t ) , z t ( t ) , v ( t ) . v t ( t ) , 6 ( t } } . The abstract deterministic disturbance «> inEqn. (1.1) is now to = [O,/, 0 ,0 ,0] , with / the disturbance in (4.10a). Finally, the s.c. semi-group eAt is uniformly stable on Y [L-T.2, Chapter 3, Sections 3.11-3.13]. (In the case offree B.C., this is due to the term (1 - p,)tv in (4.14e) with coefficient (1 — }t < 0).

Conclusion. Under the above assumptions, model (4.LO) is covered by the abstracttheory of Sections 1-3.

References[A-L.l] G. Avalos and I. Lasiecka, Differential Riccati equations for the active control of

a problem in structural acoustics, J0714 91 (1996), 695-T28.

[B-D-D-M] A. Bensoussan, G. Da Prato, M. Delfour, and S. Mitter.o/ /7%^fi%Ye Z)zmens;o»a/ ^yaêma, vols. 1 and 2, Birkhauser, 1993.

[Gr.l] P. Grisvard, Characterisation del quelques espaces d 'interpolation, /lrc/t.»(%/ Mec/i. Xnaf. 25 (1967), 40-63.

[Gr.2] P. Grisvard,

[L.2] I. Lasiecka, Optimization problems for structural acoustic models with ther-moelasticity and smart materials, Z)igc«ss2o»es Wa(/tem«f/cae

a, Con^roJ antf Op(nMz'r(i(!OM, vol. 20 (2000), 113-140.

[L.3] I. Lasiecka, Mathematical control theory of coupled PDEs systems,

[L-T.l] I. Lasiecka and R. Triggiani, Z);jêreM^z<i/ <m(f /l/<ye/;caôMs *o Eo«n(fa?'!//Fo2?i( Con^ro/ Fro6/ems;TAeory, LNICS, Springer Verlag, 1991, 160 pp.

[L-T.2] I. Lasiecka and R. Triggiani, CoM^ro/ TAeon/ /orvol. I, Cambridge University Press, Encyclopedia of Mathematics and its Appli-cations, January 2000, 668 pp.

[L-T.3] I. Lasiecka and R. Triggiani, Structural decomposition of thermo-elastic semi-groups with rotational forces, êmz^roup Forum 60 (2000). 16-66.

[L-T.4] I. Lasiecka and R. Triggiani, Two direct proofs on the analyticity of the s.c. semi-group arising in abstract thermo-elastic equations, /lâocea Z); . Eiyns. 3(3)(May 1998), 387-416.

[L-T.5] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with cou-pled hinged/Neumann B.C., v46a*mc( AppA /4»aA 3(1-2) (1998), 153-169.

[L-T.6] 1. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with freeB.C., /Inna/z' ^cuo/a /Vorma/e ûpen'ore, Pisa, Cl. Sci. (4), XXXVII (1998), 457-482.

[L-T-Z.l] I. Lasiecka, R. Triggiani, and X. Zhang, Nonconservative wave equations withunobserved Neumann B.C.: Global uniqueness and observability in one shot,Contemporary Mathematics, vol. 267, to appear.

[L-R.l] K. Liu and M. Renardy, A note on equations of a thermoelastic plates,Appl. Math. Letter.? 8 (1995), 1-6.


Solving Identification Problems for the WaveEquation by Optimal Control Methods

Suzanne Lenhart

Mathematics Department

University of Tennessee

Knoxville, TN 37996-1300

Vladimir Protopopescu

Oak Ridge National Laboratory

Computer Science and Mathematics Division

Oak Ridge, TN 37831-6355

Abstract: Inverse problems of identification type for the wave equation are approximated via

optimal control methods. The sought "unknown" coefficients are treated as controls and the goal

is to drive the model solution close to the observation data by adjusting these controls. Tikhonov

regularization is coupled with optimal control techniques and illustrated for three examples.

Research supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, under

contract No. AC05-OOOR22725 with U. T.- Battelle, LLC.

" This submitted manuscript has been authored by a contractor of the U.S. Government under Contract

No. DE-AC05-OOOR22725. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to

publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government

purpose".


0. INTRODUCTION

We survey our recent results on the approximation of inverse problem of identification

type for wave equations by optimal control methods and we illustrate this approach with

three examples.

The wave propagation problem (state system) has the following form:

utt = &u + hu + f on Q = f i x ( 0 , T )

u(x,Q) = u0(x), ut(x,0) =u1(x), on O x {t = 0} (0.1)

^ + cm = g on < 9 f t x ( 0 , T )

where fi C Rn is a bounded spatial domain with C1 boundary. The boundary conditions

are of nonhomogeneous Robin type which includes Dirichlet and Neumann as particular

cases. To streamline the presentation, we consider an isotropic homogeneous medium and

normalize the speed of propagation to one, but anisotropic, inhomogeneous media can be

treated without significant difficulty [5, 6, 7].

We denote by <j) the coefficient/function to be identified. For instance, (f> can be the source

/, the dispersion coefficient h, the surface reflection coefficient a, or other function occurring

in the problem. Starting from actual observations z of the solution of the wave problem,

u = w(0), one seeks to minimize the objective functional

((fy -z\2dxdt

over a class of functions <p in a control set t/, where W\ is a subset of Q or <9fi x (0,T),-

depending on the type of observations. The inverse problem is solved by finding 0 G U, for

which u(<p) is as close as possible to the observations z in the L2 sense.

To approximate this identification problem, we introduce the following optimal control

problem for j3 > 0:

mm ,/,(<«

with

i w2

where W-2 is a subset of Q, fi, or <9fi, depending on the type of control. This type of

approximation is called Tikhonov regularization [12]. For each j3 > 0, the coefficient to

be identified, (/>, is viewed as a control and is adjusted to get the corresponding solution,

«(</>), close to the observations z. For a fixed /? > 0, the optimal control, hp, that minimizes

Jf}(4>)^ will be explicitly characterized in terms of solutions of the optimality system, which

consists of the state problem coupled with an adjoint problem. Taking a sequence of j3n that

converges to zero, the corresponding sequence of optimal controls, <j)pn, is shown to converge

to a "solution", (/>*, of our identification problem. The interpretation of this "solution" is

as follows: If the identification problem is known to have a solution, then our method will

find a solution to that identification problem. If the observations are imprecise and thus the

identification problem may not have a solution, then our method finds a projection onto an

appropriate range space.

This interpretation holds for the case of one observation (one input yielding one output),

but in the third example, we discuss its extension to multiple observations coming form

multiple inputs.

The majority of traditional approaches to inverse identification problems [1, 2, 3] couple

Tikhonov's regularization with an optimization algorithm. Fuel and Yamamoto [10, 11, 13]

have obtained uniqueness and stability results for reconstruction algorithms using exact

controllability for various wave equation problems. Our approach has the advantage of an

explicit characterization of the approximate coefficients; moreover, this characterization leads

to a natural numerical algorithm in solving the resulting optimality system.

1. FIRST EXAMPLE

This example identifies the dispersive coefficient, h(x, t ) , from observations of the solution

of a wave equation on a set Q' C Q = £1 x (0,T). Consider the wave equation:

uu = AM + hu + f in Q

u = MO, M = MI on fi x {t = 0} (1-1)

M = 0, on < 9 f t x ( 0 , T ) ,

where / e L2(Q), MO G /^(Q), MI e L2(O) and Q C M" has a C2 boundary. The identification

problem is to find bounded h such that the corresponding solution u = u(h) of system (1.1)

is close to the observations z on Q'. We consider the control set,

U = {h e L°°(Q)| -M < h(x,t) < M},

and the approximate functional is:

Jp(h) = - ( j (u(h) - z)2 dxdt + (3 [ ti2 dx dt] .2 \JQ' JQ J

We seek hp € U such that

See Liang [8] for some further results on this type of control problem. The joint work with

Liang [9] gives more details on this particular example.

The solution space is

ueL*(Q,T;Hb($l)), uteL2(Q), utt e L2(0,T; H^

From differentiating the maps

h — > u(h)and

h -+ J0(h),

with respect to h and a priori estimates, we obtain

Theorem 1.1. There exists a control hp £ U and corresponding state up = u(hp), that

minimizes the functional Jp(h] over U. Furthermore, there exists a weak solution p in

L2(0,T;H^(n)) to the adjoint problem:

ptt = Ap + hp + (up — Z)XQ' in Q

p = Pt=0, on ttx{t = T} (1.2)

p = 0 on < 9 f i x ( 0 , T ) ,

where XQ' is the characteristic function of the set Q' , and, hp satisfies

(1-3)

This sequence hp of optimal controls approximates h, the desired coefficient as /3 — * 0.

If the measurements are inaccurate or affected by noise, then the observations z may either

come from a solution of (1.1) that does not represent the actual scenario or from a function

that is not a solution of (1.1) for any h 6 U. Thus we do not assume that z is the range of

the map

(1.4)

In this case we can prove:

Theorem 1.2. There exists:

(i) a sequence (3n —> 0

(ii) corresponding optimal controls, hpn, for the functionals Jpn(h),

(iii) h* e U and

(iv) u* = u(h*) such that

hpn -^ h* weakly in L2(Q),

u(h0n) —*• u" weakly in L (0,T;//g(fi))and

I (u - z)2dxdt = inf / (u(h) - zfdxdt.I x-,< hf JI I ,

Note the limit u* can be interpreted as a (not necessarily unique) projection of z onto the

range of the map (1.4).

2. SECOND EXAMPLE

In this example, we seek to identify the reflection coefficient a of part of the spatial

boundary, dfl. For bounded domain D C R2 with Cl boundary, define the spatial domain

n = { ( x , y , z ) \ ( x , y ) £D,w(x,y) < z < 0}

where w : D — > (— oo,0) is a C2 function. Assume the region fi contains a certain medium

(like water in a section of the ocean) and denote, as before, Q = fi x (0,T). We seek to

identify the reflection coefficient from that set

< t7(x,y) < K}.

Consider the solution, u = u(a), of the acoustic wave equation:

utt = Ait + / in Q

u = 0, on E x (0,T), sides of spatial domain

fs = 0 on 13 x {z = 0} x (0, T), top of spatial domain (2.1)

| j + au — 0 on F x (0, T), bottom of spatial domain

u = UQ, ut = HI, on f7 x {0}.

Where

r={(x,y,w(x,y))\(x,y)eD}

E = { ( x , y , z ) \ ( x , y ) e dD,w(x,y) < z < 0},

(?1i

and 7— = Vw • 77 denotes the outward co-normal derivative. To approximate the identifica-ov

tion problem, we consider the objective functional

(u-zfdxdydzdt + /3 (a(x,h))2

where G C fi is a set with positive measure. We seek to identify a from observations z on

the set G x (0,T), resulting from a single source /. To define our solution space, let

V = {v <= Hl(ty\v<0 on E}

with norm

-1 /2

|Vi>|2 dxdydz

The solution space for the state system (2.1) is defined by

e L\Q)

where V denotes the dual space of V.

Assume:

G C fi with positive Lebesgue measure

z e L2(G x (0, T)), w e C2(D), 10(1, j/) < 0.

From [6, 7], we state the combined existence and characterization result for the approxi-

mate functional Jp(a).

Theorem 2.1. There exists a unique control o~0 in U and corresponding state up = u(crg),

that minimizes the functional Jp(cr) over U. Furthermore, there exists an adjoint solution

m L2(0,T;V) such that

Pa = Ap + (up - h}xo *n Q

p = 0 on E x ( 0 , T )

| = 0 on D x {z = 0} x (0, T) (2.2)

f* + <W = 0 on r x ( 0 , T )

p = pt = 0, on n x {T},

and

r / i /-^ \ + i<rp(x,y) = mint - / upp(x,y,u(x,y),t) dt \ ,K\. (2.3)

[ \P Vo / J

VKe now /ef /? — > 0 and i/ie sequence of optimal controls <jp converge to the desired coeffi-

cient [7].

Theorem 2.2. Suppose the inverse problem has a solution, i.e., there exists a* G U such

that u* = u(a*) satisfies u* = z a.e. on G x (0,T). Then there exists O~Q €. U such that on a

subsequence j3n — > 0, we have

o~/3n — •*• O-Q in L2(T)

wpn = w((3n) -+ WQ in L2(0,T; V)

and

WQ = z a. e. on G x (0,T).

We also solved a more complicated problem along this line, namely we identified at the

same time the shape of part of the boundary, w(x,y), and the reflection coefficient, o~(x,y)

[5].

3. THIRD EXAMPLE

We apply the optimal control techniques to reconstruct the dispersive coefficient in a wave

equation from a single input of Neumann data and possibly noisy observation (output) of

Dirichlet data. For Q C Kn and Q = fi x (0, T), consider the wave equation:

utt = Au + hu + f in Qu = MQ, ut = M I , on (7 x {i = 0} (3-1)

Given Neumann data, 5 e Hl/2(d£l x (0,T)), we seek to identify the dispersive coefficient

h, from observations z on <9fJ x ( i j , i2). This reconstruction is done from a single Neumann-

to-Dirichlet type measurement.

The approximate control problem treats ft, as a control in

- M < H(x) < M}

and seeks to minimize for J3 > 0,

Jp(h) = \( I (u(h) - z)2 ds dt + /? f h2 dx] .2 \Jdnx(ti,t2) Jn J

We make the following assumptions:

<9QeC 2

z e L2(9fi x (*i,t2)) wi thO < ^ <£ 2 < T

The solution space for the state problem is

u£Hl(Q) with utteL^O.T;^1

We gave a complete characterization of the optimal control hp in [4] .

Theorem 3.1. There exists an optimal control hp and corresponding state, up = u(hp),

minimizing Jp(h) over U. Furthermore, there exists a solution p m Hl/2(Q) solving the

adjoint problem:

pu = Ap + hpp in Q

p = Q, pt = 0, on n x {T} (3.2)

where X(ti,t2) ^ the characteristic function of the time interval (ti ,^), ana

hp = min ( max ( —- / upp(x, t) dt, -M) , M ] . (3.3)V V P Jo ) )

Note that the solution space for the adjoint problem is weaker than the state solution

space due to the less regular Neumann data.

Under additional regularity assumptions of /, z, and g, we can prove the uniqueness of the

optimal control hp.

Now as /3 —> 0, we do not assume the identification problem has a solution, which allows

for noisy or inaccurate observations.

Theorem 3.2. There exists

(i) a sequence /3n —> 0

(ii) a sequence of corresponding optimal controls, hpn, for the functionals Jpn(h),

(iii) h* e U and u* = u(h*), such that

hn —^ h* weakly in

u(hpn) -^ u" weakly in Hl(Q)

u((3n)tt-^ u*tt weak* m L2 (0,T; Hl(Q)*)and

I (u' - z)2 ds dt = inf / (u(h)-zfdsdt.Jdnx(ti,t2) h&U Jdflx(ti,t2)

We do not assume z is in the range of the maps NDh:

NDh : Hl

NDh(g) =


Thus the limit u* can be interpreted as a projection of z onto the range of these maps.

Numerical illustrations related to Example 3 can be found in Ref. [4].

To consider multiple inputs g and corresponding observations z ( g ) , define the input set

G = {g 6

and the set of observations

{Z 6

where K2 depends on K\ . We assume the map

9 € G -* z(g)

is compact, i.e.

gn^g* in

implies z(gn) — > 2(5*) in L?(d£l x (0,T)). We note that the solution u of the wave equation

depends on h,g; i.e.,

Then we can prove:

Theorem 3.3. There exists u* e L2(0,T; H l ( f y ) with utt 6 L2(0,T; f/^Q)*), /i* e C/ and

g* e /f1/2(<9O x (0,T)), swc/i f/iat u* = u(h*,g*) and

f fI (u* — z(<7*))2 dsdt = minmin / \u(h, g) — z(g)\'2 dsdt.J8nx(ti , t2) 9eG h€U JdClx(ti,t2)

REFERENCES

[1] Banks, H.T. and K.K. Kunish, Estimation Techniques for Distributed Parameter Systems, Birkhauser,

Boston, 1989.

[2] Borggard, J. and J. Burns, PDE Sensitivity Equation Methods for Optimal Aerodynamic Design, ICASE

Report 96-44, NASA Langley Research Center.

[3] Isakov, V., Inverse Problems for Partial Differential Equations, Springer Veiiag, Berlin, 1998.

[4] Feng, X., S. Lenhart, V. Protopopescu, L. Rachele, and B. Sutton, Identification Problem for the Wave

Equation with Neumann Data Input and Dirichlet Data Observations, submitted to IMA Journal on

Applied Math.

[5] Lenhart, S., V. Protopopescu, and J. Yong, Identification of Boundary Shape and Reflexivity in a Wave

Equation by Optimal Control Techniques, Diff. and Int. Eqs., 13(2000), 941-972.

[6] Lenhart, S., V. Protopopescu, and J. Yong, Identification of of a Reflection Boundary Coefficient in an

Acoustic Wave Equation, ISAACS Conference Proceedings, Direct and Inverse Problems of Mathemat-

ical Physics, Kluwer Publishers, 2000, 251-266.

[7] Lenhart, S., V. Protopopescu, and J. Yong, Optimal Control of a Reflection Boundary Coefficient in an

Acoustic Wave Equation, Applicable Analysis 69(1998), 179-194.

[8] Liang, M., Bilinear Optimal Control for a Wave Equation, Math. Models and Methods in Applied

Sciences, 9(1999), 45-68.

[9] Liang, M., S. Lenhart, and V. Protopopescu, Identification Problem for a Wave Equation via Optimal

Control, Control of Distributed Parameter and Stochastic Systems, Kluwer Academic Press, Boston,

1999, 79-84.

[10] Fuel, J. P., and M. Yamamoto, Applications of Exact Controllability to Some Inverse Hyperbolic Prob-

lems C. R. Acad. Sci. Paris, 320(1995), Series 1, 1171-1176.

[11] Fuel, J. P., and M. Yamamoto, On a Global Estimate in a Linear Inverse Hyperbolic Problems, Inverse

Problems 12(1996), 995-1002.

[12] Tikhonov, A. N., and V.Y. Arsenin, Solutions of Ill-posed Problems, John Wiley, New York, 1977.

[13] Yamamoto, M., Stability, Reconstruction Formula, and Regularization for an Inverse Source Hyperbolic

Problem by a Control Method, Inverse Problems 11(1995), 481-496.


Singular Perturbations and Approximationsfor Integrodifferential Equations

.]. Liu, J. Sochacki, P. DostertDepartment of Mathematics, James Madison University, Harrisonburg, VA 22807.

Abstract

Let e > 0 and consider

andw ' ( t ) = Aw(t)+fb(t-s)Aw(s)ds + f ( t ) , t > 0, tu(0) = wa,

Join a Banach space X. Here the unbounded operator A is the generator of a strongly continuous cosinefamily and a strongly continuous semigroup, and b(-) is a continuous scalar function. We will look at thesingular perturbations when £ —> 0, and approximate the above two integrodifferential equations withtwo corresponding systems of ordinary differential equations for which the numerical solutions can becarried out easily. An application to partial differential equations with numerical solutions is given.

1 INTRODUCTION.

We study the integrodifferential equations

e2u"(t;e) + u'(t;e) = Au(t;e) + f^ b(t - s)Au(s;e)ds + f ( t ; e ) , t>Q, , ,u(0;e) = w0(e), u'(6;e) = u { ( e ) , ( ' '

and

' w'(t) = Aw(t) +^b(t-s)Aw(s)ds + f ( t ) , t > 0, ^ ^w(0) = WD,

in a Banach space X, where the unbounded operator A is the generator of a strongly con-tinuous cosine family and a strongly continuous semigroup, and & ( • ) is a continuous scalarfunction. We regard Eq.(1.2) as the limiting equation of Eq.(l.l) as e ~> 0. Now, Eq.(1.2)is of lower order in derivative of t, in this sense we say that we are dealing with the singularperturbation problems.

In an early study in Liu [9], it was shown that under some convergence conditions on theinitial data and /(i;e), one has u( - ;e ) —» w(-) as e —> 0. In this paper, we will apply thetechniques we developed in [10] to approximate the two integrodiffernetial equations (1.1)and (1.2) with two corresponding systems of ordinary differential equations for which thenumerical solutions can be derived easily. This way, we can provide a very useful procedureto numerically approximate the integrodifferential equation (1.1) arising from engineeringwith some simpler system of ordinary differential equations. Also, we will be able to see howthe singular perturbations are carried out as e —> 0. An application with numerical solutionswill be given to a partial differential equation.


2 APPROXIMATION METHODS.

In this paper we make the following hypotheses ([6]) :

(HI) . The operator A generates a strongly continuous cosine family and a strongly continuoussemigroup.

(H2). &( • ) e C2(R+,R), R+ = [0,oo).(H3). f(--e) and f&Cl(R+,X), £ > 0.

(H4). u0(e),w0 £ D(A),u0(e) -» w0, £2ui(e) -» 0, as e -> 0.

(H5). For any T > 0 , / ( • ; £ ) - > / ( • ) in L l ( { 0 , T } , X ) as e -> 0.

We say that u : R+ -> X is a solution of Eq.(l.l) if u £ C"2(fi+, X), w(t) 6 -D(A) (domainof A) for £ > 0 and Eq.(l.l) is satisfied on R+ . Solutions of Eq.(1.2) are defined in a similarway. Note that the existence and uniqueness of solutions of Eqs.(l.l) and (1.2) are obtainedin [3, 5, 7, 14, 15], and we are only interested in singular perturbations and approximationsin this paper, thus we assume that Eqs.(l.l) and (1.2) have unique solutions u(t;e) and w(t)respectively for every e > 0, when the initial data satisfy certain conditions.

From the singular perturbations results in [9], we have

Theorem 2.1. [9] Assume that hypotheses (HI) - (H5) are satisfied and let T > 0 be fixed.Then as e — > 0, one has u(t;s) -» w(t) in X uniformly for t € [0, T].

Next, we modify the techniques we developed in [10] so as to approximate Eq.(l.l) witha simpler system of ordinary differential equations.

First, as in [10], we invert Au(-) so that Eq.(l.l) can be transformed into a form withcontinuous kernel, that is, the unbounded operator A will not appear in the integral.

Theorem 2.2. Equation (1.1) is equivalent to

) + f<b(t-S)u(S;?)dS + f(t;?), t > 0, ,„ nu(0;e) = uo(e), u'(0;e) = u,(e), '

where b(-) is a continuous scalar function determined by & ( • ) .

Proof. Definefl

R*H(t)= / R(t-s)H(s)ds and 6*H = H.Jo

Then we can find a C2 solution FofF + b + F*b = 0 (see, e.g., [2, 4, 8, 11, 12]) such that

(6 + F ) * ( 6 + b)=6. (2.2)

Now. write Eq.(l.l) as

£2u"(£) + u'(e) = (6 + b)* Au(e) + f ( e ) .

Then we have(6 + F ) * \e2u"(e) + u'(£)] = Au(e) + (S + F) * f ( e ) .

Hence£2u"(e) + u'(e) = Au(e] + (6 + F) * /(e) - F * [<?V(e) + w ' (c)J .

Integration by parts yields

/"'F*u'(t;s)= /

Joft

F * u"(t- e) = / F"(t - s)u(s; s)ds + F(0)u'(t; e) - F(/:)Wl(e) + F'(0)u(t; e) - F ' ( t ) u 0 ( e ) .Jo

Therefore Eq.(l.l) can be replaced by

£2u"(t;e) + [l + £2F(Q)]u'(t;£) = [A - F (0) - e2 F' (0)}u(t; e)

[~F'(t - s) - £2F"(t - s)}u(s; £)ds

+ [(6 + F) * f ( t ; £) + F ( t ) u 0 ( e ) + e*F'(t)uQ(e)+ e 2 F ( t ) U l ( £ ) } , (2.3)

hence we complete the proof by dividing [1 + £2F(0)}. D

Next, for Eq.(2.1), we approximate the continuous scalar function &(• ) by a polynomialPn(-) of degree n, and then use

v'n(t;e) = Avn(t;e) + /„' Pn(t - s ) v n ( s ; e ) d s + f ( t ; e ) , t>0, ,„ ,= ) , v'n(Q;e) = ( '

to approximate Eq.(2.1). To do that, we need the following inequality.

Lemma 2.1.[1] Let u ( t ) be a nonnegative continuous function for t > a, and suppose that

u(t)<Cl+ k(t, s)u(s)ds + I ( h(t,s,r)u(r)dr}ds, t > a,•J Of J Ct *J Q

where C\ > 0 is a constant and k(t, s) and h(t, s, r) are nonnegative continuous functionsfor a < r < s < t. If the functions k(t, s) and h(t, s, r) are nondecreasing in t for fixed s, r,then

u(t) < d exp | / k(t,s)ds+ ( h(t, s, r)dr\ds\, t > a.J a J a J a

Now, we can prove the following

Theorem 2.3. Let TO > 0 be fixed. For any 6 > 0, there is a polynomial Pn(-) of degree nsuch that for the solution vn(-\'e) of Eq.(2.4) and the solution u ( - ; e ) of Eq.(2.1) (or Eq.(l.l)),one has, uniformly for 0 < e < K where K is any given constant,

maxte[o,T0]

\u(t-e)-vn(t;e)\\<6. (2.5)

Proof. Note that A also generates a strongly continuous cosine family, which we denote by

C(-). Then we can use the results in [6] to get, for t > 0,

u ( t ; e ) = C-t

+ { G(t - s; e) b(s - r ) u ( r ; ^)dr + /(s; e) \ ds, (2.6)Jo

vn(t;e) =

r \ fs . i+ / G(t — S',s)\ I L n\s — T)vn(r\£)dr -f j(s;e) us, (^-7)Jo l J o J

where /?(•;?), G(-;e) are linear operators denned in [6] using the Bessel functions; and theyhave the following properties: For some independent constants M > 1 and a > 0,

(PI). \\G(t-m<Meat, t>0, e > 0 .

(P2). ||e-'/2

, t>0, e > 0 .

Using (P2) and Lemma 2.1, we can verify that there is a constant C independent of n andi'such that | |wn(i;f) | | < C for 0 < e < K, n = 1,2,. . . , and t & [0,To]. (See the treatment ofu(t;e) — v n ( t ; £ ) below for details.) Therefore,

G(t-s;e)\ (b(s-r)u(r;e)-Pn(s-r)vn(r;£))dr]ds\\L7o ^ ' i

G(t - s; E) \ T {b(s - r)[«(r; e) - vn(r; e)]o LJo (

+ \b(s - r) - Pn(s - r)]vn(r\e)\dr\ds\\ (2.8)t rs _

/ Mea( '~s)|&(s - r) - Pn(s - r) | | |nn(r;e)| |rfr-dso Vo

' /"s ~/ Me \b(s — ) l ! i w ( r ' £ ^ ) — v (T'Ef]\\drds (2 9)o ,/o

< MeoT°T02C max |6(/) - Pn(l)\0 i€(o;r0] ^ ' n W I

r r a<t- ~Jo Jo

Now, we can apply Lemma 2.1 with k(t, s) = 0 and h(t, s , r ) = Me a ( s" s ) |6(s~r) j to obtain

\\u(t;e) - v n ( t ; e } \ \ <\ — ( f1 r ~ imax \b(l) - Pn(/)|)MeaToTn

2C exp-^ / / Mea(t"s)|W.s - r)|drds }. (2.11)'elo.Tol1 "/ I ./„ ./n J0 ^0

Therefore we can find Pn to approximate b to complete the proof. D

Now, for this polynomial Pn(-) of degree n, the (n + l)th derivative is zero. So we areable to rewrite (2.4) as a system of ordinary differential equations.

Theorem 2.4. Eq.(2.4) is equivalent to an ordinary differential equation

Y'(t) = GY(t) + H ( t ) , t > 0,Y(0) = Y0,

(2.12)

on Xn+3, where

G =

0 7 0 0 . . . 0 0 '=i-4 -Jj7 ^7 0 . . . 0 0

Pn(0)7 0 0 7 . . . 0 07^(0)7 0 0 0 . . . 0 0

. . . . . 0 0

. . . . . 0 0

. . . . . 7 0Pri"~1)(0)7 0 0 0 . . . 0 7P^'(0)7 0 0 0 . . . 0 0 .

, H(t) =

04r /(<•£)

o '0

00

, Vo =

" uQ(e) 'ui(e]

00

00

with / the identity operator. And G generates a strongly continuous semigroup on Xn+z.That is, Eq.(2.12) is wellposed on Xn+3.

Proof. In equation (2.4), define

Then

= ^'(«;e)1

Now define

Then

Next, define

to get

= v'n(t;e).

1 /•'4 / Pn(t -

= \ Pn(t - s)yi(s;£)ds.Jo

Pn(Q)yi(t;e)^\ P'n(t~Jo

- e)ds.

= / P'n(t - S)yi(s;e)ds,Jo

t- e)

(2.13)(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

(2.19)

(2.20)

Continuing in this way we obtaini~t

o

y'kfr?) = P(nand finally, we have

yn+3(t;£) =

y'n+3(t;£) =

(2.21)

(2.22)

(2.23)

(2.24)

Therefore, Eq.(2.4) is equivalent to Eq.(2.12). Next, G generates a strongly continuoussemigroup on Xn+3 by using the perturbation results from the semigroup theory. This com-pletes the proof. D

Now we see that by applying Theorems 2.2 - 2.4, the solution of the integrodifTerentialequation (1.1) can be approximated by y i ( - ; e ) = v n ( - ; e ] , the first component of the solutionof the system of ordinary differential equations (2.12) when 0 < 1 < K, where K is anygiven constant.

Next, the results concerning the singular perturbations in [9] implies that the solution uof Eq.(l.l) and w of Eq.(1.2) are almost the same when e ~ 0. Now, from [10], Eq.(1.2) canbe replaces by

t > o, w(0) = w0, (2.25)w'(t) = (A + b(0)I }w(t) - I F'(t-s)w(s)ds

where F is from (2.2) and J ( t ) is determined by f ( t ) . Also from [10], w(-) of (2.25) can beapproximated by z i ( - ) , the first component of the solution of

(2.26)

on Xn+2, i

Gi =

Z'(t) = G,Z(iZ(0) = Z0,

with

" A + 6(0)7 7 0 . . . 0 0 •Pn(0)I 0 7 . . . 0 073

n'(0)7 0 0 . . . 0 0. . . . . 0 0. . . . . 0 0. . . . . 7 0

73n

(n"1)(0)7 0 0 . . . 0 775

n(")(0)7 0 0 . . . 0 0 .

) + F l ( t ) , t

. Fi(t) =

> o,

r ~ -I

00

00

, Z0 =

00

00

(2.27)

where Pn is a polynomial of degree n approximating — F'. Therefore, these results implythat for e small, z\ from Eq.(2.26) can be used to approximate u of Eq.(l.l).

Summarizing the above results, we have the following methods to approximate the solutionu of Eq.(l.l): If e is small, that is, if the singular perturbation is considered, then the first

component of the solution Z of Eq.(2.26) can be used. Otherwise, if e is not small, then thefirst component of the solution 1' of Eq.(2.12) can be used. In either case, we are able toapproximate the integrodifferential equation (1.1) (which is hard to solve) by using a simplersystem of ordinary differential equations.

3 AN APPLICATION.

Let us consider the following partial differential equation from engineering,

putt(t, x; p) + aut(t, x; p) = uxx(t, x; p) + JQ b(t — s)uxx(s, x; p)ds + f ( t . x; p),u(t,0;p) = u(t,l;p)=0, (3.1)u(0,x;p) = u0(x;p), ut(0,x;p)=ul(x;p), t > 0, z e [ 0 , l ] ,

in L2[0,1], where u is the displacement of an object, p is the density per unit area, and ais the coefficient of viscosity of the medium. Divide a and change variables if necessary, wemay assume that a = 1. Therefore Eq.(3.1) is given by Eq.(l.l) with c2 = p and A = with (domain) D(A) = Wg'2[0, l]fW2'2[0,1]. Thus, the results in [9] implies that, with someconvergence conditions on the initial data and /(i,:r;e), when the density p —> 0, solutionsof (3.1) will converge to solutions of the "limiting" heat equation

awt(t,x) = wxx(t,x)+f*b(t-s)wxx(s,x)ds + f ( t , x ) ,w(t,0) = w(t,l) = Q. w ( Q , x ) = w o ( x ) , t > 0, x e [0,1].

Next, let's look at how to approximate equations (3.1) and (3.2) with systems of ordinarydifferential equations. First, from Theorem 2.2 and a corresponding result from [10], wesee that the integrals in Eq.(3.1) and Eq.(3.2) can be replaced by integrals with continuouskernels. Thus, without loss of generality, and also for computational reasons, we may considerthe following equations with a continuous scalar kernel E ( - ) ,

•£) + ut(t,x;£) = uxx(t,x-e) + J* E(t - s)u(s, x;e)ds + f ( t , x;e),u(t,0;e) = u(t,l;e) = 0, (3.3)u(0,x;e) = u o ( x ; e ) , ut(0,x;s) = u\(x\ e ) , t > 0, x e [0,1],

and

wt(t,x) = wxx(t, x) + fQ E(t — s ) w ( s , x ) d s + f ( t , x),w(t,0) = w(t,l)=Q, w(0,x)=w0(x), t > 0, x £ [0,1].

Now, applying Theorems 2.3 - 2.4 to Eq.(3.3), that is, let P n ( - ) be the approximationof £(•), we see that, for Y(e) = (yi(;-,e),y2(;-,£),...,yn+3(;-]e)) in Xn+\ Eq.(2.12) forEq.(3.3) becomes

j-ty4(t, x- £} = P^yi (t, x; e) + y5(t, x; e),(3.5)


with the corresponding initial and boundary conditions. Now the first component y i ( t , x ; £ )of the solution can be used to approximate the solution u of Eq.(3.3).

For Eq.(3.4), the kernel E ( - ) is the same as in Eq.(3.3), thus the Pn for Eq.(3.3) can beused as Pn for Eq.(3.4). Therefore, for Z = (zv(-, •), z 2 ( - , • ) , . . . , zn+2(-, •)) in -X"+2, Eq.(2.26)for Eq.(3.4) becomes

(3.6)

<L7 it T\ —gtzn+l\1',-1) — ) (0)2! (t, X) + Zn+2(t, X),

with the corresponding initial and boundary conditions. From [10], the first componentZ i ( t , x ) of the solution can be used to approximate the solution w of Eq.(3.4).

The discussion at the end of the previous section indicates that when £ is small, z\ fromEq.(3.6) can be used to approximate u of Eq.(3.3); otherwise, y\(e] from Eq.(3.5) can beused. To check these results out, we look at the following example. But note that thesolution u of Eq.(3.3) is difficult to obtain, thus we will demonstrate that for e small, z\ fromEq.(3.6) and y\(e) from Eq.(3.5) are close, which, according to Theorem 2.3, implies that z\from Eq.(3.6) and u of Eq.(3.3) are close.

Example. Let E(t) = sin t in Eq.(3.3) and in Eq.(3.4) and let f(t,x\e) = 0 in Eq.(3.3) andlet f ( t , x) = 0 in Eq.(3.4). Next, let u0(x; e) = w0(x) = x(l - x), and let Ui(x; e) = 0. Then(H4) and (H5) are satisfied. We use Pn(t) = Y^Lo1^2(~~^Yt2i+i)>. (wnen n ig °dd) as theapproximation of sin t. Now Eq.(3.5) becomes

:\e] — y2(t,x;£),

(3.7)

j,y2(t,x;e) = 4?

= yi(t,x;e)

j-tyn+2(t,x-e) =yn^

with the initial conditions

yi(Q,x;e)=x(l-x), yj(Q,x;e) = 0, j = 2,3, ...,n + 3, x e [0,1], (3.8)

and the boundary conditions y j ( t , Q ; e ) = y^t.l'.e) = 0, j = l , . . . ,n + 3, t > 0. Similarly.

Eq.(3.6) becomes

,x) =z3(t,x),,x) = zi(t,x) + z4(i,x)

(3.9)

with the initial conditions

~ (() n.\ — ~.("1 ~.\ r, (0 ™\ — n n — 9 3 n -4- 9 T C ffl 1 1 f^ 1 fTlZi Û, XJ — X[i — XJ, Z^Û,X j — U, j — Z , O , . . . , / t " T ^ , X t [ u , I J , Ô.-LUJ

and the boundary conditions Zj(t, 0) = Zj(t, 1) = 0, j = I , . . . , n + 2, t > 0.

In the following, we solve Eq.(3.7) and Eq.(3.9) numerically. We use Forward Euler int and Centered Differences in x, and apply the modified Picard method presented in [13].Therefore Eq.(3.7) becomes

yi,k(tj+i,Xi;c) = yi,k(tj,Xi-s) + /^J+1 y 2 , k - i ( s , X i ; c ) d s ,

y2,k(tj + ltxi'>~) = y2,k(tj, Xi',s) + ; (Ax)-y2,k-i(s,Xi,E) + y3,k~i(s,x,;e) ds,

Vn+3,k-ls,

and Eq.(3.9) becomes

Z2,k(tj+l, Xi) = Z2je

= Zn+1,k(t.j, Xi) + j+l 2:n+2;Jt-i(s, Xi)ds,

^Xi) +ff>+1(-l)(»-WZltk_1 (s,Xi)ds.

In the figures below we present solutions to Eq.(3.7) and Eq.(3.9) using the above Picarditerations with n = 5, xt = iAx, i — 1, ...,20, Ax = 0.05, tj+i - tj — At = 0.00125, and let


Fig. 1 e = 0

0.251

Fig. 2 e = 0.015625


Fig. 3 e = 0.0625

References

[1] D. Bainov and P. Simeonov, Integral inequalities and applications, Kluwer AcademicPublishers, Boston, 1992.

[2] W. Desch, R. Grimmer, Propagation of singularities for integrodifferential equations, J.Diff. Eq., 65(1986), 411-426.

[3] W. Desch, R. Grimmer and W. Schappacher, Some considerations for linear integrodif-ferentid equations, J. Math. Anal. & Appl., 104(1984), 219-234.

[4] W. Desch, R. Grimmer and W. Schappacher, Propagation of singularities by solutionsof second order integrodifferential equations, Volterra Integrodifferential Equations inBanach Spaces and Applications, G. Da Prato and M. lannelli (eds.), Pitman ResearchNotes in Mathematics, Series 190, 101- 110.

[5] W. Desch and W. Schappacher, A semigroup approach to integrodifferential equationsin Banach space, J. Integ. Eq., 10(1985), 99-110.

[6] H. Fattorini, Second order linear differential equations in Banach spaces, North - Hol-land, 1985, 165-237.

[7] R. Grimmer and J. Liu, Integrodifferential equations with nondensely defined operators,Differential Equations with Applications in Biology, Physics, and Engineering, J. Gold-stein, F. Kapple, and W. Schappacher (eds.), Marcel Dekker, Inc., New York, 1991,185-199.


[8] G. Gripenberg, S-O. Louden and O. Staffans, Volterra integral and functional equations,Cambridge University Press, Cambridge, 1990.

[9] J. Liu, Singular perturbations of integrodifferential equations in Banach space. Proceed-ings of the American Mathematical Society, 122(1994), 791-799.

[10] J. Liu, E. Parker, J. Sochacki, A. Knutsen, Approximation methods for integrodifferentialequations, Proceedings of Dynamic Systems and Applications, Vol. Ill, to appear.

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[12] R. MacCamy, A model for one - dimensional nonlinear viscoelasticity, Q. Appl. Math.,35(1977), 21-33.

[13] E. Parker and J. Sochacki, Implementing the Picard iteration, Neural, Parallel & Scien-tific Computations, 4(1996), 97-112.

[14] K. Tsuruta, Bounded linear operators satisfying second order integrodifferential equa-tions in Banach space, J. Integ. Eq., 6(1984), 231-268.

[15] C. Travis and G. Webb, An abstract second order semi - linear Volterra integrodifferentialequation, SIAM J. Math. Anal., 10(1979), 412-424.


Remarks on Impulse Control Problems for theStochastic Navier-Stokes Equations

J.L. MENALDI S.S. SRITHARANWayne State University US NavyDepartment of Mathematics SPAWAR SSD - Code D73H

Detroit, Michigan 48202, USA San Diego, CA 92152-5001, USA([email protected]) (e-mail [email protected])

Abstract

In this paper we will review certain recent developments in impulse controlproblems for the stochastic Navier-Stokes equation. The dynamic programmingequations for the optimal impulse control problem arises as a quasi-variationalinequality in infinite dimensions which is resolved in a weak sense using thesemigroup approach.

1 IntroductionDuring the past decade several fundamental advances have been made in optimal controlof fluid mechanics by a number of researchers [6]. In this paper we study impulse controltheory for turbulence. In optimal weather prediction the task of updating the initialdata optimally at strategic times can be reformulated precisely as an impulse controlproblem for the primitive cloud equations.

Variational technique to treat impulse control problems has been adapted to Gauss-Sobolev spaces (e.g., Chow and Menaldi [2]) with partial results. However, because ofthe technical difficulties associated with the domain of the generator we prefer to followthe semigroup approach.

The dynamic programming approach is used to discuss a simple optimal stoppingtime problem for the Navier-Stokes equation. We are forced to use sufficiently weakconditions on the data because our final objective is the optimal impulse control prob-lems.

In order to facilitate the use of the semigroup technique we first consider the 2-DNavier-Stokes equation with random (Gaussian) forcing field. Several approaches havebeen proposed in the literature (see Sritharan [7] for a complete reference list). Wethen proceed to treat the infinite dimensional quasi-variational inequality to deal withthe optimal impulse control problem in a weak sense.

Complete proofs of the results stated in this paper can be found elsewhere (cf. [4]and [5]), here only the main ideas are given.


2 Stochastic 2-D Navier-Stokes EquationLet O C R2 be a bounded domain with smooth boundary and u the velocity field. TheNavier-Stokes problem can be written in the abstract form as follows

dtu + Au + B(u) = f in L2(0,T;V), (2.1)

with the initial condition

u(0) = u0 in H, (2.2)

where UQ belong to H and the force field f is in L2(0, T; H). Let us begin by definingsome standard function spaces,

V = {v € ej(0, R2); V • v = 0 a.e. in O}, (2.3)

with the norm

v =o

and H is the closure of V in the L2-norm\ 1/2

|vo

2dx j = v . (2.5)

We will also define the following linear operators

Pm : L2(C>,R2) — > H is the Helmhotz-Hodge orthogonal projection andA : H2(O,R2) n V — > H, Au = -//P^Au, v > 0, is the Stokes operator, ^ ' '

where v is the coefficient of kinematics viscosity. The inertia term is represented bythe nonlinear operator

^ r D s C H x V — > H , 5(u,v) = Pm(u • Vv), (2.7)

with the notation B(u) = B(u, u). The domain of B requires that (u • Vv) belongs tothe Lebesgue space L2(O, R2).

Let us consider the Navier-Stokes equation subject to a random (Gaussian) termi.e., the forcing field f has a mean value still denoted by f and a noise denoted by G. Wecan write (to simplify notation we use time-invariant forces) f (t ) = f (x, t) and the noiseprocess G(t) = G(x,i) as a series dGk - Y ^ k S k ( x } t ) d w k ( t ) , where g = (gi,g2, • • • )and w = (u>i, 102, • • • ) are regarded as ^2-valued functions. The stochastic noise processrepresented by g ( t ) d w ( t ] = ^kgk(x,t)dwk(t,u>) is normal distributed in H with atrace-class co- variance operator denoted by g2 = g 2 ( t ) and given by

' ( g 2 ( t ) u , v ) =

Tr(g 2(*))=


We interpret the stochastic Navier-Stokes equation as an Ito stochastic equation invariational form

d ( u ( t ) , v) + (Au(t) + B ( u ( t ) ) , v) dt = (f , v) dt + £(gfc, v) d w k ( t ) , (2.9)k

in (0, T), with the initial condition

(u(0) ,v) = (u 0 )v) , (2.10)

for any v in the space V.A finite-dimensional (Galerkin) approximation of the stochastic Navier-Stokes equa-

tion can be defined as follows. Let {ei,C2, • • • } be a complete orthonormal system (i.e.,a basis) in the Hilbert space H belonging to the space V (and L4). Denote by H^ then-dimensional subspace of HI and V of all linear combinations of the first n elements{ei, 62, . . . , en}. Consider the following stochastic ODE in R."

(u"(f) , v) + (Aun(t) + B(un(t)), v) dt =

k

in (0,T), with the initial condition

(u(0),v) = (u0 ,v), (2.12)

for any v in the space Mn. The coefficients involved are locally Lipschitz and we needsome a priori estimate to show global existence of a solution un(t) as an adaptedprocess in the space C°(0,T,Hn).

Proposition 2.1 (energy estimate). Under the above mathematical setting let

Let un(t) be an adapted process in C°(Q,T,Mn) which solves the stochastic ODE (2.11).Then we have the energy equality

'd\un(t)\'2 + V |Vu"(Of <& - [2 (f(0,u"W) + Tr(g2(t))] dt +

k

which yields the following estimate for any e > 0

rTE{\Vun(t)\*}e~etdt <

,T , (2-15)< |u (0 ) ! 2

for any 0 < t < T. Moreover, if we suppose

then we also havefT

E{ sup \un(t}\pe-st+pv I \Vun(t)\2\un(t)\p~2e~etdt} <0<t<T JO /n -| y\

/O

for some constant Cef,T depending only on e > 0, 1 0.

Proposition 2.2 (uniqueness). Let u be a solution of the stochastic Navier-Stokesequation (SPDE) with the regularity

ueL 2 (n ; C' 0 (0 ,T ;e )nL 2 (0 , r ;V) ) , u e L4(O x (0,T)) (2.18)

and let the data f , g and UQ satisfy the condition

f 6 L2(0,T; V), g G L2(0, T;^(H)), u0 € M. (2.19)

J/v zn L 2 ( f t ; C ° ( Q , T , H ) n L2(0,T,V)) zs another solution then

u ( t ) - v(t)\2 exp f - ^ / ||u(5)||44 ds] < |u(0) - v(0)|2, (2.20)L v Jo i- (°) J

u)zi/z probability 1 /or any 0 < i < T.

Proof. Indeed if u and v are two solutions then w = v — u solves the deterministicequation

dtw + Aw = B(u) - B(v) in L2(0,T;V),

and setting r(t) = ^f L ||u(s)| 4 ds we haveO \ / ^ JO It ^ ' ' L4(O)

and integrating in i, we conclude. D

Each solution u in the space L2(H; L°°(0, T; M) n I2(0,T;V)) of the stochasticNavier-Stokes equation actually belongs to L2(ft; C°(0, T; M) n L4(O x (O,!1))) in 2-D, O C E2. Thus in 2-D, the uniqueness holds in the space L2(ft; L2(0, T; V)).

If a given adapted process u in L2(fi; L°°(0, T; H) n L2(0, T1; V)) satisfies

for any function v in V and some f in L2(0,T;V) and g in Z,2(0,T; £2(H)), then wecan findequalitycan find a version of u (still denoted by u) in L2(ft; C°(0, T; H)) satisfying the energy

d|u(0|2 = [2{f(0, u(0) + Tr(g2(i)] dt + 2 (g (<) , u ( f ) ) dw(t) (2.22)

e.g. Gyorigy and Krylov [3].

Proposition 2.3 (2-D existence). Let f, g and UQ be such that

f eL p (0 ,T;V') , g G L p ( 0 , T ; £ 2 ( H ) ) , u0 € H, (2.23)

/or some p > 4. TVien i/ie?'e is are adapted process u(t,x,u>) with the regularity

u e LP(Q; C°(0, T; H)) n L2(0; £2(0, T; V)) (2.24)

which solves the stochastic Navier-Stokes equation and the following a priori boundholds

rE{ sup u(t)\p+ I Vu(t)\2 u(t)\p~2dt} <

o<i<T Jo

/or some constant Cp — C(T,v,p) depending only on the numbers T > 0, v > 0P > 2. a

The proof of this result can be found in our previous work [4, 7]. Our proof is basedon the L4-monotonicity of the nonlinear Navier-Stokes operator (and it generalizes toother cases, including multiplicative noise). If we denote by Br the (closed) L4-ball in

r}, (2.26)

then the nonlinear operator u i— > Au + B(u) is monotone in the convex ball Br i.e.,

QOr4

(Aw, w) + (B(u) - 5(v),w) + —— |w|2 > - H w l l 2 , (2.27)

Vu 6 V, v e Br and w = u - v.

3 Markov-Feller ProcessIn what follows for the sake of simplicity we assume that the processes f ( x , i , w ) andg(x, i ,w) are independent o f t , i.e.,

f € V and g 6 4j(H) (3.1)

and we denote by u(f ; UQ) the semiflow, i.e., the solution of Navier-Stokes equation.Also usually we substitute UQ with v.

Proposition 3.1 (continuity). Under the previous conditions the stochastic semiflowu(t; v) is locally uniformly continuous in v, locally uniformly for t in [0,oo). Moreover,

for any p > 0 and a > 0 there is a positive constant A sufficiently large such that thefollowing estimate

E{e-at(X + \u(t; v)|2)p/2} < (A + v 2)p/2, (3.2)

Vi > 0, v G H holds, also for any stopping time t = T. Furthermore, iff and g belongto H and ^(V) respectively, then the semiflow is also locally uniformly continuous ini, locally uniformly for v in V. D

The Navier-Stokes semigroup ($(*), t > 0) defined by $(t)h(v) = E{h(u(t;v))},is indeed a Markov-Feller semigroup on the space Cb(W) (of continuous and boundedreal function on El endowed with the sup-norm). Since the base space H is not locallycompact, the Navier-Stokes semigroup is not strongly continuous.

In our approach, it is convenient to work with unbounded functions. Let CP(W) bethe space of real uniformly continuous functions on any ball and with a growth boundedby the norm to the p > 0 power, in another words, the space of real functions h on Hsuch that v i— > h(v)(l + |v|2)~p/'2 is bounded and locally uniformly continuous, withthe weighted sup-norm

where A is a positive constant sufficiently large to so that

a > a0(p), p > 0. (3.4)

It is clear that Cb(M) = C0(W) and Cq(W) C CP(H) for any 0 < q < p.Then for any a > 0, (linear) Navier-Stokes semigroup ($„(<), t > 0) with an a-

exponential factor is defined as follows

$„(*) : CP(H) —— CP(W), M<)A(v) - E{e-ath[u(t; v)]}. (3.5)

Proposition 3.2 (semigroup). Under the above assumptions the Navier-Stokes semi-group ($„(£), t > 0) is a weakly continuous Markov-Feller semigroup in the space<7P(H). a

Since the Navier-Stokes semigroup is not strongly continuous, we cannot considerthe strong infinitesimal generator as acting on a dense domain in CP(M). However,this Markov-Feller semigroup ($0(t), t > 0) may be considered as acting on real Borelfunctions with p-polynomial growth, which is Banach space with the sup-weighted normand denoted by BP(W). It is convenient to define the family of semi-norms on Bp(W)

(s-v))\e^°s}, V v e H , (3.6)s>0

where A is sufficiently large. Now, if a sequence {hn} of equi-bounded functions inBP(M) satisfies po(hn — /i,v) — > 0 for any v in H, we say that hn — > h boundedlypointwise convergence relative to the above family of semi-norms. It is clear thatPo($a(t)h — h, v) — > 0 as t — > 0, for any function h in C'P(H) and any v in EL

Definition 3.3. Let C'P(H) be the subspace of functions h in BP such that themapping given by t H- > h[u(t\ v)] is almost surely continuous on [0, +00) for any v inand satisfies

\\mpo(<ba(t}h- h,v) = 0, V v e

where p o ( - , •) is given above. D

This is the space of function (uniformly) continuous over the flow u ( - , v ) , relativeto the family of semi-norms and it is independent of a. Hence, we may consider theNavier-Stokes semigroup on the Banach space C'p(H), endowed with the sup-weightednorm. The weak infinitesimal generator — Aa with domain T>p(Aa) (as a subspace ofC'p(H)) is defined by boundedly pointwise limit [h — $a(t)h]/t — > Aah as t — > 0, relativeto the family of semi-norms. Notice that p0($a(t)Ji, v) < p0(h,v) for any t > 0, Ji inCP(W) and v in M.

Proposition 3.4 (density). // the above assumptions hold, then CP(K) C CP(S),the Navier-Stokes semigroup leaves invariant the space CP(S) and for any function hin Cp(Irl), there is _a equi-bounded sequence {hn} of functions in the domain T>p(Aa)satisfying po(hn — /z, v) — » 0 for any v in EL D

^From above results it is clear that given a > 0, p > 0, A sufficiently large and afunction h in CP(W) there is another function u in T>p(Aa) such that Aau = Ji, where

$a(t)hdt. (3.8)

The right-hand side is called the weak resolvent operator and denoted by either Tia =A~l or *R,a — (Aa + Q/)™ 1 . Moreover, if a0 = «o(A) then for any p > 0 we havea0(A) — > 0 as A — > oo, and for any stopping time r,

|Vu(f; v) |2(A

u r ; 2 p / 2

Vv G H, and then for any a > a0 we obtain

||<MO/»|| < £-(«-«»)' P||, po(<f>a(t)h, v) <

for any t > 0, and

\\Ka~h\\ <

for an v in

a — QO

and where the

a — a0

|| and the semi-norms p o ( - , v).

(3.10)

(3.11)

4 Impulse Control ProblemLet us now consider the problem of sequentially controlling the evolution of the sto-chastic process u(t; v) by changing the initial condition v. For this purpose we considera controlled Markov chain qi(i) in H with transition operator Q(k) and a control pa-rameter k which belongs to a compact metric space K. For a sequence ((",-, i = 1 , 2 , . . . )of independent identically distributed H-valued random variables we have

' '"' ' (4.1)), Vv e H,

for any initial value q(l), any bounded and measurable real-valued function h on HI andany k in K. For the sake of simplicity, this Markov chain (i.e., each random variable (",)is assumed to be independent of the Wiener process w — (wi,Wz,. • •) used to modelthe disturbances in dynamic equation.

A sequence {T;, fc,-; i = 1 , 2 , . . . } of stopping times T,- and decisions fc; such that r,approaches infinity is called an impulse control. At time t =• T; the system has animpulse described by the (controlled) Markov chain qjt(z') with k — A:;. Between twoconsecutive times T, < t < r,-+i, the evolution follows the Navier-Stokes equation:

fu( i ) = uft.Ti-uM), if Ti < « Ti+1,< ~ (4.2)[and u(r;) =q(u(r t—),C; A:;),

where u(t, s; v) is the solution of Navier-Stokes eqaution with initial value v at time s.Since T,- —> oo, we can construct the process u(i) by iteration, for any impulse control{r,-, fcj-; z = 1 ,2 , . . . } and initial condition v in H. It is clear that r,- is an stoppingtime with respect to the Wiener process enlarged by the cr-algebras generated by therandom variables Ci, ( 2 , . . . , Ci-i- Also, the decision random variables A;,- are measurablewith respect to the cr-algebra generated by T,-.

To each impulse we associate a strictly positive cost known as cost-per-impulseand given by the functional L(v,k). The total cost for an impulse control {T,-,^,-; i =1,2 , . . .} and initial condition v is given by

/

oo

F(u(t))e-aidt + Y] L(u(Tt-), k,)e-aT'} (4.3)~^

and the optimal cost

t /(v)= inf J(v,{r,-,fc,-}), (4.4){r,,kt}

where the infimum is taken over all impulse controls, and u(t) is the evolution con-structed as above, with initial condition v.

Let us follow a hybrid control setting as in Bensoussan and Menaldi [1]. Thedynamic programming principle yields to the following problem. Find U in CP(M) suchthat

U < MU, AaU < F, and AQU = F in [U < MU], (4.5)

where Aa is interpreted in the martingale (semigroup or weak) sense and M is thefollowing nonlinear operator on C(1HI) given by

Mh(v) = m f { L ( v , k ) + Q ( k ) h ( v ) } , Vv e H, (4.6)

where Q(k)h(v) = E{h(q(~v,(^i | k ) ) } is the transition operator. This problem is calleda quasi-variational inequality (QVI).

To solve the QVI we define by induction the sequence of variational inequalities(VI)

(Un+l £ C'p(M) such that Un+l < MUn , AaUn+l < F and

(AaUn+1 = F m [Un+1 < MUn],

where U° — U° solves the equation AaU° — F. This VI can be formulated as a maximumsub-solution problem

Un+l e CP(H) such that Un+l < MUn, AaUn+l < F, (4.8)

for any n > 0. In view of the Theorem for the VI in [5], we need only to assume that Moperates on the space Cp(lrl) to define the above sequence U" of functions. This meansthat first, we impose the condition

(\\L(-,k)\\<C, VkeK,

V v € H, and next

I lim s u p { p 0 ( $ a ( t ) Q ( k ) h - Q ( k ) h , v ) } = 0, t4'1^

y k e K, v € H, V h e C"p(H), for any m > 0, some positive constant Cm and wherethe norm || • || and the semi-norms po(-, v). One of the main differences between impulseand continuous type control is the positive cost-per-impulse, i.e., the requirement

L(v, k) > t0 > 0, Vv eH, k e A', (4.11)

which forbids the accumulation of impulses. We also need

F e C'p(H), F(v) > o, Vv e e, (4.12)to set up the above sequence.

An important role is played by the function U° — U° which solves AaU° = F, andby the function UQ = UQ. which is defined as the solution of the following variationalinequality

' U0 € Cp(lHI) such that UQ<miL(-,k), AaU0 < F andk (4.13)

[aU0 = F in [U0<mtL(;k)],

or as the maximum sub-solution of the problem

t/o 6 CP(H) such that U0 < inf L ( - , k), AaU0 < F. (4.14)

Consider the quasi-variational inequality (QVI)

f t / £ Cp(Irl) such that U < MU, Ajj < F and

[AaU = F in [ U < M U ] ,

or the maximum sub-solution of the problem

U e Cp(H) such that [/ < Mf/, A,t/ < F. (4.16)

Theorem 4.1 (QVI). Let the above assumptions hold. Then the VI defines a (point-wise) decreasing sequence of functions Un(v) which converges to the optimal cost U(v),for any v in H. Moreover, if the condition

there exists r £ (0,1} such that r U°(v) < U0(v), Vv £ H (4.17)

is satisfied then we have the estimate

0 < (jn - Un+l < (1 - r)n L>°, Vn = 0 , l , . . . , (4.18)

i/ie automaton impulse control {f,-, fc;}, generated by the the continuation region [U <MU} and defined by f0 = 0,

Ti = M{t> f,-.! f/lu^ju^!))] - Mf/[u(«;u(r,-_1))] },

fcE A

zs optimal, i.e., U(v) = J(v, {f,-, &,}), arac? ^/te optimal cost U is the unique solution ofthe QVI or the maximum sub-solution of the problem. D

If we impose

L(v, k) > I0(l + v|2)p/2 > 0, Vv e H, jfc e K, (4.20)

then assumption (4.17) holds for any 0 < r < 1 such that r \\F\\ < I0(a — a0)-

References[1] A. Bensoussan and J.L. Menaldi, Hybrid Control and Dynamic Programming, Dy-

namics of Continuous Discrete and Impulsive Systems, 3 (1997), 395-442.

[2] P.L. Chow and J.L. Menaldi. Variational Inequalities for the Control of StochasticPartial Differential Equations, in Proceedings of the Stochastic Partial DifferentialEquations and Application II, Trento, Italy, 1988. Lecture Notes in Math., 1390(1989), 42-51.

[3] I. Gyongy and N. V. Krylov, On stochastic equations with respect to semimartin-gales Ito formula in Bariach spaces, Stochastics, 6 (1982), 153-173.

[4] J.L. Menaldi and S.S. Sritharan, Stochastic 2-D Navier-Stokes Equation, Appl.Math. Optim., submitted.

[5] J.L. Menaldi and S.S. Sritharan, Impulse Control of Stochastic Navier-Stokes Equa-tions, Nonlinear Analysis, Methods, Theory and Application, submitted.

[6] S.S. Sritharan, (Editor) Optimal Control of Viscous Flow, SIAM, Philadelphia,1998.

[7] S.S. Sritharan, Deterministic and Stochastic Control of Navier-Stokes Equation withLinear, Monotone and Hyper Viscosities, Appl. Math. Optim., 41 (2000), 255-308.


Recent Progress on the Lavrentiev Phenomenonwith Applications

Victor J. Mizel Carnegie-Mellon University, Philadelphia, Pennsylvania

The Lavrentiev phenomenon is associated with the sensitivity of the infimum ofa variational problem to the smoothness of the class of admissible functions. Sincethe determination by Tonelli that the class of absolutely continuous functions is anappropriate class of admissible functions with which to obtain existence of minimizersto variational problems on a real interval by direct methods, it was shown that manyof the classical problems yield to this approach. Moreover, in many of the classicalproblems the minimizers were actually Lipschitz or better, so that the problems wereinsensitive to the particular subclass of absolutely continuous admissible functionschosen for applying the direct method for existence. Therefore it was quite surprisingwhen in 1926 M. Lavrentiev published, in response to a challenge issued by Tonelliduring a lecture in Moscow, an example [L] of a functional of the form

,6J[y] = I f(x,y(x),y'(x))dx with y- K b\

J a

subject to certain constraints and smoothness conditions, in which the infimum of Jover the class of all absolutely continuous functions subject to certain boundary con-ditions at a and b is strictly smaller than its infimum over the class of all Cl functionsmeeting the same boundary conditions. Thereafter in 1934 B. Mania published anexample [Ma] involving a much simpler (polynomial) integrand. The presentation ofan example involving a functional ,7 possessing a strictly elliptic integrand occurredonly in 1985 [B&M] during the course of an investigation stimulated by the possiblerelevance of such questions to the theory of (multidimensional) hyperelastic materials.

The present article reports on recent progress in the study of this phenomenon.We adopt the following notations for clarity. For each p £ [1, oo]Wl'p[a,b] = {y : [a,b] -> R\y G AC[a,b], ij G Lp(a, &)}, so that for example with[a, b] = [0,1] and y(x) = x0, ft G (0,1), y G W^O,!] <^> p G [1,1/(1 - /?)). ForJ as above we put i(p] = mf{J[y]\y G Wl'p[a,b\ + EC's], p G [l,oo], so thatPi < "Pi =*• i(p\) < i(pi)- Then if / is such that i(p\) < i-(pi) for some pi < p? wehave the Lavrentiev Phenomenon A. (Cf [Bu&M],[Bu&B] for a relaxation view.)

I. The first topic we study is a description of the possible boundary conditionswhich can lead to A, as well as the issue of whether this can occur when the integrandf in J is strictly elliptic and coercive. To facilitate this discussion we introduce fora,b,A,BE.M., p £ [1, oo] the following notation:

(i) Both ends pinned [Lagrange problem] A-2(p) ={yeW^[a,b]\y(a)=A, y(b) = B}

(ii) One end pinned Aj(p) = {y G Wlf[a,b}\y(a) = A}(iii) No ends pinned Ao(p) = Wl*[a, b],

and we denote the respective infima by i - i ( p ) , i \ ( p } , and ?'o(p). The matter will beclarified by consideration of the following:

'Research partially supported by the US National Science Foundation.

A. We take as our integrand f(x,y,z) = (if — x3)'2(l + z"20) so thatJ[y] = /oV-^3)2(l + (2/)20KT. and we take y e A:(p) = {y 6 W^[Q, l ] \ y ( 0 ) = 0}.Clearly, J[y] > 0, while for yo(x) = x3/5 J[y0] = 0. Hence by the computation on thepreceding page, i\(p] = 0, if p E [I, 5/2). We now show that i](5/2) > 0 whence Aoccurs. The argument consists in treating two cases (cf[Ml],[DHM]):Case 1: For some x* & ( Q , l ] , y ( x ) < [i(z)3/2]1/5, x <= (0,x*),y(x*) = [f (z*)3/2]1/s.Then by use of Holder's and Jensen's inequalities and the chain rule J[y] > k > 0,for k = (2/3)20(l/2)6;Case 2: y(x) < [^(x)3/2]1/5 for all x £ (0, 1}. In this case a direct computation yields,with k as above, J[y] > k > 0.Although the integrand / is convex in z ["elliptic"] it is neither strictly elliptic norcoercive in z, but use of a device from [B&M] leads to the construction of a perturbedintegrand f*(x,y,z) = f(x,y,z) + £22 which satisfies both requirements and retainsthe Lavrentiev phenomenon A i*(p) < z*(5/2), for all p e [1, 5/2).

B. With / as above we take J~(y] = j'^(rj5 - z3)2(l + (?/)20)cte, y e Ao(p).By considering separately Case 1 y(0) > 0, Case 2 y(0) < 0, one deduces thatA z~(p) = 0, if p 6 [1, 5/2), % (5/2) > k > 0 (with k as in A). Once again onecan construct a modified strictly elliptic coercive integrand /* with the Lavrentievphenomenon A i*0(p) < £[(5/2), p e [I, 5/2) (cf. [DHM]).

II. The second topic concerns the role of A for autonomous integrands /, for inte-grands f which have no y-dependence, and for integrands which are jointly convex in(y,z). For first order problems it has been shown under various regularity assump-tions that A cannot occur in any of these cases (cf. [C&V],[A&C], [Da], [S&M]).

Therefore we begin by considering an autonomous / which involves second order aswell as lower order derivatives. We take as our integrand f(x,y,z,w) = f(y,z,w) —[(§2/)2 ~ (z + 4)2(|(-2 ~ 6))3]2-u;16, so that with obvious notation for the derivative ofthe functions y'&AC[0, 1} J[y] = /^[(fy)2 - (y' + 4)2(|(y' - 6))3]2(y")16dx,

,42>2(p) - {y 6 W^[0,l}\y(0) = 0,^(0) = 6,y(l) = 7,^(1) = f }. ClearlyJ[y] > 0, while J[yo] = 0, for yo(x) = 6x + x5/3. A direct computation shows that2/o € W^fO, 1], p € [1,3). However it can be shown by the use of Holder's andJensen's inequalities and the chain rule (cf. [M1],[H]) that i^(3) > 0. Thus we do haveA ii(p) = 0,p E [1,3), ^2(3) > 0 in this autonomous second order context. Here, too,one can construct a modified strictly elliptic coercive autonomous integrand /* whichretains the Lavrentiev phenomenon A (cf [H]). On the other hand, one can raise thequestion of whether in higher dimensions and first order variational problems any oneof the three restrictions on the integrand mentioned above retains its role of excludingA. To the contrary it can be shown that there is an integrand / = f(F) for whichall three restrictions mentioned above hold, such that the corresponding variationalfunctional exhibits A with an elementary set of boundary conditions. This exampleis very closely related to that discussed in IV below (cf. [FHM2] for a detaileddescription).

III. The next topic concerns the possible structure of infimum functionsi(-) : [l,oo] — > [0,oo) exhibiting the Lavrentiev phenomenon. Although the simplestexamples discussed above possessed right continuous jump discontinuities, it is not

evident that this is the only possibility. We consider first the followingf ( x , y, z) = (y2 - I6x)2{cz6 + [exp(-^;-1/2)2/2 - 16 exp(-x-1/2).T/(log.'r)2]2 exp(13z8)}foixe [0,l/e]Thus we haveJ[y]= /0

1/e(2/2

for y e A-i(p) = {y E W*<p[Q,l/e]\y(Q) = 0, y(l) = 4e^1/2}. It can be proved thatthis satisfies A z2(j>) = 0,p G [1,2), i2(2) 6 (0,1), i2(p) = l,p e (2,oo), so that z2

is neither right nor left continuous at the jump point p = 1. Again it is possible toproduce a modified strictly elliptic coercive integrand /* for which the lack of onesided continuity of i% at p = 2 is retained (cf.[S] for details and related results).

Next we consider a two-dimensional example. Given po,pi G (1,°°) withPo < pi and a function a € W1>1[1, oo) satisfyinga(p) = Q,p £ [l,p()],a'(p) > Q,P 6 \po,P\},a(p) = a(pi) > 0,p e bi:°°); consider theintegrandf(x,y,u, z) = c(y)o'(y)|u|(m-3")/(»-1)(|«|''/(''-1) - x)2 z m

withm e [3Pl,oo),c(y) =(m-3)(m-2)/2 [((m-l)/m)(j//(y-l))]m, a; 6 [0,1], y e [p0,Pi]and take

J[M] = /E c(y)a'(y)\u (m-^/(y-^(\u\yl(y-^ - x)2|w,x \mdxdy, with

EPO,P1 = [0,lfx[po,Pi] and « 6 A(p) = {u 6 ^^(Epo^JKO, •) = 0,«(!,-) = 1}.The infimum for this problem satisfies A i(p) — a(p),p £ [1, oo), so that the infimumfunction can be absolutely continuous. The argument makes use of a 1988 result (cf.[H&M]) involving Noether's theorem for invariant one- dimensional variational prob-lems (cf. [F] for details and related results). An effort to present a one-dimensionalvariational problem possessing properties of both types encountered above is currentlyunder way ([M2]).

IV. Our final discussion has to do with the topic which originally motivated thework by Ball and Mizel on the Lavrentiev phenomenon (cf. [B&M]), namely the rel-evance of this issue to the behavior of multidimensional nonlinearly elastic materials.Here we consider a homogeneous material in two dimensions with stored energy in-tegrand W = W€ : Lin+(K2) -> [0, oo) of the formW(F) = (||F||2 - 2detF)4 + c[(det^)-1 + (1 + ||F||2)"/2], for q > 2,e > 0, whereLin f(M2) C Lin(R2) denotes the set of linear operators on 2-dimensional space withpositive determinant.

It is easy to verify that W has the following properties:(a) W is smooth,(b) W is objective and isotropic, i.e. W(QF) = W(F) = W(FQ) for each

orthogonal operator Q E Lin+(M2),(c) W is polyconvex, i.e. there is a convex mapping g : Lin(K2) x (0, oo) —•> R

such that W(F) = g(F, detF), for F e Lin+(M2),(d) W(F) -^ +00 as detF -» 0+,(e) W(F)>c, | | ^ | | ' -c 2 , c i >0,(f) W(F) = V(«i ,w 2 ) = h-'>2 8 +£[ l / (u iV2) + (l + (wi)2 + (u2)2)« /2], where the v's

are the singular values of /'' [principal stretches of a deformation]. Thus the functionA defined by A(6) = i/>(\/S, \/fi) is convex throughout (0, oo).

The variational problem to be considered involves the total stored energyE[u] = jn W(Vu)dx for fi = {x e K2|zi2 + z2

2 < 1, xz > 0}, with u e .A(p) ={u e W^^R^Vu G Lin+ a.e. w^O) e [0,1] x {0},Xi 6 [0,1]; w(a:i,0) 6{0} x [0,1], .T! e [-l,0];M(x) = (cos(9/2),sm(6/2)),x = (cos0,sm0),0 e [0,7r]},p > 2, so that w maps the unit upper half disk fi into the unit quarter disk fi'. Wehave the following result

Theorem There is an £o(<?) > 0 such that if e < £o(g) and 2 < pi < 4 < p2 thenA i(pi) = inf{£[w]|w e -A(pi)} < i(pi) = \nl{E[u]\u e A(pi}}

Notice that by the Sobolev imbedding theorem the displacements in both A(p\)and A(pi) are continuous on fi, so that no cracks are created. Indeed, the convexityof A in (f) ensures that the stored energy of continuous deformations from fi into fi'cannot be lowered even by enlarging class A to include discontinuous deformationsof bounded variation with the same average density (cf. [M3]). That is, this materialhas the property that opening "cracks" in a deformation cannot result in a loweringof the energy.

The proof depends critically on the following properties of the integrand W0 [i.e.,W£ with e = 0]:

(i) Wo > 0 is convex on Lin(M2)(ii) solutions of the Euler-Lagrange system associated with WQ are known explicitly

via complex analysis. In fact, u*(x) = [(a?])2 + (x2)2]1/4(cos(6»/2), sin(0/2)), where 0is the polar angle to x, yields Wo(Vw*) = 0 and u* e A(p), for p e (2,4), whileu**(x) = ((x^ + (a;2)2]u/28(cos(<V2),sm(6>/2)) satisfies «** e A(p), for p 6 (2, f)and _£/o[M**] > 0, with u* and u** both satisfying the Euler-Lagrange system forE = EQ (cf. [FHM1] for further details and related results).

Bibliography

[A&C] G. Alberti and F. Serra Cassano, Nonoccurence of gap for one-dimensionalautonomous functionals, Istituto di Matematiche Applicated, Universita di Pisa(1994)

[A&M] G. Alberti and P. Majer, Gap phenomenon for some autonomous functionals,J. Conv. Anal. 1 (1994), 31-45.

[B&M] J. M. Ball and V. J. Mizel, One-dimensional variational problems whose mini-mizers do not satisfy the Euler-Lagrange equation, Arch. Rational Mech. Anal.90 (1985) 325-388.

[Bu&M] G. Buttazzo and V. J. Mizel, Interpretation of the Lavrentiev phenomenon byrelaxation, J. Funct. Anal. 110 (1992) 434-460.

[Bu&B] G. Buttazzo and M. Belloni, A survey on old and recent results about the gapphenomenon in the calculus of variations, in Recent Developments in Well-PosedVariational Problems, R. Lucchetti and J. Revalski eds.. Kluwer 1995.

[Ce] L. Cesari, Optimization-theory and Applications, Springer-Verlag, New York1983.

[C&V] F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basicproblem in the calculus of variations, Trans. Amer. Math. Soc. 289 (1985)73-98.

[Da] A. M. Davie, Singular minimizers in the calculus of variations in one dimension,Arch. Rational Mech. Anal. 101 (1988) 161-177.

[DHM] K. Dani, W.J. Hrusa and V. J. Mizel, On the Lavrentiev phenomenon for totallyunconstrained variational problems in one dimension, Nonlinear Diff. Equationsand Appls. (in press).

[F] M. Foss, Examples of the Lavrentiev phenomenon with continuous Sobolevexponent dependence, Center for Nonlin. Anal. Dept. Math. Sciences CMURes. Report OO-CNA-013 (10/2000).

[FHM1] M. Foss, W.J. Hrusa and V.J. Mizel, Occurrence of the Lavrentiev phenomenonin two dimensional nonlinear elasticity (in preparation).

[FHM2] __________________, On types of integrands exhibiting the Lavrentievphenomenon in dimensions greater than one, (in preparation).

[H] W.J. Hrusa, Lavrentiev's phenomenon for second-order autonomous variationalproblems in one dimension (preprint).

[H&M] A.C. Heinricher and V.J. Mizel, The Lavrentiev phenomenon for invariant vari-ational problems, Arch. Rational Mech. Anal. 102 (1988) 57-93.

[L] M. Lavrentiev, Sur quelques problemes du calcul des variations, Ann. Mat.Pura Appl. 41 (1926) 107-124.

[Ma] M. Mania, Sopra un esempio di Lavrentieff, Boll. Un. Mat. Ital 13 (1934)147-153 .

[Ml] V.J. Mizel, New developments concerning the Lavrentiev phenomenon, Tech-nion 1998, in Calculus of Variations and Diff. Equations Chapman and Hall/CRCResearch Notes in Math. #410,2000, A. loffe, S. Reich, I. Shafrir, eds.

[M2] __________________, The Lavrentiev phenomenon in one dimensionwith general monotone exponent dependence (in preparation).

[M3] __________________, On the ubiquity of fracture in nonlinear elasticity,J. of Elasticity 52 (1999) 257-266.

[S] A. Siegel, Two examples of Lavrentiev's phenomenon, Master's Thesis, Dept.of Math. Sciences, Carnegie Mellon U. 1999.

[S&M] M. Sychev and V.J. Mizel, A condition on the value function both necessaryand sufficient for full regularity of minimizers of one-dimensional variationalproblems, Trans. Amer. Math. Soc. 350 (1998) 119-133.


Abstract Eigenvalue Problem forMonotone Operators and Applications to

Differential Operators

Silviu Sburlan

Department of Mathematics, Ovidius University,Bd. Mamaia 124, 8700-Constantza, Romania

E-mail: <[email protected]>

Abstract

In this work we extend the multiple orthogonal sequence method, developed in[3], to the energetic space of an abstract linear monotone operator. This methodleads to an abstract eingenvalue problem that it produces orthonormal bases in somenested Hilbert spaces, that they are suitable to develop abstract Fourier or projectionmethods. Some examples to diceerential operators are also given.

One also considers the abstract semilinear eingenvalue problem and some results,known for compact operators, are extended for monotone type operators with appli-cation to diceerential operators.

Let X be a real Hilbert space with inner product (- , •) and the induced norm1 1 - 1 1 -

Consider a linear operator B : D(B) C X —»• X, with D(B) in0nite dimen-sional, which is symmetric, i.e.,

(Bu,v) = (u,Bv), Vu,v e£>(.B) (1)

and strongly monotone, that is, there exists c > 0 such that

(Bu,u)>c\\u\\2, Vu£D(B) (2)

We induce on D(B) the energetic inner product

(u,v)E := (Bu,v), Vw,v <E D(B)

and the energetic norm

\U\\E := (u,u)lj>\

Denote by E the completion in X of the linear subspace D(B) withrespect to the energetic norm an call it the energetic space of the operatorB. It contains all u G X that are the limit points of Cauchy sequences{un} C D(B) with respect to the energetic norm || • ||.E. Extending by continuitythe energetic inner product, i.e.,

(u,v)E :=lim(un,vn)E , Vu,v e E,

the energetic space E becomes a real Hilbert space containing D(B] as a densesubset and the embedding E <—+ X is continuous, namely

The duality map J : E — + E* , de0ned through

< Ju,v >:= (U,V)E, Vt / ,v e E,

is a linear homeomorphism with

\\Ju\\E. = \\U\\E, VueE,

(see D. Pascali and S. Sburlan [7, p. 112]) and it is an extension of B, i.e.,

Ju = Bu, Vw £ D(B).

The Friederichs extension A : D(A) C X — *• X of the operator B is de0nedthrough

Au := Ju, Vw € D(A), (3)

where D(A) := {u 6 E\ Ju £ X}. Observe that u 6 D(A) if and only if thereexists an / G X such that

<Ju,v>=(f,v)E, Vv&E

and D(B) C E C X C E* , (see E. Zeidler [11, p. 280]).

Remark that the Friederichs extension is in fact the maximal monotoneextension of B in X since D(A) is dense in X and A is closed, self-adjoint,bijective and strongly monotone, i.e.,

(Au,u)>c\\u\\2, VuGD(A),

(see A. Haraux [2, p. 48]). Also, the inverse operator A~l : X —>• X is linearcontinuous self-adjoint and compact, whenever the embedding E <^ Xis compact. Therefore applying the Fredholm theory we can state follow-ing variant of the multiple orthogonal sequence theorem (G. Morosanu and S.Sburlan [3]):

Theorem 1: If the embedding E t— >• X is compact, then there exists thesequences {en} C E and {An} C (0,+oo) that are eingensolutions of A, i.e.,

(Aen,v) = Xn(en,v),\/v£X, n&N (4)

and such that:

(i) {en} is an orthonormal basis in E;(ii) {\/An

en} is an orthonormal basis in X;(in) {Anen} is an orthonormal basis in E*;(i) {An} is increasingly divergent to +00.

Proof, (i) Suppose that the embedding E <^-> X is compact and let {un} 6 Ebe a bounded sequence, ||wn||,E < c. Then, passing eventually to a subsequence,we can suppose that un —> M in X,and u £ E, because {un} is strongly and alsoweakly convergent in E. Then by the compacity of the embedding

p-1^ - A-lu\\E = sup{< A~l(un - u},v>;\\v\\E < 1} =

= sup{(un - M,u) |H|B <l}- c\\un - u\\ -> 0

and, thus, A~l maps bounded sets into relative compact sets. By Hilbert -Schmidt theorem A~l has a countable set of eingenvalues {kn} C (0, +00), kn >> kn+i —»• 0, and the corresponding eingenvectors, say {en},

A~len - knen, Vrz e N,

form a Hilbertean basis in E (see e.g. S. Sburlan & co [9, p. 186]). Hence

Aen - \nen, Vn <E N

where the characteristic values, \n = •£-, are such that \n < Xn+i —*• +00, asstated in (iv), and therefore (4) holds.

iii) To show that {V%Ten} is a Hilbertean basis in X it suffices to prove thatit is complete in X because

&mn = (em ,en)fi = A n (e m , e n ) (5)

as it results from (4). For this let h G X be such that

(h, v^en) = 0, Vn e N (6)

by the Lax-Milgram theorem, with

a(u, v) := (u, V}E, Vu, v G E,

it exists only one u £ E such that

(u,v)E = (h,v), \fveE (7)

Therefore, from (6) and (7), it results

and, since {en} is complete in E, we have the implication u = 0 =>• h — 0, thatis, the completeness in X of the system {\fX^en}.

(iii) To show that {Anen} is a Hilbertean basis in E* we must prove that itis an orthogonal system complete in E*.

Note that E* is a Hilbert space with the norm

and the inner product de0ned by

For orthogonality we use the equation (4), namely

\^nen)\Sn) '—< ^n6n,en >= (Ane, j ,enJ = (Aen,en) =

= (en ,en)B = \\en\\E = 1and

/ i i „ \ „ \ —(j-i(\ f \ r-i(\ P \\^—\AnKn i Amem)£,, — {J \^ncn)^ *> \Am^m))E —

= (en,em)E — 0, Mm n.

For completeness let v* 6 E* be such that (v*, Anen)£, = 0 for all n £ N. Itthen results

(v*,h)E' =0, Mh &X

because {\/\^en} is a complete system in X. Since X is dense in E* , for all€ > 0 there exists an h£ 6 X such that

£ > |K - Ml, = IKIII- - 2(«*,MB. + IIMI- = IHII* + ll lll..Hence, we have the implication

|K|||. <£, Ve>0=^* =0

as required. •

Direct consequence: Denote by En := 5p{ei,e2, ....,en} C £", Xn == 5p{v/Aie1,v/A262, • • - , V^nen} C X and £"* := 5p{Aifii, A 2 e 2 , ...,Xnen} CC E* , the 0nite dimensional subspaces generated by the 0nite sequence {e\, 62, ..., en}.Then En,Xn and E*n are projectionally complete in E,X and E* , respectively,that is Trnu — > u in each space, where

n

irnu := ' a k < f > k , Vn > 1,

with {<fl,<f2, ...,<pn] one of the above mentioned basis and ak, 1 < k < n, thecorresponding Fourier coefficients.

These coordinate systems can be used either for abstract Galerkin projectionmethod or for abstract Fourier series method (see S. Sburlan and G. Moroanu[10]).

Remark: Since D(Bk) C D(Bk~l) C ... C D(B) and Bk is also symmetricwe can apply the above method to produce orthogonal basis in the energeticspaces associated to Bk, k G N, and the corresponding dual spaces.

De0ne the spaces Ek '•= D(Ak/2), A; G N, where A is the Friederichs extensionof B. It is easily seen that Ek are Hilbert spaces with the inner products:

V u , u e £ f c .

Of course, El = E and (u, u)i = (u, v)E, Vw, v G E (see E. Zeidler [11, p. 296]).As above we can easily show that {A^~~ + " en} is an orthonormal basis in Ekand

(8)

Since {\n} is an increasing sequence of positive numbers it results, by (8),that the embedding Ek+i Ek is continuous. Also, identifying X by its dualspace we get the inclusions

...- C Ek+l CEkC...CE1=ECXCE*=E^C ... C E\ C E*k+1 C ...

As the embedding E <^-» X is compact, it then follows that the embeddingsEk+i °—> Ek, X <—>• E* and E^ c—>• E^+1 are also compact. The duality mappingJk : Ek —>• £"| is de0ned by

< JkU, v >— (u, v ) k , Vu,v£Ek

and, obviously, J\ = J. It is easily seen that

Vn k P M (£>},, v u, K, t IN. \y)

and then {A^+ 1- ) /en} is an orthonormal basis in E^.

Example 1. Let fi be a bounded domain in W1 with its boundary d£lenough smooth to apply the Green's formula. Take X := L2(f2), that is aHilbert space with the inner product

( u , v ) := I u ( x ) v ( x ) d x , \/u,v Gn

and let B := —A be a de0ned on

D(B) :- {u e C2(fi) n Cl(ti)\u = 0 on F C dtt, //(F) > 0}.

Then B is symmetric and strongly monotone having

E:= {v eH1(fy\v = Qon F}

as energetic space. Note that E is a Hilbert space with the inner product

r(u, V)E '•— I Vw(x) • Vv(x)dx, Vw, v 6 E

Jn

and the embedding E ^-> X is compact when N > 2 (see J. Neas [6,p.117]).Therefore we can apply the Theorem 1 and write (4) in the form

Ven(ar) • Vv(x)dx - \n I en(x)v(x)dx, Vv £ E (9')n a

Then by Green's formula we deduce that en are the weak solutions of thefollowing eingenvalue problem

-Aen(x) = Xnen(x),x 6 fi,en(x) = 0 ,x£T, (10)

Remark the limit case F = (ft when E = Hl(£l) is a Hilbert space withanother inner product

f(u, V)E '•— I [u(x)v(x) + Vw(ar) • V v ( x ) ] d x , Vw, v e E

that it modi0es (9') as follows

/ Ve n (ar ) - Vv(x)dx = (An - 1) / en(x)w(a;)da; (11)</ Jn n

and, thus, en are the weak solutions of the problem

-Aen(ar) = (A n_i)e n (z) , xett,^(x) = 0 , x G 9J2. (Uj

Application: The deformation of a body B, that occupies a bounded regionf2 C ]$>N (N = 2 or 3), is characterized by the displacement vector u ifi^IR^ andthe corresponding strain tensor s = e(u). In the case of the small deformations,e(u) reduces to the symmetric part of the displacement gradient, i.e.,

l < M - < ^ } . (13)

The constitutive relation that characterizes the elasticity is a generally non-linear dependence of the stress tensor

a := {o-y |ffy = ff,-,-, 1 < i, j < N} (14)

on the strain, namelya = a(s) = A£ + o(s2), (15)

where A :— {a^j/ 6 K|a, jfc/ = a,jik\ = akuj , I < i,j,k,l < N} are the elasticcoefficients. In the linear case, adopting the Einstein's summing convention(i.e., the repeated index means summing over that index), we have the followingrelations

(16)

known as the Hook's law. The coefficients A must depend continuously on thepoint in 17 or they are constants (the hyperelastic case) and they satisfy theellipticity condition

aijk,£ij£kl > c\e 2, Ve e RN*N,e = £T , (17)

where |- | means the Euclidean norm.Let X := [L2(Q)]N be the Hilbert space of square integrable vectorial func-

tions endowed with the inner product

[(u,v) = / UiVi

Jnand de0ne on

D(B) :={ve[c2(the operator B of the linear elasticity

B\ = — divcr(£(v)),

where the components of <r are given by (16).By Green's formula we have

(5u, v) := / divff(£(u)) • vdx =7, aijki£ki(u)£ij(v)dx =n n

o / a k i i j £ i j ( v ) £ k i ( u ) d x = - diva(£(v))udx =: (u,Bv), Vu,v € D(B).n

Moreover, we have

(5u,u) = - / aijki£ij(u)£ki(u)dx > - I £ij(u)£ij(u)dx = c I |Vu|2cfe.n n n

The energetic space E is the completion on D(B) with respect to the norm

I f I(18)

*•

and it contains all vector functions from [//^(fi)]^ that vanishes on T.Moreover, the norm (18) is equivalent with the following one

and thus, by Sobolev-Kondrashov theorem, the embedding E <^-> X is compact.Hence the duality mapping J : E —* E* is de0ned by

1 r< Ju,v>=- I aijkieij(vi)£ki(v)dx (19)

fi

and, by the Theorem 1:

I f f- I aijk,£ij(en)£k,(v)dx = \n I (en)iVidx.

</ «/n n

By Green's formula we deduce that en are the weak solutions of the eingen-value problem:

—divu(£(en)) = Anen in fi,en = 0 on T, (20)<rij(e(en)}vj = 0 on <9Q\r. •

Consider now the semilinear eingenvalue problem in X

Lit + yuTV(w) = /, (21)

where L G L(X) is compact and positive, TV is a nonlinear perturbation of Leray-

Schauder type (i.e., TV :— I — T with T a compact operator), and / G X a givenelement. Since L is, in particular, hemicontinuous and monotone it is maximalmonotone and, thus, / + AL is invertible for each A > 0 and ||(7 + AL)"11| < 1(see e.g. D. Pascal! and S. Sburlan [7, p.106]).

Consequently we can write (21) equivalently as

(I-M(X))u = g (22)

where M(A) := (/ + AL)~ :(7 — TV) is a compact operator, A = — G ffi+ andg = (7 + \L)~lf. This it allows to introduce the coincidence degree for the pair(L, TV), simply, as follows:

If D C X is an open bounded set such that

N(u) + XLu ^ f V u G 3D, A G M+ (23)

then the coincidence degree of the pair (L, TV) in D relatively to / is de0ned by

dx ((L, TV), D, /) := dLS (I - M(A), D, g } , (24)

where d^s denotes the Leray-Schauder degree. Of course, this degree has allthe properties of Leray-Schauder degree, because of the above mentioned equiv-alence, and we can apply all classical results to (21). We point out here somein the linear case (see Mortici [4]):

Suppose that TV G L(X). We say that A G M+ is a characteristic value forthe pair (L, TV) provided that ker(TV + AL) 7^ {0} and we say that A is a regularcovalue of (L,N) if the resolvent 7?(A) := (TV + AL)"1 exists on X and it iscontinuous.

Since, from the above equivalence, we have

ker(TV + XL) = ker(/ - M(X)) (25)

it results that each A > 0 is a characteristic value or a regular covalue for(L,N).

Denote by C(L, TV) the set of characteristic values of (L, TV) and by La thespectral operator for (L, TV)

La := (N + aL)~lL = R(a)L, (26)

and observe that La is compact whenever L is compact. From the identity

(X-a)La] (27)

we see that TV + XA is invertible if and only if a — A is not a characteristicvalue of La. Hence from Fredholm theory it results that C(L, TV) is at most acountable set with only one possible acumulation point at in0nity, as we haveseen in the theorem 1.

Suppose now that a is a 0xed covalue of (L, TV) and de0ne the multiplicitym(A) of the characteristic value A 6 C(L, TV) to be the algebraic multiplicity ofthe eingenvalue (a — A)"1 of the compact operator La.

Let a G D be an isolated solution of the equations (21) and choose anopen ball B(a, r) such that R(X)f fl B(a, r) = {a}. By the excision property ofthe coincidence degree we can de0ne the coincidence index of (L,N) in a withrespect to / to be

ix((L, TV), a, /) := d x ( ( L , N),B(a, r), /) (28)

and, by the above equivalence, it is true the following tranversality property:

Proposition 1 (Leray-Shauder) // [Ai, A2] f~l C(L, TV) = {A}, then

iXl ( ( L , TV), a, /) = (-l)">WiAa((i, TV), a, /).

In the semilinear case it apperas naturally the following question: Are thewell known results of Krosnoselskii and Rabinowitz, concerning bifurcation the-ory for compact operators, true for monotone type operators? The answer isYes! and was given in [5] in the following way:

Let X be a re/Eexive uniformly convex Banach space and X* be its dualBanach space. As usually we denote by || • || the norm in both spaces, by< •, • > the duality pairing and by J : X —>• X* , the duality map.

Consider the eingenvalue problem

Jx + fj,Ax + R(n,x) = 0, (29)

where A : X — > X* is a linear continuous operator and R(/J,, •) : X — -*• X* isa nonlinear perturbation such that R(n,Q) = 0, V/u G K. Note that only J ismonotone.

In this case (//, 0) are solution of (29) for all n G M-named trivial solutionsand the set of all trivial solutions are denoted by C.

A point (/J.0,0) 6 C is said to be a bifurcation point for (29) provided thatthere exists solutions (/J,a^), x^ ^ 0 in each neinghbourhood of (n0_,0). Letus denote by So the set all of these nontrivial solutions and let 5 := So be itsaderence in M x X.

The key step in our extension is the Browder-Ton theorem concerning thecompact imbedding property for separable Banach spaces (e.g. D. Pascali andS. Sburlan [7,p.302]):

Theorem (Browder-Ton). Let X be a separable rejEexive Banach spaceand let S be a countable subset of X. Then there exists a separable Hilbert spaceH and a compact one-to-one linear operator ^ : H — »• X such that S C tl>(H)and ijj(H) is dense in X.

De0ne now the adjoint operator <f> : X* — »• H by using the inner product ofH denoted by (•, •) :

<<{>u,v>=(u,il>v),Vu<EH, v&X*. (30)

Then the operator L := —il><f>A : X —>• X is linear and compact as thecomposition of a continuous map with a compact one. Since the spectrum cr(L')is discrete we can choose 6 > 0 such that

tr(L) n (/i U 72) = </>,where I\ := (en0 — 6,e/J0) and 72 := (ep0,£fi0 + 6) and /j,0 G <r(L).

Let /j,l 6 7i and //2 G h be arbitrary 0xed. Then the mappings

, i=l,2are regular and thus there exists M > 0 such that

— i a ; > M \\x\\, \/x<=X, i= 1,2. (31)

Suppose that A is bounded from below in the sense

<Ax,x>>- —— - — -||*||2, MX 6 x (i)

and the complementary part is asimptotical zero, i.e.,

uniformly in [i on bounded sets. Using the following Leray-Schauder homotopy

h t ( x ) := x + -ipipJx + -uibipAx, t G fO, 11, x e X. (32)e s

we easily obtain

__ . I I 7* I p <* A™ T> \<^ __ . l l - r l l ^C ^ /T-ti* . iL s* —

which is a contradiction, (C. Mortici and S. Sburlan [5]).By homotopy invariance of the Leray-Schauder degree we can conclude that

/ I \ ( \ \dis 7 + -/uV"M, 5 , 0 = dis 7 + -^$(3 + p.A), 5 , 0 .

Let now use the Berkovitz de0nition of degree for (5+) mappings (e.g. S.Sburlan and G. Moroanu [10])

ds(J + nA, B, 0) = dLS(I + -^M, B, 0) (33)

and consider the mapping TL de0ned by

Since L is a compact map and, by (33)

where m(/u) is the sum of algebraic multiplicity of all eingenvalues A > - of L.It then follows that f(n^) = — <^(//2) whenever the eingenvalue AO = -^- hasodd algebraic multiplicity because it appears only in the expresion ofTherefore in this case

ds(J + KA, B, 0) ds(J + n2A, B, 0). (34)

Take now the Leray-Schauder homotopies H\ : [0, 1] x X -— > X, i — 1, 2

Hi(x) = (I-^L)x + -^(Jx + R(^,x)), t € [ 0 , l ] , x€X,£ C

and therefore,

dLS(I - ^L, B, 0) = dLS(I + i( J + A + R(^, •)), B, 0), » = 1,2.

Combine this equalitty with (34) to produce

ds(J + A + R ( V I , - ) , B, 0) ds(J + n2A + R(»2, •), B, 0).

Join linearly fj,l with /K2 by nt := (I — t)^l + t//2 and consider the (5+)

homotopyX t x ) := Jx

Hence it could exist T G [0, 1] and xr G dB such that

JxT + nTAxT + R(nT , XT) = 0,

that is the equation (29) has nontrivial solutions in any neighbourhood of(e/ /o> 0) G M x J5^, and thus (efiQ, 0) is a bifurcation point for (29).

Thus we have proved an analogous of Krasnoselskii theorem for monotonetype operators (e.g. S. Sburlan [8]).

Theorem 2. (Krasnoselskii). Let /j,0 be a characteristic value with oddalgebraic multiplicity of the linear compact operator L G L(X).

If there exist £, 6 > 0 such that (i)-(ii) hold, then (£/i0,0) G R G X is abifurcation point for (29).

Example 2. If consider R ( f i , x ) := / j , \ \ x \ \ 2 J x , then

,x),x >= H\\x\\2 < Jx,x >= H\\x\\4.

The above results can be applied in Sobolev space X :— H1^). The corre-sponding operator is — Aw + /i | |u| |Aw or p-laplacian operator in the general caseX := Wl>p(tt). m

Let us denote

i- :=ds(J +LiA,B,Q) = dLS(I + -^(j)A, B,0), fi G h,

i+ :=ds(J + nA,B,Q) = dLS(I+-^(t>A,B,Q), n 6 72,

and observe that these degrees are constant in /i1 G /i and //2 € h-For any 0xed r > 0 de0ne the mapping Hr : M+ x X — » M+ x X* as follows

#r(//, x) := (\\x\\2 - r2, Jx + nAx + R(n, x ) ) , V(//, x ) < E R x X

and we have a formula similar to Ize's formula:

d.(Hr,B,Q) = i--i+, (35)

where B = {(v, x) £ R x .\>2 + ||a;||2 < <52 + r2}.Indeed, by de0nition of (S+) degree (e.g., S. Sburlan and G. Moroanu [10])

we have

where

Ur(fi, x) := (\\x\\2 - r2, (I - pL)x - N(n, x)), V(//, x) e M x X,

and, x ) ) = o(\\x\

We can now prove a global result concerning the bifurcation under monoto-nocity condition similar to those under compactness condition proved by Rabi-nowitz.

Theorem 3 (Rabinowitz). If S is a connected component containing thebifurcation point (e//0, 0) G S, then we have one of the following two posibilities:

(j) £ is unbounded in M x X .(jj) £ contains a 0nite number of bifurcation points (e/^-,0) where

?t e "W-Moreover, the number of these points, including (£/^0 ,0), is even.

Proof. Suppose that £ is bounded in M x X and let G be a, bounded domainthat contain £ and a 0nite number of points (e^0,0). Moreover, suppose thatthere is no solution of (29) on 3D. In this case d(Hr,G,0) is well de0ned andindependent of r > 0. For r enough small

where Gj are disjoint and each Gj contains only one point (s/j,j,0). It thenfollows from (35) that

where the sum has only an even nonvanishing terms becausei*- (j) — J+ G {0, ±2}. If mj is the algebraic multiplicity of e^ij , then

i+C/) = (-l)m''-(j),that is i-(j) ^ i + ( j ) , which is possible if and only if ep,j has an odd algebraicmultiplicity. •

We conclude our work with the following

Example 3. Let g : I x M2 — » M be a bounded continuous function

\g(t,t,ri)\<M, w e / , (£,??) e M2

and consider the eingenvalue problem

u"(t) + \u(t) + g(t, u(t), «'(<)) = f ( t ) , i € /(£«)(*) = 0, i e <9/.

where J := [0, 1] C M, B denotes either Dirichlet boundary conditions

u(0) = w(l) = 0

or Neumann boundary conditions

u'(0) = «'(l) = 0,

or periodic boundary conditions

w(0) = «(1), u'(Q) = u'(l).

Such problems was extensively studied in the last time (see e.g. Drbek [1]).Take X := L2(I), L : D(L) ->• X with D(L) := {u e C2(I); (5w)(i) = 0,

< G $/} de0ned by LM := —u", N(u) : X —> X the superposition operator(N(u)(t) := g(t,u(t),u'(t)) and / £ C(/). With these notations we arrive to thefollowing equation in X.

Lu + \u + N(u) = fwith L G L ( X ) symmetric and positive.

Since L is invertible withi

(Au)(t) = L~lu(t) := f G(t, s)u(s)ds, u £ X, t € [0, 1]o

where G : [0, 1]2 — »• M is the continuous function

it results that A is compact and the above equation can be written as

u + XAu + N(u) = Af.

Now we can apply the above results. •

REFERENCES

[1] P. Drbek, Solvability and Bifurcations of Nonlinear Equations, PitmanRes. Not. Math., 264, Longman, London, 1992.

[2] A. Haraux, Nonlinear Evolution Equations. Global Behaviour of Solutions,Lect. Notes Math., Vol. 841, Springer - Verlag, Berlin, 1981.

[3] G. Morosanu and S. Sburlan, Multiple Orthogonal Sequence Method andApplications, An. St. Univ. jOvidiusj Constanta, 2 (1994), 188-200.

[4] C. Mortici, Bifurcations for Semilinear Equations with Compact Nonlin-earities, Bull. Appl. Com. Math., BAM-1714/'99XC-B, pp.265-272.

[5] C. Mortici and S. Sburlan, Bifurcations for Semilinear Equations of MonotoneType, An. Univ. Ovidius Constanta, vol. 7, (1998) fasc.2, pp.

[6] J. Neas, Les Mthods Discretes en Thorie des quations Elliptiques, Ed.Academia, Praque, 1967.

[7] D. Pascal! and S. Sburlan, Nonlinear Mappings of Monotone Type, Sijhoce& Noordhoce Int. Publ., 1978.

[8] S. Sburlan, Gradul Topologic. Lecii asupra Ecuaiilor Neliniare, Ed.Academiei, Bucureti, 1983.

[9] S. Sburlan, Luminita Barbu and C. Mortici, Ecuaii Difereniale, Integratei Sisteme Dinamice, Ex Ponto, Constanta, 1999.

[10] S. Sburlan and G. Morosanu, Monotonocity Methods for Partial Dioeeren-tial Equations, MB-11/PAMM, Budapest, 1999.

[11] E. Zeidler, Applied Functional Analysis, Appl. Math. Sci., 108, Springer-Verlag, Berlin, 1995.


Implied Volatility for American Optionsvia Optimal Control and Fast Numerical Solutions of

Obstacle Problems

Srdjan StojanovicDepartment of Mathematical Sciences

University of CincinnatiCincinnati, OH 45221-0025

[email protected]

http://math.uc.edu/~srdjanhttp://CFMLab.com

December 2000

• 1. Statement of the Problem

Let S(t) denote the price S of a particular stock at the time /. We suppose that the price evolves on the stock marketaccording to the stochastic differential equation

dS(t) = 5(0 (a(t, 5(0) dt + cr(t, 5(0) dB(t)}

where a(t, S) is the appreciation rate, <r(t, S) is the volatility, and B(t) is the standard Brownian motion. An Ameri-can call option is a contract that entitles the holder to buy the stock at the prescribed strike price, denoted k, upuntil the prescribed expiration time, denoted T. An American put option is a contract that entitles the holder to sellthe stock at the prescribed strike price k, up until the prescribed expiration time T. Although an option does nothave to be exercised, if it is, the time of the actual exercise is a stopping time r < T. We shall assume for simplicitythat the underlying stock does not pay the dividend. It turns out that under such an assumption the optimal exercisetime for the American call option is T = T (or alternatively, option is let to expire). Therefore we concentrate ourattention to American put options. Suppose the holder of the put option does not own the stock, and wishes toexercise at some time T. That means he/she buys the underlying stock on the open market at the price S(T) and thensells the same stock at the strike price k. The payoff for the holder is equal to k - S(r). Moreover, since option doesnot have to be exercised, the payoff is never negative - i.e., the payoff is equal to 0(S(r)) = Max(0, k - S(T)).

According to the celebrated Black-Scholes theory [2] (treating European options), the fair price of an American putoption with fixed strike price k, and fixed expiration time T, as a function of the current running time t and thecurrent price of the underlying stock S, is the unique solution of the (degenerate) backward parabolic obstacleproblem:

8i/i(t, S) + 1 s2 d <),(t S) 2 +/.5 d<K<' fl _,^ ( < > S) = Q in# = ((/, 5); W, 5) > Max(0,*-5)}

Ot 2 (7O OO


http://math.uc.edu/~srdjan/

http://cfmlab.com/

tf/(t, S) > Max(0, k-S)

<fi(t, S) = Max(0, k-S) on T = dN f~\ {(I, S);t<T,S>0]

together with the terminal condition

i/i(T, S) = Max(0, k-S)

where r is the interest rate, Max(0, k - S) is the obstacle as well as the terminal condition, N is the non-coincidenceset, T = dN n {(t, S);t<T,S> 0} is the free boundary. Notice that no boundary condition is imposed, due to thedegeneracy of the governing partial differential equation. The option trading significance of the free boundary isthat if the price of the underlying stock drops below certain threshold (the free boundary T), then it does not makefurther sense to hold (or trade) the particular put option, and instead, such an option should be exercised.

On the other hand, according to the Dupire theory (see e.g. [3] for the case of European options), the fair price ofan American put option with fixed underlying stock price S, at fixed time t, as a function of the expiration time Tand the option strike price k, is the unique solution of the (degenerate) forward parabolic obstacle problem (writtenin the free boundary problem form):

V(T,k)>Max(0,k-S)

V(T,k)= Max(0, k-S) onF = dNf] {(T, k); t0 < T, k > 0)

dV(T,k) , „-1 = 10n r

together with the initial condition

V(tt>, k) = Max(0, k-S)

The trading significance of the free boundary now is that for the given current price of the underlying stock S, onlyoptions with strikes below the free boundary should be considered for trading. It is remarkable that the samevolatility function or, with different arguments, appears in both problems (as well as in the underlying stock priceevolution SDE above). Also notice that in either one of the problems the underlying stock price appreciation ratea(t, S) does not appear (but instead the known interest rate r does), while volatility cr is of the deciding importance(options with different underlying volatilities have significantly different prices). It is then of paramount importanceto have a precise estimate of the volatility cr.

Volatility of the underlying stock price cr can be estimated from the historical statistical data accurately. Neverthe-less, the volatility changes, and in the above equations it is the future volatility that appears and not the current orpast. Moreover, knowing or having a good estimate of the future volatility may be useful in trading for otherreasons, as well. Therefore it is of great importance in trading options or even stocks, to have a reliable and effi-cient technology for estimating future volatility cr based on the current prices of the whole variety of the correspond-ing call and put options. For example, for the underlying QQQ (Nasdaq-100 Index Tracking Stock), AS OF DEC26, 2000 10:46:20 AM (E.T.), the collection of sufficiently liquid put options with expiration in January and

February of 2001 (T = 1.05593, 1.13259; time is measured in units of years, and we start at the year 2000), withdifferent strike prices (the second column), had the following market prices (the third column):

.05593

.05593

.05593

.05593

.05593

.05593

.055931.055931.055931.055931.055931.055931.055931.055931.055931.055931.055931.055931.055931.055931.055931.055931.05593,1.05593

485051525354555657585960616263646566697071758386

0.7812511.031251.18751.3751.6251.875

2.18752.3752.75

3.1253.68754.31254.68755.18755.81256.3756.8757.6259.87510.7511.5

14.87522.7525.75.

c 1.132591.132591.132591.132591.132591.132591.132591.132591.13259.1.13259

48505255565960657580

1.62512.1252.6253.68754.06255.31255.81258.812515.812520.1875,

(Notice, by the way, how much more trading activity is performed for options that expire sooner rather then later.)The question is what can be said about the (market consensus about the) future volatility cr of the underlying stockprice based on these recorded prices. The present paper describes a methodology for answering that question basedon the optimal control theory for obstacle problems, as well as on a fast numerical solution of the Dupire obstacleproblems. Some alternative methods can be found in [3].

• 2. Some Remarks on Obstacle Problems

We recall some known facts about obstacle problems, introduce some new (see also [6,8]), as well as a method forsolving obstacle problems, employed in our fast numerical solution of the Dupire obstacle problem.


• 2.1. Various Formulations of Obstacle Problems

Consider, for the sake of simplicity, an obstacle problem for the Laplace's operator and with zero obstacle. Every-thing that follows can be generalized to the case of arbitrary, sufficiently regular obstacle, as well as to the case ofan arbitrary elliptic (or parabolic) operator with smooth coefficients. Obstacle problem can be formulated in manydifferent ways. Arguably, the most popular way is the variational inequality formulation. Let fl c R" be a boundeddomain with regular boundary, let (-/) e £2(fT) be the right hand side, and let g e //'(H), g > 0 be the boundaryvalue. Denote also //g(H) = g + //o(fi), and K = (v e //g(H), v > 0}. The variational inequality formulation of suchan obstacle problem is:

Find v e K, such that

for any ip e K. It is worthwhile emphasizing that the obstacle appears explicitly as a constrain! imposed on allfunctions considered as possible solutions.

As it is very well known, such a problem has an unique solution. The variational inequality formulation is usefulbecause it allows an easy variational proof of existence of the weak solution, as well as its uniqueness. Further-more, the higher regularity of the weak solution can be established posteriori via additional arguments. Indeed,under above assumptions v e 7/£c(fl) (see e.g. [5]). Increasing the regularity of the right hand side yields higherregularity of the solution, but not beyond ^O'"(fl) (unless obstacle does not affect the solution). The solution v,being non-negative, determines two distinctive regions in ft: the coincidence set A = (v = 0) D Q and me non"coincidence set N = {v > 0) f| ft. The boundary separating the two T = d{u > 0( f| H is called free boundary. Ingeneral, free boundary is not a smooth surface no matter how smooth is the right hand side (see [4] for fundamentalresults ensuring smoothness of the free boundary under additional assumptions; see also [5]).

• 2.1.2. Semi-Linear PDE Formulations of Obstacle Problems

Here are couple, not so well known, formulations of the same problem, introduced and utilized by the author in[6,8].

1, x e ALet IA(X) be a characteristic function of the set A, i.e., let IA(x) = { . Also let1 0, x $ Ah+ = Max(/z, 0), h~ = -Min(h, 0) be the positive and negative parts of h, so that h = h* - h~. The above obstacleproblem can be formulated as:

Find v e H*(£l) f| H^(fl) such that

-Av + //(v>0) = 0/" /(v=0| = 0

All the equalities are to be understood as equalities almost everywhere.

The above obstacle problem can also be formulated as:


Find maximal v e //j(ft) f| //,2c(ft) such that

-Av + //,v>0) =0

Alternatively, let

-W.* = f w e tf](n) 0 tfi2oc(«)> -A v + / /,,,>0) = 0)

then

v(x) = MaxÂ-, AU w(x).

X\fljg is typically a small set. All the elements of X\ fljg are non-negative functions due to the maximum princi-ple, and if/ > 0 then X\fijg is a singleton consisting of the solution of the obstacle problem. Notice also that thenon-negativity follows although no constraint is imposed explicitly in either of the above two formulations.

For example, let ft = (0, 1), g= 1, f(x) = -87T2 cos(47rx). Then î,n,/,g has two elements:wiM = 7 (cos(4 n x) + 1) and w2 (x) = j (cos(4 nx) + 1) <±)U(jt>i)> me fifst one being larger, and therefore beingthe solution of the obstacle problem. Notice also /" /(W2=o) = -8^2cos(4wx)/(|<.t<||W * 0.

• 2.1.3. Classical Free Boundary Problem Formulation

Assume / is sufficiently regular. The free boundary problem can be formulated as:

Find an open set N c ft and v e C2(AO f| C\N), such that

v > 0

and

- A v + / = 0

in N, such that

v = g

on a N 0 3ft, and such that

and

Vv = 0

on dN fl ft- The last two conditions are called free boundary conditions.

Notice that if / is sufficiently regular, any element of î,n,/,g is a solution of the free boundary problem. Thisimplies in particular that the free boundary problem formulation does not yield necessarily an unique solution. Inthe above example, both w\ and W2 are solutions of the free boundary problem.

• 2.2. Maximum Boundary Value Formulation of the Obstacle Problem

For each open N c R", let Aw = R" - A'. Now let

g,w |AV = 0}

Notice that w e H^£N(fl) has more then one extension, i.e., representation in //^(R"). Further, let WN be theunique function in //(jgA,(i"l) such that

-AWJV +/ = 0

Notice that if N f~) fi = 0 then vf// = 0. On the other hand, in order to see non-trivial examples of WN, letn = (-l, 1), g = 1 ,/(*) = -10+ 20/(,>(» and let ty = (-co, }), jV2 = (-l, oo). Then

t.~ . •, . •, 141x 21\ .28 28^

which looks like

while w^OO = (10 /(r>0) x2 - 5 - - |i - ±) , which looks like


It turns out that the pointwise maximum of functions such as these, yields the solution of the obstacle problem,which by the way in this example is equal to

V(JC) =

and looks like

i ]+ rrn + (5*2 + 2 (-5 + ^ 't«'-vr)

-0.5 0.5

So, formalizing, let

XHJJS = (WN, N c R")

Compared to X\fijf, Xfij^ is much larger. Also, one should notice that, like in the examples above, depending onN, as opposed to X\fijg the elements of X&jf take negative values, i.e., values below the obstacle, as well.

As announced, the obstacle problem can be formulated, and/or solved, simply as

v(x) = Max.reA^ w(x).


Before proving this claim, it is instructive to visualize this maximization (on the same example), together with thesolution, and its free boundary:

The following simple facts may have practical implications for numerical searching of the free boundary

( v > 0 ) = l I ( w > 0 )\JweXnsj

or alternatively,

(v = 0} = n (w<0)I \wsXn.u

and consequently, for any Y c Xn

{v > 0) D (J (w > 0)

and

In particular, for any Y c Xnj-g, the set fXe)' {w s 0} contains the free boundary d{v > 0} (~) ^- In order to provethe above claim, since v e Xnjg, we need to show only jv > 0} D Uwe^n, fw > "'• ^et * e (w > 0} for somew e Xftj£. This implies v(x) > w(x) > 0, and therefore x e {v > 0}.

This formulation-solution of the obstacle problem can be compared - judged against the classical free boundaryproblem formulation. The free boundary problem can be viewed as the first order necessary condition for the abovemaximization problem. The computational efficiency of the present method lies in the fact that for each N, comput-ing the corresponding WH is a linear problem. Still, in general, especially in higher dimension, choosing proper A"s,i.e., choosing sufficiently representative finite family Y c ^n,/,g mav indeed be difficult, or at least additionalissues will need to be addressed there. Nevertheless, in special cases such as the Dupire obstacle problem, when thegeometry of the solution and of the free boundary is well understood a priori (in spite of being unknown), the abovemaximization can be done very efficiently.

• 2.3. A Proof of the Equivalence

We shall prove that if

v(x) = MaxweXn.fg w(x).

and if z e H^(Cl) f) H?x(n) satisfies

/(z>o) = 0

then v = z. Indeed, since z e An_/,£, all what is needed to prove is that if WN e A'n./.g, then z > w^. To that endrecall

IJnrw

for any ip e //0' (H f| AO, and

f(Jn

for any y e H^(Cl). Consider (WN - z)+ e H^(fi). Notice also (ww - z)+ lnnAA. = 0. Therefore

f (Vww-V(ww -z)+ +//A, (WAf -z)+)dx = 0Jn

and

f (Vz- V(ww -z)+ +//(z>0) (wN -zDdx = 0Jn

Subtracting

0= f( |V(w^-z)+ |2 +(/+-/-)(/w-/(r>o))(wA,-z)+)^ =Jn

f ( I V(WN - z)+ |2 +/+ (1 - ;,z>0)) (WAT - z)+ - /" (/„ - 1) (WN - z)+) d^ > f ( | V(wN - z)+ |2) dxJn Jn

since /" /(_->o) = /" /(:>oi = /" , yielding V(wN - z)+ = 0, and therefore (WN - z)+ = 0, and finally z > WN.

m 3. Fast Numerical Solution of the Dupire Obstacle Problem

Back to the Dupire obstacle problem (free boundary problem formulation):

Y(T, k) > Max(0, k-S)


V(T, k) = Max(0, k-S) on T = <9 W fl ((T, k); ta<T,k>0}

d V(T, k)dk • = 1 on F

together with the initial condition

V(t0, k) = Max(0, k-S)

Under some theoretical assumptions, that we shall not be going into here, the Dupire obstacle problem admits anunique solution.

This problem is approximated by the time finite difference variant:

• V(T, k)-V(T-dT,k)\ 1 , d2 V(T, k) , d V(T, k)• v ' ' rr(T ]f\ r I- —~ , ~——— u \* > >*) ~ t K. ———~——— —ok2- ok-\+-k2

V dT I 2 6P0 inN = {(t, S); V(T, k) > Max(0, k-S)}

V(T, k) > Max(0, k-S)

V(T,k)= Max(0, k-S) onT =

dV(T,k)

, k); ta < T, k > 0)

dk = 1 onT

i.e., by the family of 1-dimensional elliptic obstacle problems, that are solved (numerically) in succession. Forexample, if <r(T, k) = o-stat = 0.531349 (QQQ statistical historical volatility based on daily closse between March10, 1999 and December 22, 2000), then it rums out that the corresponding solution looks like

150

100

50

while, more precisely, its free boundary looks like

80

Free Boundary

1.02 1.04 1.06 1.

The geometry of the solution is very simple: the free boundary is a graph of a function, i.e.,T = {{T, g(T)}, {T, to, t\}}; moreover function g is strictly increasing (initiating at the price of QQQ AS OF DEC26, 2000 10:46:20 AM (E.T.)) . Those two facts can be exploited in the search for the maximal boundary valuesolution of each 1-dimensional elliptic problem, like in 2.2.

• 4. Optimal Control Problem

The above Dupire obstacle problem can be written more precisely, for example in its complementarity form

d V(T, k} 1 2 d2 V(T, k) 2

— — d T ~ + 2 k dV <T(T'k) ~'

y(T,k)>Max(0,k-S)

together with the same initial condition above. Also, for the purpose of numerical solutions, we need to restrict theconsideration on a finite domain, say {(T, k);t$<T < t \ , k f , < k < k \ } , for some 0 < &o < k\, with some appropriatetruncation boundary values at k = k$ and k = k\.

Let the class of considered volatilities be:

AT = (cr e i2((r0, 0, f ' f [( dT dk

For each <r e K let V(cr) denote the unique solution of the Dupire obstacle problem (this is formal, and motivatedby the numerical scheme; for the sake of theoretical results it might be necessary to impose more restrictions on theset of admissible volatilities, which in turn would make the numerical scheme less natural). So, from the discreteoption price observations, we construct the target functions vt(k), i = 1, .., n, that the solution of the Dupireobstacle problem is supposed to match. So n is the number of different expiration times considered. We adopt thecost functional

JW = T-/do-(T, k)\2

,(-^)+,2«r(r,

where e\, £3, qi(k) > 0 are given and crslat is the statistical estimate of the constant volatility.

The optimal control problem is to find

a-aft e K

such that

•/(^opt) = Mm(J(cr), creK)

m 5. Derivation of the Minimization Algorithm

The state equation being what it is, the functional J(<r) is not differentiable, but rather only Lipschitz continuous (inan appropriate sense). This is a very well known phenomenon in the theory of optimal control of variationalinequalities (see e.g. [1,7],). Nevertheless from the practical point of view, from the point of view of designing aworking numerical optimization algorithm, this is not a problem at all. Indeed, for a fixed <T<=K, such that J isdifferentiable at cr (see [7]) and a such that a- + So" e K, compute the directional derivative of J in the direction a":

J' (a-; ZT) = lira w / v ' = > (K(r/, k) - v,(sj) w(T,, k)q,(k)dk +s_>0 s / , i /i/i. v

+

/o h, \ dk dk I

where

,T M i-w(T, k) = Mm<5->0 S

is the unique solution of the forward equation

dw(T,k) I 62w(T,k) , dw(T,k) l 2 d 2 V ( o - ) ( T , k ) , „„_ ,„„——dT~ + 2 k —M- a(T> k)~rk —ST- = ~* —— dit —— k)<T(T> k}

in the non-coincidence set # = {(T, k); V(cr) (T,k)> Max(0, k - S)} [~] {(T, k); I0 < T < t\ , kg < k < k\ }, and

w(T, k) = 0

in the coincidence set A = {(T, k); V(cr) (T, k) = Max(0, k - S)}. Let for any !<( '<« , p,(T, k) be the uniquesolution of the backward equations (not obstacle problems) in the non-coincidence region:

(a-(T, kf +k(4 o-(OJ)(7", k) + k o-(0'2)(7", k)) cr(T, k) + k2 o-(0'[\T, k)2 + r) p,(T, k) = 0

for N 0 {(T, k); t0 < T < T,}, with the terminal condition

Pi(Ti, k) = -(V(<r)(T,, k) - v,(k))q,(k)

and (non-cylindrical) boundary condition

on the lateral boundary of N f| {(T, k); t0 <T < 71,}. Each p, is furthermore extended as zero outside of the non-coincidence region. By means of integration by parts, this terminal-boundary value problem is equivalent to itsweak formulation:

JJ _____). + l

Nr\{(T,k);l,<T<T,\

Ti, k) - Vi(k))qi(k) <t>(T,, k)dkJka

- r<f>(T:,k)Pl(T,k)dk= f ' (Jka Jka

for any smooth test function (/> such that 0 = 0 on the backward parabolic boundary of the non-coincidence set.

In particular, if 0 = w,

Z JJ

JJaV(r, k) .

e, ——-^— + to (a-(T, k) - o-5tat) v(T, k)dkdT +ffJl0 Jka \

I r" da-(T, k,) r"eU — — H T — — c r ( T , k i ) d T -U,0 dk j,0 dk

f fPrrtT k\', k) - o-stat)

On the other hand, there exists z such that

W7.2

Since /'(cr;a°) = VJe(cr)-o° for any admissible a, we conclude that, for any T, z(T, •) is the solution of, andconsequently can be computed as a unique solution of the boundary value problem for an ordinary differentialequation (regularization equation):

d2z(T, k)dk2 eo z(T, k) = -k2 -o-(T,k)}_i(Pi(T,k)X(,{,,n.

/=i

ff-<r(T, k)H to (<r(T.', k) - crslat) I

with boundary conditions

dz(T, kg) = da-(T, kg)dk ~ dk

dz(T,k,) = da-(T,k,)dk ~ dk

We summarize the steepest descent minimization algorithm. The single iterate is done in 4 steps:

1) for given cr(T, k), V(cr)(T, k) is computed as the unique solution of the Dupire obstacle problem;

2) pi(T, k), i = l , ..., n are computed using V(cf)(T, k) as solutions of the adjoint equations in the non-coincidence region;

3) V(<r)(T, k) and all of the p,(T, k)'s are used in the regularization ODE's computing the gradient z(T, k);

4) a-nextC?1, k) = a-(T, k) -pz(T, k), for some p > 0.

• 6. Example: Put Implied Volatility for QQQ (Nasdaq-100 IndexTracking Stock)

We proceed with the data shown in Section 1. The first step is to construct the target functions v,(A), i = I, 2, thatthe solution of the Dupire pde is supposed to match. They look like

vilkl (Expiration) T = 1.0559325

55 60 65 70 75 80 85k (Strike)

V2 [ k ] (Expiration) T = 1.1325920

17.5

15

12.510

7 .5

60 65 70 75—— k (S t r ike )

The steepest descent method is initialized by selecting the volatility function to be a statistical volatilitycr(T, k) = o-stat; see above. After a number of iterations, on two different grids, the optimal volatility function, as afunction of strike and expiration time looks like

150

100

50

The corresponding solution of the Dupire obstacle problem, together with the free boundary, and together with theobserved prices, looks like

1.1


One can notice two features above. One is the reduced smoothness of the free boundary, as compared to the onepresented before in the case of the constant volatility. The second feature is that the observed option market pricesare all below the computed free boundary, as theory postulates: options with strikes above the free boundary needto be executed, and not be held or traded.

Proceeding, the corresponding gradient (the smallness of the gradient is the measure of the success of the minimiza-tion procedure; more iterations on the same grid, as well as going down on the finer grid and performing a numberof iterations there, would reduce the gradient, and the precision of <r) was

150

100

50

while the solution of the Dupire obstacle problem was matching the targets quite well:

25

20

15

55. 65 70 75 80 85

20

15

75 80

Finally, in order to concentrate our attention on time dependence of the implied volatility, we average the computedimplied volatility for each time. The average, superimposed with points representing times of expiry (January andFebruary 2001), looks like

It is interesting to compare the term structure of the put implied volatility, with the one computed for calls (insteadof the Dupire obstacle problem, Dupire partial differential equation and it's optimal control was used) on the sameunderlying (the constant statistical volatility is plotted, as well). The comparison, i.e. the mutual relationshipbetween the two may suggest some insight about the option traders short term outlook of the (Nasdaq) market:

AS OF DEC 26, 2000 10:46:20 AM ( E . T . ) $60.25

(Indeed, if an option trader expects QQQ price to increase, he/she would like to buy a call option (sell a putoption). If many option traders share that opinion, the price of the call will go up, and price of the put (on the sameunderlying) will go down, and consequently its implied volatilities, as well. So, the difference in implied volatilitiesfor calls and puts on the same underlying can be attributed to such a scenario, and consequently, can be exploited.)Although the market outlook sometimes changes suddenly, later that same day, the outlook did not look muchdifferent,


AS OF DEC 26, 2000 4 :15:22 PM ( E . T . ) $60 .88

Many more details, including the complete implementation of the above algorithm, can be found in [9].

• References

1. V. Barbu, Optimal Control of Variations! Inequalities, Pitman, 1984.

2. F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Pol. Econ. 81 (1973), 637-659.

3.1. Bouchouev and V. Isakov, Uniqueness, Stability, and Numerical Methods for the Inverse Problem that Arisesin Financial Markets, Inverse Problems, 15 (1999), R1-R22

4. L. A. Caffarelli, Regularity of free boundaries in higher dimensions, Acta Math. 139(1977), 155-184.

5. A. Friedman, Variational Principles and Free Boundary problems, Wiley, 1982.

6. S. Stojanovic, Remarks on W^-solutions of bilateral obstacle problems, IMA preprint #1318, University ofMinnesota (1995).

7. S. Stojanovic, Perturbation formula for regular free boundaries in elliptic and parabolic obstacle problems, SIAMJ. Control AOptimiz., 35 (1997), 2086-2100.

8. S. Stojanovic, Modeling and minimization of extinction in Volterra-Lotka type equations with free boundaries, J.Differential Equations 134 (1997), 320-342.

9. S. Stojanovic, Computational Financial Mathematics, (book and CD-ROM in preparation).


First Order Necessary Conditions ofOptimality for Semilinear Optimal

Control Problems

M.D. VoiseiDept. of Math., Ohio University,

Athens, OH, 45701, USA

Abstract

This paper is concerned with the providing of necessary conditionof optimality for optimal pairs (y*,u*) with respect to the cost func-tional g(y) + h(u) subject to Ay + Ly = Bu + /, where A, B are linearand L is Lipschitz continuous. One example of applicatios of our nec-essary condition to a semilinear elliptic equation is presented in termsof Clarke's generalized gradient.

1. Preliminaries

In this article we investigate first order necessary conditions of optimality foroptimal pairs of problem

(P) Minimize

g(y) + h(u)

on all (y,u) G V x U subject to state equation

f .

Here (U, (•, •){/), (H, ( • , •)# are Hilbert spaces, V is a Hilbert space whichis not identified with its dual V* such that V C H C V* algebraically andtopologically with dense inclusions, V is compactly imbedded in H,


g : H —> IR and h : U —> 1R U {00} are given functions, L : H —» H isLipschitz continuous, A 6 L(V, V*), B 6 L(C7, #) and f € H.

We denote by • || the norm in V, with {•, •} the pairing between V andV*, with | • * the norm in V*, with | • H the norm in H and with | • \u thenorm in U.

Assume (P) has an optimal pair (y*,u*) £ V x U. The main idea is tolinearize the state equation by introducing a new control variable and bypenalizing the cost functional with an appropriate function we construct asequence of approximative problems depending on (y*,u*) and a parametere > 0, given by

(Pe) Minimize

g(y] + h(u) + ±\v-Ly2H + ±\u-u*2

u + ±\y- y*

on all (y, u, v) e V x U x H subject to state equation

f.

2Hi

Under some suitable assumptions on g, h, B, A and L problem (Pe) has atleast one solution. We will derive the approximative necessary conditions ofoptimality and prove that these conditions converge in some sense, for e — >• 0,to the necessary optimality condition of (y*,u*).

Previous treatments of this problem for L = d<f> the subdifferential of aproper convex lower semicontinuous function tp, are due to Barbu [2] and usethe penalization function (p(y) + <f>*(v) — (y,v) and the same state equationfor the approximative problem (Ps). Here we study the case of Lipschitz con-tinuous L followed by an example governed by a semilinear elliptic equation.

Related results can be found in Aizicovici, Motreanu & Pavel [1] and Tiba[7]-

In what follows, for a locally Lipschitz continuous function F denned ina Banach space X, we define its directional derivative by (see Clarke

F°(x-v) =limsup (l/t)(F(x + tv)-F(x)}, x,v e X. (I)

The Clarke generalized gradient dF : X — > 2X* of F is defined by

y € dF(x) iff (y,w)x*xx < F°(x;w), Vw e X, (2)

where X* is the dual of X and (-.-}x*xx denotes the dual product betweenX and X*.

When (X, (- , -)x) is a separable Hilbert space, we can construct the reg-ularizations of F given by

) = / F(Pnx - \Anr)pn(T)dr, x 6 X, (3)J TRn

where A > 0, n = [A"1], {em}m=i ig an orthonormal basis in X, Xn is thefinite dimensional space generated by (em}^=1, Pn is the projection of X

ninto Xre, that is, Pnx = £ (x, em)xem, An : Kra — > Xn is denned by An(r) =

m=l

£ rmem, r = (n, .. ,</-„) and pn e C0°°(IRre), ^(0) = 0 form=l

Pn > 0, />^ = 1, Pn(9) = Pn(-6), W e M ".We recall that FA is Frechet differentiable and if we denote its Frechet

derivative with VF\, then the sequence (V-F\) is uniformly bounded onbounded subsets of X, lim F\(x) = F(x), Vx 6 X, and if x\ — >• x strongly inX and VFA(xA) -> ^ weakly in X*, then ^ e 5F(x) (see Barbu [2]).

For a proper, convex, and lower semicontinuous function h : U — > IR U {00}the directional derivative of h is given by

h'(u;u) =lim (l/t)(/i(w + tu) - h(u)), u,ueU (4)

and the subdifferential dh : D(dh) C U — > 2U is denned by

77 e 9/i(w) iff (77, u; - u)v + h(u) < h(w), Vw e U. (5)

We refer to Motreanu & Pavel [6] for results of convex analysis used inthis paper.

We have considered the function SL : H x H — > IR, defined by

SL(y,v) = \\v - Ly\\ y,v £ H. (6)

In the following Lemma we characterize its generalized gradient.Lemma 1.1 Assume L : H — » H is Lipschitz continuous. For z € H we

define /z : tf -»• IR by

,x£H. (7)

Then the following statements are truei) SL is locally Lipschitz continuous in H x H and fz is Lipschitz in H

for every z G H,ii) SLO(y,v;y,v) = (v - Ly,v)H + fQ

(v_Ly)(y;y}, Vy,v,y,v e H,iii) (w, z) 6 dSL(y, v) iff z = u — Ly and u> e dfz(y),iv) HH^t fk / f .VK^eaS^y ,* ; ) , (8)

where K > 0 is the Lipschitz constant of L.v) If (wn,zn) 6 dSL(yn,vn), n>l and

wn —* w weakly in .fiT,zn —> 2: weakly in /f,un —> v weakly in H,

and

yn -> y strongly in H,

then (w,z) E dSL(y,v).Proof. Easy calculations show that

SL °(y, v; y, v) = _hmsup ±(\v + tv - L(y + ty)\2H - \v - Ly\2

H) =(y,v)->(y,v),tlO

=hmsup (v -Ly,v- \(L(y + ty) - Ly))H = (v - Ly,v)H + ffv_Ly)(y,y),y-*y,tlo

for every y, v,y, v G H, so, ii) is true and it implies iii).iv) For each (w, z) £ dSL(y, v) we have z = v — Ly and w 6 dfz(y] that is

(w,y)H <limsup -±(z,L(y + ty) - Ly))H < K\z H\y\H, Vy e H,

or \w < K\Z\H, where K > 0 is the Lipschitz constant of L.

v) We have zn — vn — Lyn and wn € dfZn(yn}. Therefore the given hy-potheses imply

zn —> z = v — Ly weakly in H.

From

(wn,v)H <limsup -±(zn,L(y + tv) - Ly)H, Vv e H,

we infer that for arbitrary fixed v <G H there exist (yn), (tn) such that\yn-yn < n>° <tn< n and

(wn, v)H < -£(*«, L(yn + tnv) - Lyn)H + £, Vn > 1. (9)

We can assume, by Mazur Theorem, that

wn = V] Oilnwni —> w strongly in H and zn = Y^ a^zTli —> z strongly in H

(Here /n is a finite set of positive integers, a\ > 0 and^ o^ = 1).ie/n

Then (9) can be written as

(£n,v)H < -±(zn,L(yn + tnv) - Lyn)H + i, Vn > 1. (10)

If we let n — >• oo in (10) then we get

<limsup -±(zn,L(yn + tnv) - Lyn)H =re— >oo

=limsup -f (2, L(yn + *nu) - Lyjn <n—xx

<limsup -}(z,L(y + tv) - Ly)H,

because yn —> y strongly in .ff.Since f is arbitrarily chosen we conclude that w £ fz(y) and (w,z) G

55L(y,t;).BRemark 1.1 Let us notice that in the previous Lemma we have proved

the following

wn G Jzn (yn)wn —>• w weakly in H,zn —s- z weakly in H,2/n -»• y strongly in H, ,

2. The Main Result

Let us consider the assumptions(Ai) V,U,H are Hilbert spaces, H is separable, V is dense in H and theinclusion of V into H is compact,(A2) g is locally Lipschitz continuous in H and g(y) > —C\y\n + D, Vy e H,where C > 0, D 6 H,(A3) /i : U — > IR U {00} is proper convex and lower semi continuous,

(A5) L : H H is Lipschitz continuous, that is

\Lyi - Ly2\ < K\yi - y2\H, VVl,y2 e H (K > 0), (12)

(A6) A € L(V, V*) satisfies

(Av,v)>u\\v\\2,VveV (u>0), (13)

(Ay) A dominates L, i.e.,

uj\i > K,

\\v\\2

where A! = infi11-^-; z; e V. v = 0).-71 Mvl/f

In the following we suppose that (Ai— A7) are fulfilled.Theorem 2.1 Let (j/*,u*) £ V x C/ be a solution of (P).

Then there exist p G V, £ G # such that

-A> - £ e ^(y*), B*p e dh(u*}, £ e 0/_p(y*).

Proof. Let us prove first that (Pe) has at least one optimal solution (ye, ue, ve}.We define

AH : D(AH) = {y£V;Ay£H}cH->HbyAHy = Ay,y£ D(AH)

and, since A is a linear homeomorphism between V and V*, we can considerD(AH) as a Hilbert space endowed with the inner product

,Vu,v G D(AH}.

Then ( D ( A H ) , ( - , •}D(AH)} is compactly embedded in V and A G L(D(AH),H).The above assumption imply inf(P£) > — oo and if (yn^univn} is a mini-

mizing sequence, that is Ayn + vn = Bun + f and

inf(Pe) < g(yn) + h(un] + ± vn - Lyn 2

H + \\un - u*fv + \\yn- y*\2H <

< mf(Pe) + I (14)

then (yn,un,Vn) is bounded in H x U x H.The state equation Ayn+vn = Bun+f shows, in fact, that (yn) is bounded

in D(AH) so, we can suppose that on a subsequence (denoted in the sameway for simplicity) we have

yn -»• y strongly in V,vn ~^ v weakly in H,un —> u weakly in U,

and Ay + v = Bu + f .If we let n — > oo in (14) we infer that (y, u, v) G D(AH) x U x H =: V is

a solution of (P£). We denote this solution by (y£, u£, ve).Next, we show that

(y£, ue, v£) -> (y*, u*, Ly*) strongly in V (15)

Since (y£,u£,v£) is optimal for (Pe) we have

g(ye) + h(u£) + ±\v£ -Ly£2H + \\ue - u*\2

v + \\y£ - y*\2H <

(16)

Hence, (ys,u£,v£] is bounded in V and on a subsequence it satisfies

(y£,u£,v£) -» (y,u,v) weakly in V,y£^y strongly in V,

h(u£) — »liminf h(ue) > h(u),

and Ay + v = Bu + /.By passing to limit in (16) we get

h(u) <liminf h(ue) < inf(P) - g(y) < oo, (17)ej.0

limsup j-\v£ - Ly£\< inf(P) - g(y) - h(u) < oo, (18)ej.0

lim v£ - Ly£\H = Q,v = Ly, (19)ej.0

g(y) + h(u] + \ limsup [\u£ - u*\l + \\y£ - y* ZH} < inf(P) <

<eg(y) + h(u), (20)

because Ay + Ly = Eu + /. This final inequality proves that

us — >• u* strongly in U,y£ — > y* strongly in LT.

But (19) implies w£ — > Ly* strongly in H, since Lys — > Ly* strongly in Lf.Therefore

+ f- Ly£ -^ Bu* + f - Ly* = Ay* strongly in H,

i.e.

y£ ->• y* strongly in D(AH),

so (15) is proved.For A > 0, we define

Ix(y, u, v) = gx(y] + h(u] + \SL(y, v) + \\u - u*\l + \\y - y* 2H+

where g\ is the regularization of g in the separable Hilbert space H , given by(3).

Using a similar argument as above, for every A > 0 problem(R\) Minimize

on all (y, u, v) e V x U x H subject to

Ay + v = Bu + /,

admits a solution (y\, u\,v\) 6 V and we prove that (on a subsequence)(?/A, UA, vx) -» (ye,ue, v£) strongly in V x U x V* (21)

Indeed, since Aye + v£ = Bu£ + f and (yx,u\,v\) is optimal for (^A), wehave

M(RX) = I$y\,ui,vx) < Ix(y£,u£,vs) < C£ < oo (22)

and this provides us with the boundedness of (yx,u\, v$ in V.Hence, on a subsequence we can assume that

(yx, ux, vx) -+(y, u, v) weakly in Vy\ — > y strongly in V, v\ — >• v strongly in V* , Ay + v = Bu + f

, h(ux) -^liminf h(ux) > h(u)AJ,U

^) > SL(y,v}.

Then, by passing to limsup in (22) we get

+| limsup {\ux - u*\l; + \yx - y*\*H + \ux - u£\2H + \yx - y£\2

H} <AJ.O<limsup/A (yx,ux,vx) <lim Ix(y£,ue,ve) = inf(.R.\) <

A|0 Al°< g(y) + A(S) + ^SL(y,v) + lu-u*l + ±\y-y*\2

H,

and so

l i m s u p { u A - M £ ^ + \y\-ye2

H} < 0. (23)A|0

This implies UA — >• u£ = u strongly in U, y = y£, v = v£ and

Ayx — » Ay£ = Ay weakly in H

since v\ — > we weakly in ff.Hence yx — > ye weakly in _D(A# ) and eventually on a subsequence yA — >• y£

strongly in V, thereby (21) is proved.The necessary condition of optimality for (y\,u\,vx) can be easily de-

ducted according to the principle that if F : X — > IR is a given functiondenned on a Banach space X and F ( z ) =inf F(x) then F°(z,v] > 0 forevery v e X.

In our case, for problem (R\) we find

(Vgx(yx),y)H + hf(ux,u) + l(S^+ (2ux -u*- u£,u)u + (2yx - y* - ye,y)H > 0 (24)

for every (y, u, v) G V x U x H such that Ay + v = Bu, where

dSL(yx,vx] C H x H .

According to Lemma 1.1, we have S^(yx, v\) = vx — Lyx,(yx,vx} e df(vx-Lyx)(yx) and S$(yx,vx)\H <Relation (24) can be equivalently written as

vx) + 2yx - y* - ye,y}H - \(vx - Lyx,Ay)H > 0, (25)

for every y £ D(An), and

h'(ux,u] + (±B*(vx - Lyx) + 2ux - u* - u£,u)u > 0, Vu e U,

or,

dh(ux) 3 B*px - 2ux + u* + u£, (26)

where p\ = -^(vx- Lyx) e H.We have

P\ -> PE •= -~(v£ ~ Ly£) weakly in H,Sy(yx, vx) -> w£ weakly in H,

and since Sy(yx,vx) e df(Vx_Lyx)(yx) then by Remark 1.1 we know that

If we let A -> 0 in (25), (26) then we get

(T£ + \w£ + y£- y*,y)H - \(v£ - Ly£, Ay)H > 0, Vy G D(AH), (27)

where r£ e dg(ye), and

dh(ue) 3 B*p£ + u* -u£. (28)

Let ?]£ 6 D(AH} = {y <EV\ A*y e H} be such that

A*r]e = r£ + ^w£ + ys-y*

Then relation (27) can be stated as

(A^£,y}H = -(p£,Ay)H, \/y 6 D(AH) (29)

But

(A*ri£,y)H = (A*rj£,y} = (r]£,Ay) = (r]£,Ay)H.

Therefore (29) is equivalent to rj£ = —p£.In this way we proved that p£ G D(A*H) and

-A*pe - \ws + y*-y££ dg(y£). (30)If we multiply the last relation with — pe then we get, according to (12),

that

U\\Pe\ 2 - Pe||(d + 1^1*) < 0, (Ci < 00),

since {Udg(ye}} is bounded in H.Hence

wv^Tbe H < u\\Pe\\ < \±we\* + Cl<^

We have uj\i > K, therefore, (pe) and (~w£) are bounded in H so, even-tually on a subsequence we infer that

p£ —> p weakly in H, -w£ —> I weakly in H.

We can pass to limit in (29) and (30) because A* is weakly-strongly closedin H and B* e L(H, U) to get

-A*P-eedg(y*),B*P£dh(u*). (31)

In order to determine a relation between p and I we start recalling thatw£ edf(Ve-LyE)(y£), i.e.,

(we,h)H <limsup -\(vs - Ly£,L(y + th) - Ly)H,Mh e ^,y->2/e,*io

and so for arbitrary but fixed h E H we have

(±w£,h)H <limsup j ( p £ , L ( y + th) - Ly)H.

Therefore for every e > 0, there exist y'e e H, t£ e (0, e) such that\y'e ~ y£ H < £ and

(~£w£, h)H < £(pe, L(y'e + tsh] - Ly'£) + e.

Because ^w£ =: l£ — > i weakly in H and p£ — > p weakly in H we mayassume by the Mazur theorem that on a convex combination we have

£n = E c>44; -» £ strongly in H and pn = E a£pei — > p strongly in #•Je/n J6/71

(Here Ira is a finite set of positive integers, aln > 0 and E an = !)•

This allows us to find (£n, h}n < j~(pn, L(y£+t£h) — Ly£)H+£, and passingto limit with e — > 0 (n — >• oo) we obtain

(£, /i)H <limsup £(p,

<limsup ^ ( p £ , L ( y + th) - Ly)H,Vh

because t£ [ 0, y^ — >• y* strongly in H. This final inequality shows thati e df-p(y*}. The proof is complete. •

3. An Example to Optimal Control of a Semilinear EllipticEquation

We investigate the following particular case of (P)(Pi) Minimize

h(u),

on all (y,u) 6 HQ(&.) x [/, subject to{ -Ay + P(y) = Bu + f in ft,

(SE} \y = 0 in ft,where ft is a bounded open domain of class C2 in IRn, V = -ffo(ft), f/ isa given Hilbert space, H = I/2(ft), g,h,B,f satisfy hypotheses (A2), (A3),(A4) and (3 : IR -^ 1R is a Lipschitz continuous function, i.e.,

P(r2-)\ < K\ri~r2 , Vra ,r2 e 1R (K > 0). (32)

We may see (SE) as Ay+Ly = Bu+f, where A = -A : H%(tyand L : I/2(f2) — > L2(J1) is given by

Theorem 3.1 Suppose, in addition to the above hypotheses, that (y*, M*)is a solution of (Pi) and

\i>K (33)

where A = inf{|Vi>|i,2(£))/|i>|L2(Q); v = 0} is the first eigenvalue of —A in Q.Then there exist p e H^ft), I e L2(O) such that

(ii) B*p e(iii) £(x) e p(x)d0(y*(x)) a.e. x e fi.

Proof. Since all the hypotheses are fulfilled, we may apply Theorem 2.1to find p e #£(n), ^ G L2(ft) such that

+ 1 3 -^*P, 5*p € dh(u*),

The final relation is equivalent to

(t, h}H <limsup (1/t) / (0(y + th) - {3(y}}pdx, Vv ey->y*,tlQ J

and it can be described as I G dG(y*) where G : L2(fi) — > IR is defined by

Thus i(x) e p(x)d(3(y*(x)) a.e. x G O (see Clarke [4] or loffe and Levin[5]). Since —A* = A, the proof is complete. •

References

1. Aizicovici, S., Motreanu D. & Pavel N.H., Nonlinear ProgrammingProblems Associated with Closed Range Operators, Appl. Math. Op-tim., Vol. 40, pp. 211-228, 1999,

2. BARBU, V., Analysis and Control of Nonlinear Infinite DimensionalSystems, Academic Press, Boston, 1993,

3. BARBU, V., Partial Differential Equations and Boundary Value Prob-lems, Kluwer Academic Publishers, Dordrecht, Boston, 1998,

4. CLARKE, F. H., Optimization and Nonsmooth Analysis, John Wileyand Sons, New-York, 1983,

5. IOFFE, A. D. and LEVIN, V. I., Subdifferential of Convex Functions,Trudy Moskov. Mat. Obshch. (translated in Transaction of theMoscow Mathematical Society), Vol. 26, pp. 3-13, 1972,

6. Motreanu D., Pavel N.H., Tangency, Flow Invariance for DifferentialEquations and Optimization Problems, Monographs and Textbooks inPure and Applied Mathematics, Vol. 219, Marcel Dekker, New-York -Basel, 1999,

7. TIBA, D., Lectures on the Optimal Control of Elliptic Equations, The5th International Summer School Jyvaskyla, Lecture Notes 32, Finland,1995.


Lyapunov Equation and the Stability ofNonautonomous Evolution Equations

in Hilbert Spaces

Quoc-Phong VuDepartment of Mathematics

Ohio UniversityAthens, OH 45701, U.S.A.

andSiu Pang Yung

Department of MathematicsUniversity of Hong Kong

Hong Kong

Abstract. We apply the method of Lyapunov equations A*P + PA = —I to thequestion of exponential stability of the differential equation u'(t] = A(t)u(t),t > 0,in a Hilbert space E. Under some suitable conditions, we show that the solutionsare exponentially stable provided that A(t) generate exponentially stable semigroupswith exponential types < a < 0, and are slowly varying in some sense. Estimates arealso given for the rates of convergence of the solutions to zero.

1. Introduction

Let A(t) be closed linear operators on a Banach space E such that the differentialequation

u'(t) = A(t)u(t),t>0, (1)

is well-posed, i.e., for each XQ from a dense subset T> C E, Eq.(l) has a unique solutionwith u(0) = x0, which depends continuously on the initial value x0. In this paper, weare concerned with conditions which imply that solutions to Eq.(l) are exponentiallystable, i.e. \\u(t)\\ < Ne-fft\\u(Q)\\ for every solution u ( t ) of Eq.(l), where N and aare positive constants.

Note that if dim E < oo and Eq.(l) is autonomous, i.e. A(t) = A are independentof t, then its solutions are exponentially stable if and only if all eigenvalues of A have

negative real parts, i.e., there exists a positive number a such that

tr(A) C {A €C : Re A < -<r},

where u(A) is the set of eigenvalues (the spectrum) of A.

However, it is well known that a similar statement does not holds for nonau-tonomous equations. Indeed, assume that A(t) = V'~l(t)A^U(t), where

. / — 1 — 5 \ ,,,.,. / cost s'mtA°=( 0 - l j ' ^=( - s in t cost

Thena(A(t}} = {-!}, for all t > 0,

but the solution matrix is

m _ ( e*(cos ^ + | sin t) e~3t(cos t - \ sin t)( ' ~ { e'fsin t - | cos t) e-3t(sm t + f cos t)

and therefore is unstable (see [2]).Various conditions have been obtained for equations (1) in a finite dimensional

space £, that, together with the condition of uniform exponential stability of thematrices A(i),i.e.

||esA((>|| < Ne'", ((7 >0, N > I are independent of <) , (2)

imply the exponential stability of solutions of Eq.(l) (see [1,3,5,7,10]). All theseconditions express the property that A(i] are slowly varying matrices in a suitablesense. To our knowledge, analogous results have not been obtained for Eq.(l) in aninfinite dimensional space.

In this paper, we consider the problem of exponential stability of Eq.(l) in infi-nite dimensional Hilbert space, and we extend some results of the above mentionedpapers to this case. Namely, we prove that if A(i) are uniformly bounded (i.e.supt>0 ||A(i)|| < oo) and slowly varying in an appropriate sense, and if (2) holds,then solutions to Eq.(l) are exponentially stable. We also obtain estimates of therate of convergence of the solutions u(i) to zero (Theorems 7 and 8).

The method we use to obtain these results is based on the Lyapunov's equationA*(i)P + PA(i] = —I with the variable operators A(t). This method also allows usto extend the results to some cases involving unbounded operator coefficients. Forother recent applications of Lyapunov equations in the stability theory of evolutionequations in infinite dimensional spaces see [13-15].

Throughout the paper T>(A) denotes the domain of an operator A, and a (A]denotes the spectrum of A.

2. Main Results.

Assume that A(i] are closed linear operators on a Hilbert space E which satisfythe following conditions:

HI. A(t) = iA0 + Ai(t), where A0 : E — » E is a self-adjoint, generally unbounded,operator, A\(t] are bounded, and A0 commutes with Ai(t) for all i, i.e. from x 6V(Ao) we have Ai(t}x e T>(A0] and AQAi(t}x = Ai(t)A0x, for all t.

H2. sup^o ||Ai(i)|| < oo, and

||e^i(0|| < jVe~CTS, (a > 0,7V > 1 are independent of t).

H3. For every x G T) := "D(Ao), the function t H-> Aj(i)a; is differentiable andA\(t]x is a bounded operator from T> to E1, so that it can be extended by continuityto a bounded operator on E.

Note that conditions H1-H3 include the case of Eq.(l) with bounded operatorsA(t), in an infinite dimensional Hilbert space (set A0 = 0). It also follows from thetheory of evolution equations (see e.g. [8], Chapter 5, Theorem 2.3 and 3.1) thatunder these conditions there exists an evolution system U(t, s)(0 < s < t < oo)associated with Eq.(l), such that solution of Eq.(l) with the initial value u(0) = x isgiven by u(t) - U(t,Q)x,t > 0.

Our method of investigation of the asymptotic behavior of Eq.(l) is based on theoperator equation of the following form

PA + BP = C, (3)

where A and B are closed linear operators on E and C : E — > E is a boundedlinear operator. A bounded linear operator P : E — > E is called solution of Eq.(3) ifPV(A) C V(B] and P Ax + BPx = Cx for all x € T>(A). The following propositionis well known (see, e.g. [9], [12]).

Proposition 1. Let A and B be generators of exponentially stable Co-semigroups{esA}s>o and {e5B}s>0, respectively. Then the integral

fOO

P = - I esBCesAds (4)Jo

converges in the uniform operator topology and is the unique solution of Eq.(2).

Now let A(i] satisfy conditions H1-H3. We will need the following lemma.

Lemma 2. An operator P(t) is a solution to equation

P(t)A(t) + A*(t)P(t) = -I (5)

if and only if it is a solution of

-I. (6)

Proof. Assume that P(t) is a solution to (5). By Proposition 1, and since AQ andcommutes with Ai(t) as well as with Ai(t)*, we have

fOO fOO

P^ = ~Jo e ' 6 l S~~Jo 6 6 S'

hence P(t) is a solution to (6). The converse is proved analogously, d

Thus, for each t > 0,/•oo

p(i\ .— I fsA"i(ty sAi(t) j fj\f ( t ) .— e e as (I)Jo

is a bounded solution of Eqs(5)-(6).

Proposition 3. There exist a, (3 > 0, independent of t, such that

/?IN|2 < (P(t)x,x) < a\\x\\\ for all x e E. (8)

Moreover, one can choose a = -7= and fl = -^.

Proof. From (7) and condition H2 it follows that/•oo

I I P r / W I I 2 < / /V 2p~ 2 < 7Vsll r l l 2 — n - 2 l l r l l 2j -f 1 t /•*' _^~ I I V C UO 1 1 kO 1 1 —— Ct " i

Jo

where a = -j=. Hence {P(i)x,x} < a||x||2. On the other hand

/•oo yoo

' J o ~~ J o ~/•oo 1

jf e-2sM^||x||2 = ||a:||2 = /?||x||2. d

Proposition 4. The operator function P(t] is differentiate and satisfies

||P'(i)|| < K\\A((t)l

where K =

Proof. That P(t)x is differentiate for every x G E follows from (7). Since

P(t)Ai(t)x + Ai(t)P(t)x = -x,

it follows that

x + AWP'Wx = -[P(t)A((t)x + A'1(tyP(t)x].

Since by H2 A((t) is bounded, and generates an exponentially stable semigroup, wehave by Proposition 1

/*OO

P'(t)x= \ esA^[-P(t}A((t}-A\(tYP(t)}esA^xds.Jo

Therefore, from (8) we have

/•oo\\P'(t)\\ < JQ JV2e-2<"[2a||Ai(i)||]<fe < K\\A((t)\\,

where K = *£ = _|L . Box

Consider the Cauchy problem

u'(t}= A(t)u(t)«(0)= x0

(y>

where the operators A(i) satisfy conditions H1-H3. We need the following simplelemma.

Lemma 5. Let (p(t) and f ( t ) be real functions on [0, oo) such that ip(t) > 0 for allt > 0, is differentiate, satisfies y'(t) < f ( t ) for all t > 0. Then

Proof. The required inequality follows from

Jo ds ~ Jo

Proposition 6. Let u(t) be a solution of Eq.(9) and P(i) be solutions of Eqs.(5)-(6).Then the following estimate holds.

(P(t)u(t),u(t))<exp{-\--2MK f K(s)||dsl } (P(Q)x0,x0}, for all t > 0.I La Jo J )

(10)

Proof. Since Eq.(l) is well posed, if u(io) = 0 for some to then u ( t ) = 0 for allt > 0. Hence, we can assume that u(i] ^ 0 for all t > 0. By Proposition 3,(P(t)u(t},u(t}} > 0 for all t > 0. We have

= (P'(t)u(t),u(t)) + (P(t)u'(t},u(t)) + (P(t)u(t),u'(t)).

On the other hand,

( P ( t ) A ( t ) u ( t ) , u ( t ) ) + ( P ( t ) u ( t ) , A ( t ) u ( t ) ) =( P ( t } A ( t ) u ( t ) , u ( t } } + ( A * ( t ) P ( t ) u ( t ) , u ( t ) ) =

-(u(t),u(t))<-(P(t)u(t),u(t}).

Moreover, from Propositions 3-4 it follows

= 2 M K \ \ A ' 1 ( t ) \ \ ( P ( t ) u ( t ) , u ( t ) ) .

Therefore

d f 11— ( P ( t ) u ( t ) , u ( t } ) < \2MK\\ A'^H - - ( P ( t ) u ( t ) , u ( t ) ) ,

and (10) follows from Lemma 5. d

We remark that Proposition 6 gives better estimates than those in [4]. FromPropositions 3 and 6 we obtain the following result.

Theorem 7. Let u(i) be a solution of Eq.(9). Then

IKOH 2 < 2aMexp f- f- - 2MK f \\A!,(s]\\ds\ } \\x0\\\I La Jo JJwhere a and K are constants defined in Propositions 3-4.

From Theorem 7 we obtain the following stability result which are extensions ofresults in [1-5], [7-10].

Theorem 8. Assume that A(t) satisfy conditions H1-H3. In addition, assume thatAi(t) satisfy one of the following conditions:

7 :=-sup | |Ai ( S ) | |<——|——, or (11)s>o ZaMK

Then Eq.(l) is exponentially stable, i.e. ||u(t)|| < Le~u/'||ti(0)|| for every solution u ( t )and some positive constants L,u.

Proof. If (11) holds, then

HOI!2 < 2aMe-ia \\XQ\\\ where a = - - 2MKf > 0.a

Assume now that (12) holds. Choose £ > 0 such that

1aM

By (12), there exists to such that, for every t > to, we have

ff HA'^Uds <et.Jo

Therefore,\\u(t}\\2 < 2aMe-ta, for all t > t0. D

We remark that the conclusions in Theorems 7 and 8 remain valid for mild solu-tions, i.e., for functions u(i] = t/(i,0)x, thanks to the well-posedness of Eq.(l) in theconsidered situation. In other words, there exist positive constants L, a such that

\\U(t,s)\\ <Le-"(t-s}\\, for al l* > s.

We also remark that the following condition (considered by Cesari for ordinary dif-ferential equations)

° < oo (13)

implies that (12) holds. On the other hand, if (13) holds, then, for every x G £>, thelimit lim^oo Ai(i]x exists. In fact,

f t yoo\\mAi(t)x= lim / A((s)xds + A^x = / A',(s)xds + Ai(0)x, for all x <5 V.t-^ca t-Kx JQ JQ

Hence A(i)x converges as t —>• oo, for all x € T>. However, even in this case Theorem8.1 of [8, p. 173] is not applicable since A(t) do not generate, in general, analyticsemigroups.

Examples.

1. Let E := L2(IR) and b(t) be a bounded differentiable scalar valued functionsuch that b(t) < — a for some a > 0, and assume that one of the following conditionsis fulfilled:

(a) supt>0 \b'(t)\ is sufficiently small;(b)f~\V(t)\dt«X>;

Define operators A(t) on E by

d2uA(t}u(x) := i—— + b ( t ) u ( x ) , u ( t ) € E.

Then, as is easily seen, A(t) satisfy conditions H1-H3 with AO := J^j and Ai(t}u : =b(t)u. Consequently, solutions of the equation

d2

ut(t,x) = i——u(t,x) + b(t)u(t,x),t > 0,x € IR,(J JL

converge to zero exponentially (in L2(IR)-norm).

2. We generalize the above example by putting A0 := A, where A is a Laplaceoperator on E := L2(1R3), or E := L2(Q) where fi is a bounded domain in IR3 withsmooth boundary, and A is defined with an appropriate boundary condition. It iswell known that AO is a self-adjoint operator (see e.g. [6, Chapter 5]). Furthermore,let Ai(t) be a uniformly bounded family of operators on E such that

||e^i(<)|| < Ne-"s, (a>Q,N >1 are independent of t ) ,

and assume that A\(i) satisfies (11) or (12). Then solutions to the equation

ut(t, x) = iAu(<, x ) + A^(t}u(t, x), t > 0,

are exponentially stable.

References

1. L. Cesari, Un nouvo criteria di stabilita per soluzioni delle equazioni differenzialilineari, Ann. Scoula Norm. Sup. Pisa (2)9 (1940), 163-186.

2. W.A. Copell, Dichotomies in Stability Theory, Lecture Notes in Mathematics,vol. 629, Springer, Berlin, 1978.

3. J.K. Hale and A.P. Stokes, Conditions for the stability of nonautonomous dif-ferential equations, J. Math. Anal. Appl. 3 (1961), 50-69.

4. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notesin Mathematics, vol. 840, Springer, Berlin, 1981.

5. C.S. Kahane, On the stability of solutions of linear differential systems withslowly varying coefficients, Czechoslovak Math. J. 42 (117) (1992), 715-726.

6. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.

7. L. Markus and H. Yamabe, Global stability criteria for differential systems,Osaka Math. J. 12 (1960), 305-317.

8. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differen-tial Equations, Springer, New York, 1983.

9. C.R. Putnam, Commutation Properties of Hilbert Space Operators and RelatedTopics, Springer, Berlin, 1967.

10. G. Sansone and R. Conti, Nonlinear Differential Equations, McMillan, NewYork, 1964.

11. H. Tanabe, Evolution Equations, Pitman, London, 1979.

12. Vu Quoc Phong, The operator equation AX—XB = C with unbounded operatorsA and B and related abstract Cauchy problems, Math. Z. 208 (1991), 567-588.

13. Vu Quoc Phong, On the exponential stability and dichotomy of Co-semigroups,Studia Math. 132 (1999), 141-149.

14. Vu Quoc Phong and E. Schiller, the operator equation AX — XB = C, admissi-bility, and asymptotic behavior of differential equations, J. Differential Equations145 (1998), 394-419.

15. E. Schiiler and Vu Quoc Phong, The operator equation AX — XV2 = —<50 andsecond order differential equations in Banach spaces, Semigroups of operators:theory and applications (Newport Beach, CA, 1998), 352-363, Progr. NonlinearDifferential Equations Appl., 42, Birkhauser, Basel, 2000.

Least Action for N-Body Problems withQuasihomogeneous Potentials

Shih-liang WenDepartment of Mathematics, Ohio University,Ohio 45701,USA

Shiqing ZhangMathematical Department, Chongqing University, Chongqing 400044, China

Abstract Using variational minimization methods, we study the existence of a

noncollision or a generalized periodic solution for N-body problems with

quasihomogeneous potentials, specially for N-body and N+l-body problems in R2k(k

^ 1), we study the geometric characterization for variational minimization solutions.

1. Introduction and Main ResultsN-body problems with quasihomogeneous potentials ([6], [12]-[14], [22]) are related

with the motion of N point masses mi, •••, WN in RK (KS=1) under the action of the

potential -W(q) given by

±- fo , - ? ) (1.1)

Where U(q) = - jT ———'— (1.2)li4*j±N q.-q.

b v-i tn,mt

^iWhere a,b>Q,a2 +b2 *0,a,/3>Q,a2 + p2 ^0,q = (ql,---,qN),qi e RK.

The equations of the motion for the N-body problems with a potential -W(g) are

given bydW(q) .m,<li = „ *,i = l,---,N (1.4)

8q,

Note that -W(q) is the classical Newtonian potential when a =0, P =1, or

b =0, a=1 or a = P =1 ,and is a homogeneous potential when a =0 or 6 =0 or a = p.

References [1H5], [8], [10H12], [15], [19]-[23] used variational methods to

study the periodic solutions of N-body problems.The true motions of the celestial bodies should obey the Least Action Principle of

Fermat-Maupertuis in some sense. Hence, we use variational minimizing methods tostudy the periodic solutions of N-body problems.

Serra-Terracini ([15]) used variational minimizing methods to study the existence ofa noncollision periodic solution for 3-body problems with the classical Newtonianpotential and a radial perturbation potential in R3.

Long and Zhang ([10-12]) and Zhang-Zhou ([22]) and Chenciner and Desolneux([3]) studied the shape of the solution minimizing the Lagrangian action integral on theT/2-antiperiodic or zero mean loop of class W1'2 (RJTZ, K ) for some N-body problems.

Zhang and Zhou ([23]) studied the variational characterization of Lagrangianelliptical solutions with equilaterial triangle configurations for planar Newtonian3-body problems, they proved that the regular minimizers of the Lagrangian actionintegral restricted on the periodic orbit space with three nonzero relative windingnumbers are exactly the Lagrangian elliptical solutions.

In this paper, we study the existence of a noncollision or a generalized periodic

solution for the systems (1 . 1 )-(! .4).

For (1.1)-(1.4) in R2k (k^l), we also study the shape of the orbit minimizing its

Lagrangian action integral defined on the periodic orbit space with the same integral

mean for each body during one period.Definition 1.1 ([2], [4]) Given T>0, Let q, , eWl-2([0,T],RK') ,we say

q - (<?!,•••, ? #) is a generalized periodic solution (or a noncollision periodic solution)

of (1.1)— (1.4),if there hold:( i ) S(q) = f e [0, T] J31 <i*j<N, si., qt (t) = qs (t)} has zero Lebesgue measure

(or is an empty set)(ii)Forall te[0,T]\S(q) (or t& [0,r]), q(t) satisfies (1.4).

(iii)For all / e [0,T]\S(q) (or / e [0,r]), there holds

Theorem 1.1 For any given T>0,I f ( i ) a > 2 , /?>2

or(ii) a>2, 0<y(?<2

or(iii) /?>2, 0 < a < 2

holds, then the system (1.1)-(1.4) has a noncollision periodic solution.If (iv) 0<a<2 , 0</7<2 , a2 +/32 # 0 Then the system (1.1)— (1.4) has a

generalized T-periodic solution.

Theoreml.2 Given energy heR, we consider N-body problems (1 .!)-(! .4) and

If one of the following three conditions holds:

( i ) a>2, p>2, (a, P) * (2, 2)

(ii) a>2, 0 < p<2, (a, f t ) ^(2, 2)

(iii) p>2, 0<a<2, (a, P) #• (2, 2)

Then there is a T>0 such that (1.1)-(1.4) has a noncollision T-periodic solution

with energy ho

If the following condition holds( i v ) 0 < « < 2 , Q<p<2, &*+/?*()

Then there is a T>0 such that (1.1)-(1.4) has a generalized T-periodic solution with

energy h .Theorem 1.3 For N-body problem (1.4) with a quasihomogeneous potential (1.1) in

), we define

=f If »i|9,|2* + {>(?)*.« e A (1-6)

Then the global minimum point q(f)={q\(f), qi(t\ —, <?#(0) ofj(q) exists and mustbe the non-collision relative equilibrium T-periodic solution with minimal period Tofthe N-body problem (1 .4) in B?k (£>1) whose configuration is the central configurationminimizing all configurations with potential f/and Fat any moment t and whose masspoints rotate along circular orbits around the center ( the common integral mean) withthe same constant angular velocity on a fixed plane.

Theorem 1.4 Let ml =m2 =--- = mN >0, mN+l >0,

miqt (t) = 0, qt (t) * qj (t), \<i*j<n+\ (1.7)

mtm.

^<w*qi-qj

Then the minimizer q=(qi(t),---,qN(t),qN+l(t)) of I (q) on S is a classical

T-periodic solution of

Where W(q) = U(q) + V(q) (1.10)

m'm< (1.12)-Kitjt.N^q.-t

and satisfies that qt(i = l,---,N) moves on the circle centered at qN+l = 0 with a fixed

angular velocity and a fixed motion plane and the configuration of ml,---,mN is the

central configuration minimizing all configurations with potential U and V at anymoment.

2. The Proof of Theorem 1.1Given T>0, Let A = j? = (ql,-,qN)\ql,e Wl'\RITZ,RK) ; <?,« *<?,(» ,

(2.1)

i=l

Lemma 2.1 The functional fiq) in (2.2) attains its global minimum value on theclosure A ofA, and the minimizer q(t) = (ql(t\---,qN(t}) is a generalized T-periodic

solution for the system (!.!)-( 1.4)

Proof Since flq) is coercive and weakly lower-semi-continuous on A, so f(q) attauis

the global minimum value on A, furthermore, similar to the proof of [1], [2], [4], [8],

we know that the minimum point q(t) is a generalized solution for the system

(1.1H1.4).Lemma 2.2 In the systems (1.1)-(1.4),

Then there holds the Gordon's strong force condition ([8]): there exists a functionGe C\RK - (0},R) and a neighborhood N of 0 in R* such that

Lemma 23 ([8] [2])Let {q"} be a sequence in A and q"—*qed\ ,then

Now we can prove Theorem 1.1:Under the assumption ( i ) or ( ii ) or (iii) of Theorem 1 . 1 , by Lemma 2.2, Wy ( £ )

satisfies the Gordon's strong force condition, then Lemma 2.3 implies the generalized

solution q(t) obtained by Lemma 2.1 is a noncollision periodic solution, otherwise, thenthere is a minimizing sequence q" such that f(q") -> f(q) = -H» , this is contradictionwith inf \f(q\q e A} < +00 .

3. The Proof of Theoreml.2We define

, VxeM

Where = \x = (jelf-, xN)e T, ^[-W(x)--W'(x)-x\dt = f

1], (Rk)N\ xt(t) * X](te '

j<. N, [0, 1], = 0}

(3.1)

(3.2)

(3.3)

Similar to the proof in [1] , we have

Lemma 3.1 If x is a non-constant critical point for g(x) on M, then we can define TX):

T =- \W\x)xdt

(3.4)

and q(t)=x(t/T) is a T-periodic solution of (1.1)-(1.5)0

Lemma 3.2 ([12])Fora>0 /3>0;a2 +fi2 * 0,he R, if g has a non-constant

critical point then we have(i) if (a, 13) € [2, + oo)2 \ {(2, 2)} then h>0,

(ii) if («,/?) = (2, 2) thenh = 0 ,

(iii) if (a,/3) e [0, 2]2 \ {(2, 2), (0, 0)} then h<0.

Lemma 33 For any integer k>0, N>2 and positive masses m\, •••WN, power index o>0

and q = (g,, • • - , qN) g (Rk)N the following homogeneous function with degree 0:

/(3.5)

attains global minimum on (Rfc)N, we denote this minimum by Da(N, m,C)a

Lemma 3.4 (Wirtinger's inequality) For any mj>0 and jc, e Wl-2([0,l],Rk) , (i=l,

N)if

then/=! J=l

In the following , we'll prove Theorem 1 .2:

(i)o>2, P>2, but a, p can't be 2 simultaneously

By Lemma 3. 2, wehaveh>0» Hence

(3.6)

(3.7)

2 dt

By Lemma 3.4, We know that g(x) is coercive on M.

(ii)0<a<2, P>0or0<p<2, a>0

byLemma3.2,h<0, but

Hence by Lemma 3.3, We have

=2, 0<P<2orp=2, 0<a<2,

By Wirtinger's inequality (Lemma 3.4) we have

i-—

Hence g(x) is coercive on M. Under the above two cases (i) and (ii), It's easy to know

that g(x)>0 and g(x) is weakly lower semi-continuous on M, and M is a weakly closed

subset. Hence g(x) attains the global minimum on M . The minimum value point is the

required periodic solution.

4. The Proof of Theorem 1.3In order to prove Theorem 1 .4, we also note that/^) is invariant under translations.

Assume q = (ql,q2,-,qli)e A, Let

l(t) = ql(t)-[qlli = lA-,N (4.1)

Then [x,]=0 and x=(x\, X2,'" ,XN) satisfies that Xi(t)ttj(t) (l<i*j<N) if and only if

By Wirtinger's inequality and Coti Zelati's type inequality (Lemma 3.3) andJensen's inequality we have

» 1 /-2;ri2 f Vm \v-•r(^r) J,2,*/l*/

Mi \x, |

(4 2)

(4.3)

Let

Then

(4.4)

Where <p(s) = (~)2 -s2 +Da(N,m,a)Tl+a/2 -s^ +Df(N,m,b)T1+^2 -s^ (4.6)

We note that <p(s) is a strictly convex smooth fiinction on s>0 and

<p(s) -* +00, as s -> 0+ or s -> +ao , so it has a unique global minimizing point s=So>0,

hence fix) attaining its infimum if and only if the inequalities of Wirtinger and CotiZelati's type and Jensen take equalities simultaneously. Hence we have

(i) xt(t) = alcos~t + b,&Ja—t,ai,bteRu,

(ii) U(x) \2mi \xiv/=i

simultaneously,

a/2

2 and8'2

attains their minimum

Similar to the proof of [10, 11, 22], we deduce that the orbits Xi (t) (i=l, 2, —, N)locate the same fixed plane and must be circles whose centers are at the origin by (i) and(ii) and (iii).

Hence qi(f)(i=\, 2,—, N) have the properties of Theorem 1.1 by (4.1).

+2

5. The Proof of Theorem 1.4N m m ( N 2

Lemma 5.1 Let 1=0, then % ' »+ ' >\ JTm, •«„+,i=i |qt,|v U=i

and inequality (5.1) takes equality if and only if

Now we prove Theorem 1.4:By the inequalities of Wirtinger and Lemma 3.3 and Lemma 5.1 we have

(5.1)

(5.2)

+ I —

By Jensen's inequality we have

Da(N,m,a)

-a/2

Where note that ^(5) is strictly convex smooth function on S>0 and ^(5) ->• +00 as


S ->• +00, hence ip(S) has a unique global minimizing Point S=So>0 and I(q) attains its

infimum if and only if all inequalities which we have used take equalitiessimultaneously, hence we have

(0 <?,(

(ii) U(q)\J\mt\qin and

N ^'2

^i=l

attains their minimum simultaneously,

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