[Alexander J. Zaslavski] Turnpike Properties in Th(BookFi.org)

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TURNPIKE PROPERTIES IN THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL

Nonconvex Optimization and Its Applications

VOLUME 80

Managing Editor:

Panos Pardalos University of Florida, U.S.A.

Advisory Board:

J. R. Birge University of Chicago, U.S.A.

Ding-Zhu Du University of Minnesota, U.S.A.

C. A. Floudas Princeton University, U.S.A.

J. Mockus Lithuanian Academy of Sciences, Lithuania

H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A.

G. Stavroulakis Technical University Braunschweig, Germany

H. Tuy National Centre for Natural Science and Technology, Vietnam

______________________________________

TURNPIKE PROPERTIES IN THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL

By

ALEXANDER J. ZASLAVSKI The TechnionIsrael Institute of Technology, Haifa, Israel

1 3

Library of Congress Cataloging-in-Publication Data

Zaslavski, Alexander J. Turnpike properties in the calculus of variations and optimal control / by Alexander J.

Zaslavski. p. cm. (Nonconvex optimization and its applications ; v. 80)

Includes bibliographical references and index. ISBN-13: 978-0-387-28155-1 (alk. paper) ISBN-10: 0-387-28155-X (alk. paper) ISBN-13: 978-0-387-28154-4 (ebook) ISBN-10: 0-387-28154-1 (ebook) 1. Calculus of variations. 2. Mathematical optimization. I. Title. II. Series

QA316.Z37 2005 515.64dc22

2005050039

AMS Subject Classifications: 4902

ISBN-10: 0-387-28155-X ISBN-13: 978-0387-28155-1 e-ISBN-10: 0-387-28154-1 e-ISBN-13: 978-0387-28154-4

2006 Springer Science+Business Media, Inc.

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Contents

Preface ixIntroduction xiii

1. INFINITE HORIZON VARIATIONAL PROBLEMS 11.1 Preliminaries 11.2 Main results 31.3 Auxiliary results 71.4 Discrete-time control systems 171.5 Proofs of Theorems 1.1-1.3 20

2. EXTREMALS OF NONAUTONOMOUS PROBLEMS 332.1 Main results 332.2 Preliminary lemmas 372.3 Proofs of Theorems 2.1.1-2.1.4 542.4 Periodic variational problems 592.5 Spaces of smooth integrands 622.6 Examples 69

3. EXTREMALS OF AUTONOMOUS PROBLEMS 713.1 Main results 713.2 Proof of Proposition 3.1.1 763.3 Weakened version of Theorem 3.1.3 793.4 Continuity of the function Uf (T1, T2, x, y) 833.5 Discrete-time control systems 883.6 Proof of Theorem 3.1.2 903.7 Preliminary lemmas for Theorem 3.1.1 94

vi TURNPIKE PROPERTIES

3.8 Preliminary lemmas for Theorems 3.1.3 and 3.1.4 993.9 Proof of Theorem 3.1.4 1063.10 Proof of Theorem 3.1.3 1123.11 Proofs of Theorems 3.1.1 and 3.1.5 1143.12 Examples 114

4. INFINITE HORIZON AUTONOMOUS PROBLEMS 1154.1 Main results 1154.2 Proofs of Theorems 4.1.1-4.1.3 1194.3 Proof of Theorem 4.1.4 150

5. TURNPIKE FOR AUTONOMOUS PROBLEMS 1535.1 Main results 1535.2 Proof of Theorem 5.1.1 1585.3 Proof of Theorem 5.1.2 1695.4 Examples 172

6. LINEAR PERIODIC CONTROL SYSTEMS 1736.1 Main results 1736.2 Preliminary results 1766.3 Discrete-time control systems 1836.4 Proof of Theorem 6.1.1 1866.5 Proof of Theorem 6.1.2 1886.6 Proof of Theorem 6.1.3 1906.7 Proof of Theorem 6.1.4 193

7. LINEAR SYSTEMS WITH NONPERIODIC INTEGRANDS 1977.1 Main results 1977.2 Preliminary results 2017.3 Discrete-time control systems 2037.4 Proof of Theorem 7.1.1 2047.5 Proof of Theorem 7.1.2 2097.6 Proofs of Theorems 7.1.3 and 7.1.4 215

8. DISCRETE-TIME CONTROL SYSTEMS 2238.1 Convex innite dimensional control systems 2238.2 Preliminary results 2268.3 Proofs of Theorems 8.1.1 and 8.1.2 230

Contents vii

8.4 Nonautonomous control systems in metric spaces 2368.5 An auxiliary result 2398.6 Proof of Theorem 8.4.1 248

9. CONTROL PROBLEMS IN HILBERT SPACES 2579.1 Main results 2579.2 Preliminary results 2619.3 Proof of Theorems 9.1.1-9.1.3 2629.4 Proof of Theorems 9.1.4 and 9.1.5 2739.5 Systems with distributed and boundary controls 277

10. A CLASS OF DIFFERENTIAL INCLUSIONS 28310.1 Main result 28310.2 Preliminary results 28810.3 Sucient condition for the turnpike property 29410.4 Preliminary lemmas 29810.5 Proof of Theorem 10.1.1 31010.6 Example 318

11. CONVEX PROCESSES 32111.1 Preliminaries 32111.2 Asymptotic turnpike property 32211.3 Turnpike theorems 32411.4 Proofs of Theorems 11.3.1 and 11.3.2 32511.5 Stability of the turnpike phenomenon 33411.6 Proofs of Theorems 11.5.1, 11.5.2 and 11.5.3 337

12. A DYNAMIC ZERO-SUM GAME 34912.1 Preliminaries 34912.2 Main results 35112.3 Denitions and notation 35212.4 Preliminary results 35312.5 The existence of a minimal pair of sequences 35412.6 Preliminary lemmas for Theorem 12.2.1 35712.7 Preliminary lemmas for Theorem 12.2.2 36612.8 Proofs of Theorems 12.2.1 and 12.2.2 372

Comments 381

viii TURNPIKE PROPERTIES

References 387

Index395

Preface

This monograph is devoted to recent progress in the turnpike the-ory. Turnpike properties are well known in mathematical economics.The term was rst coined by Samuelson who showed that an ecientexpanding economy would for most of the time be in the vicinity of abalanced equilibrium path (also called a von Neumann path) [78, 79].These properties were studied by many authors for optimal trajecto-ries of a NeumannGale model determined by a superlinear set-valuedmapping. In the monograph we discuss a number of results concern-ing turnpike properties in the calculus of variations and optimal controlwhich were obtained by the author in the last ten years. These resultsshow that the turnpike properties are a general phenomenon which holdsfor various classes of variational problems and optimal control problems.

Turnpike properties are studied for optimal control problems on -nite time intervals [T1, T2] of the real line. Solutions of such problems(trajectories) always depend on the time interval [T1, T2], an optimalitycriterion which is usually determined by a cost function, and on datawhich is some initial conditions. In the turnpike theory we are inter-ested in the structure of solutions of optimal problems. We study thebehavior of solutions when an optimality criterion is xed while T1, T2and the data vary. To have turnpike properties means, roughly speaking,that the solutions of a problem are determined mainly by the optimalitycriterion (a cost function), and are essentially independent of the choiceof time interval and data, except in regions close to the endpoints of thetime interval. If a point t does not belong to these regions, then thevalue of a solution at t is closed to a trajectory (turnpike) which isdened on the innite time interval and depends only on the optimal-ity criterion. This phenomenon has the following interpretation. If onewishes to reach a point A from a point B by a car in an optimal way,then one should enter onto a turnpike, spend most of ones time on itand then leave the turnpike to reach the required point.

The turnpike phenomenon was discovered by Samuelson in a spe-cic situation. In further numerous studies turnpike properties wereestablished under strong assumptions on an optimality criterion (a costfunction). The usual assumptions were that a cost function is time in-dependent and is convex as a function of all its variables. Under these

x TURNPIKE PROPERTIES

assumptions the turnpike is a stationary trajectory (a singleton). Thesimple form of the turnpike with a convex cost function allowed oneto discover the turnpike property in this case. Since convexity plays animportant role in mathematical economics, turnpike theory has manyapplications in this area of research. It should be mentioned that thereare several interesting results concerning turnpike properties withoutconvexity assumptions. In these results convexity was replaced by otherassumptions. The verication of these assumptions was rather dicultand they hold for a narrow class of problems. Thus the turnpike phe-nomenon was considered by experts as an interesting property of somevery particular problems arising in mathematical economics for whicha turnpike was usually a singleton or a half-ray. This situation haschanged in the last ten years. In this monograph we discuss resultswhich were obtained during this period and allow us today to thinkabout turnpike properties as a general phenomenon which holds for var-ious classes of variational problems and optimal control problems. Toestablish these properties we do not need convexity of a cost functionand its time independence.

It was my great pleasure to receive on October 2000 the following let-ter from Paul A. Samuelson, the discoverer of the turnpike phenomenon.

Dear Professor Zaslavski:

I note with interest your long paper The Turnpike Property ...Func-tions in Nonlinear Analysis 42 (2000), 1465-98.

It may be of interest to report that this property and name originatedjust over half a century ago when, as a Guggenheim Fellow on a 1948-49 sabbatical leave from MIT, I conjectured it in a memo written atthe RAND Corporation in Santa Monica, California. In The CollectedScientic Papers of Paul A. Samuelson, MIT Press, 1966, 1972, 1977,1986, it is reproduced. R. Dorfman, P.A. Samuelson, R.M. Solow, LinearProgramming and Economic Analysis, McGraw-Hill, 1958 gives a pre-Roy Radner exposition. I believe that somewhere Lionel McKenzie hasgiven a nice survey of the relevant mathematical-economics literature.

With admiration,Paul A. Samuelson

Our studies are based on the following ideas. A turnpike is notnecessarily a singleton or a half-ray. It can be an absolutely contin-uous time-dependent function (trajectory) or a compact subset of Rn.To establish a turnpike property we consider a space of cost functions

PREFACE xi

equipped with a natural complete metric and show that a turnpike prop-erty holds for most elements of this space in the sense of Baire categories.We obtain a turnpike theorem in the following way. We consider an op-timality criterion (a cost function f) and show that for a problem withthis criterion there exists an optimal trajectory, say Xf , on an innitetime interval. Then we perturb our cost function by some nonnegativesmall perturbation which is zero only on Xf . We show that for our newcost function f the trajectory Xf is a turnpike, and that optimal solu-tions of the problem with a cost function g which is closed to f , are alsomost of the time close to Xf .

Alexander J. Zaslavski

June 2005

Introduction

Let us consider the following problem of the calculus of variations: T0

f(v(t), v(t))dt min, (P0)

v : [0, T ] Rn is an absolutely continuous functionsuch that v(0) = y, v(T ) = z.

Here T is a positive number, y and z are elements of the n-dimensionalEuclidean space Rn and an integrand f : RnRn R1 is a continuousfunction.

We are interested in the structure of solutions of the problem (P0)when y, z and T vary and T is suciently large.

Assume that the function f is strictly convex and dierentiable andsatises the following growth condition:

f(y, z)/(|y|+ |z|) as |y|+ |z| .Here we denote by | | the Euclidean norm in Rn and by < , > thescalar product in Rn. In order to analyse the structure of minimizers ofthe problem (P0) we consider the auxiliary minimization problem:

f(y, 0) min, y Rn. (P1)It follows from the growth condition and the strict convexity of f thatthe problem (P1) has a unique solution which will be denoted by y.Clearly,

f/y(y, 0) = 0.

Dene an integrand L : Rn Rn R1 byL(y, z) = f(y, z) f(y, 0) < f(y, 0), (y, z) (y, 0) >

= f(y, z) f(y, 0) < (f/z)(y, 0), z > .

xiv TURNPIKE PROPERTIES

Clearly L is also dierentiable and srictly convex and satises the samegrowth condition as f :

L(y, z)/(|y|+ |z|) as |y|+ |z| .Since f and L are strictly convex we obtain that

L(y, z) 0 for all (y, z) Rn Rn

andL(y, z) = 0 if and only if y = y, z = 0.

Consider the following auxiliary problem of the calculus of variations: T0

L(v(t), v(t))dt min, (P2)

v : [0, T ] Rn is an absolutely continuous functionsuch that v(0) = y, v(T ) = z,

where T > 0 and y, z Rn. It is easy to see that for any absolutelycontinuous function x : [0, T ] Rn with T > 0, T

0L(x(t), x(t))dt

= T0

[f(x(t), x(t)) f(y, 0) < (f/z)(y, 0), x(t) >]dt

= T0

f(x(t), x(t))dt + Tf(y, 0) < (f/z)(y), x(T ) x(0) > .

These equations imply that the problems (P0) and (P2) are equivalent:a function x : [0, T ] Rn is a solution of the problem (P0) if and onlyif it is a solution of the problem (P2).

The integrand L : Rn Rn R1 has the following property:(C) If {(yi, zi)}i=1 Rn Rn satises limi L(yi, zi) = 0, then

limi yi = y and limi zi = 0.Indeed, assume that

{(yi, zi)}i=1 Rn Rn and limi

L(yi, zi) = 0.

By the growth condition the sequence {(yi, zi)}i=1 is bounded. Let (y, z)be a limit point of the sequence {(yi, zi)}i=1. Then,

L(y, z) = limi

L(yi, zi) = 0

INTRODUCTION xv

and (y, z) = (y, 0).

This implies that (y, 0) = limi(yi, zi).Let y, z Rn, T > 2 and a function x : [0, T ] Rn be an optimal

solution of the problem (P0). Then x is also an optimal solution of theproblem (P2). We will show that T

0L(x(t), x(t))dt 2c0(|y|, |z|)

where c0(|y|, |z|) is a constant which depends only on |y| and |z|.Dene a function x : [0, T ] Rn by

x(t) = y + t(y y), t [0, 1], x(t) = y, t [1, T 1],x(t) = y + (t (T 1))(z y), t [T 1, T ].

It follows from the denition of x and x that T0

L(x(t), x(t))dt T0

L(x(t), x(t))dt

= 10

L(x(t), y y)dt + T11

L(y, 0)dt + TT1

L(x(t), z y)dt

= 10

L(x(t), y y)dt + TT1

L(x(t), z y)dt.

It is not dicult to see that the integrals 10

L(x(t), y y)dt and TT1

L(x(t), z y)dt

do not exceed a constant c0(|y|, |z|) which depends only on |y|, |z|. Thus T0

L(x(t), x(t))dt 2c0(|y|, |z|).

It is very important that in this inequality the constant c0(|y|, |z|)does not depend on T .

We denote by mes(E) the Lebesgue measure of a Lebesgue mesurableset E R1.

Now let be a positive number. By the property (C) there is > 0such that if (y, z) RnRn and L(y, z) , then |y y|+ |z| . Thenby the choice of and the inequality

T0 L(x(t), x

(t))dt 2c0(|y|, |z|),mes{t [0, T ] : |(x(t), x(t)) (y, 0)| > }

xvi TURNPIKE PROPERTIES

mes{t [0, T ] : L(x(t), x(t)) > }

1 T0

L(x(t), x(t))dt 12c0(|y|, |z|)and

mes{t [0, T ] : |x(t) y| > } 12c0(|y|, |z|).Therefore the optimal solution x spends most of the time in an -

neighborhood of the point y. The Lebesgue measure of the set of allpoints t, for which x(t) does not belong to this -neighborhood, does notexceed the constant 21c0(|y|, |z|) which depends only on |y|, |z| and and does not depend on T . Following the tradition, the point y is calledthe turnpike. Moreover we can show that the set

{t [0, T ] : |x(t) y| > }is contained in the union of two intervals [0, 1] [T 2, T ], where0 < 1, 2 21c0(|y|, |z|).

Under the assumptions posed on f , the structure of optimal solutionsof the problem (P0) is rather simple and the turnpike y is calculatedeasily. On the other hand the proof is strongly based on the convexity off and its time independence. The approach used in the proof cannot beemployed to extend the turnpike result for essentially larger classes ofvariational problems. For such extensions we need other approaches andideas. The question of what happens if the integrand f is nonconvex andnonautonomous seems very interesting. What kind of turnpike and whatkind of convergence to the turnpike do we have for general nonconvexnonautonomous integrands? The following example helps to understandthe problem.

Let

f(t, x, u) = (x cos(t))2 + (u + sin(t))2, (t, x, u) R1 R1 R1

and consider the family of the variational problems T2T1

[(v(t) cos(t))2 + (v(t) + sin(t))2]dt min, (P3)

v : [T1, T2] R1 is an absolutely continuous functionsuch that v(T1) = y, v(T2) = z,

where y, z, T1, T2 R1 and T2 > T1. The integrand f depends on t,for each t R1 the function f(t, , ) : R2 R1 is convex, and for eachx, u R1 \ {0} the functon f(, x, u) : R1 R1 is nonconvex. Thus thefunction f : R1 R1 R1 R1 is also nonconvex and depends on t.

INTRODUCTION xvii

Assume that y, z, T1, T2 R1, T2 > T1 + 2 and v : [T1, T2] R1 is anoptimal solution of the problem (P3). Note that the problem (P3) hasa solution since f is continuous and f(t, x, ) : R1 R1 is convex andgrows superlinearly at innity for each (t, x) [0,)R1.

Dene v : [T1, T2] R1 byv(t) = y + (cos(1) y)(t T1), t [T1, T1 + 1],

v(t) = cos(t), t [T1 + 1, T2 1],v(t) = cos(T2 1) + (t T2 + 1)(z cos(T2)), t [T2 1, T2].

It is easy to see that T21T1+1

f(t, v(t), v(t))dt = 0

and T2T1

f(t, v(t), v(t))dt T2T1

f(t, v(t), v(t))dt

= T1+1T1

f(t, v(t), v(t))dt + T2T21

f(t, v(t), v(t))dt

2 sup{|f(t, x, u)| : t, x, u R1, |x|, |u| |y|+ |z|+ 1}.Thus T2

T1f(t, v(t), v(t))dt c1(|y|, |z|),

where

c1(|y|, |z|) = 2 sup{|f(t, x, u)| : t, x, u R1, |x|, |u| |y|+ |z|+ 1}.For any (0, 1) we have

mes{t [T1, T2] : |v(t) cos(t)| > }

2 T2T1

f(t, v(t), v(t))dt 2c1(|y|, |z|).

Since the constant c1(|y|, |z|) does not depend on T2 and T1 we concludethat if T2T1 is suciently large, then the optimal solution v(t) is equalto cos(t) up to for most t [T1, T2]. Again, as in the case of convextime independent problems we can show that

{t [T1, T2] : |x(t) cos(t)| > } [T1, T1 + ] [T2 , T2]where > 0 is a constant which depends only on , |y| and |z|.

xviii TURNPIKE PROPERTIES

This example shows that there exist nonconvex time dependent inte-grands which have the turnpike property with the same type of conver-gence as in the case of convex autonomous variational problems. Thedierence is that the turnpike is not a singleton but an absolutely con-tinuous time dependent function dened on the innite interval [0,).This leads us to the following denition of the turnpike property forgeneral integrands.

Let us consider the following variational problem: T2T1

f(t, v(t), v(t))dt min, (P )

v : [T1, T2] Rn is an absolutely continuous function

such that v(T1) = y, v(T2) = z.

Here T1 < T2 are real numbers, y and z are elements of the n-dimensionalEuclidean space Rn and an integrand f : [0,) Rn Rn R1 is acontinuous function.

We say that the integrand f has the turnpike property if there existsa locally absolutely continuous function Xf : [0,) Rn (called theturnpike) which depends only on f and satises the following condi-tion:

For each bounded set K Rn and each > 0 there exists a constantT (K, ) > 0 such that for each T1 0, each T2 T1 + 2T (K, ), eachy, z K and each optimal solution v : [T1, T2] Rn of variationalproblem (P), the inequality |v(t) Xf (t)| holds for all t [T1 +T (K, ), T2 T (K, )].

The turnpike property is very important for applications. Supposethat the integrand f has the turnpike property, K and are given, andwe know a nite number of approximate solutions of the problem(P). Then we know the turnpike Xf , or at least its approximation, andthe constant T (K, ) which is an estimate for the time period requiredto reach the turnpike. This information can be useful if we need tond an approximate solution of the problem (P) with a new timeinterval [T1, T2] and the new values y, z K at the end points T1 andT2. Namely instead of solving this new problem on the large interval[T1, T2] we can nd an approximate solution of problem (P) on thesmall interval [T1, T1 + T (K, )] with the values y,Xf (T1 + T (K, ))at the end points and an approximate solution of problem (P) on thesmall interval [T2 T (K, ), T2] with the values Xf (T2 T (K, )), zat the end points. Then the concatenation of the rst solution, thefunction Xf : [T1 + T (K, ), T2 T (K, )] and the second solution is an

INTRODUCTION xix

approximate solution of problem (P) on the interval [T1, T2] with thevalues y, z at the end points.

We begin our monograph with a discussion of the problem (P). InChapter 1 we introduce a space M of continuous integrands f : [0,)Rn Rn R1. This space is equipped with a natural complete metric.We show that for any initial condition x0 Rn there exists a locallyabsolutely continuous function x : [0,) Rn with x(0) = x0 such thatfor each T1 0 and T2 > T1 the function x : [T1, T2] Rn is a solutionof problem (P) with y = x(T1) and z = x(T1). We also establish thatfor every bounded set E Rn the C([T1, T2]) norms of approximatesolutions x : [T1, T2] Rn for the problem (P) with y, z E arebounded by some constant which does not depend on T1 and T2.

In Chapter 2 we establish the turnpike property stated above for ageneric integrand f M. We establish the existence of a set F Mwhich is a countable intersection of open everywhere dense sets in Msuch that for each f F the turnpike property holds. Moreover we showthat the turnpike property holds for approximate solutions of variationalproblems with a generic integrand f and that the turnpike phenomenonis stable under small pertubations of a generic integrand f .

In Chapters 3-5 we study turnpike properties for autonomous prob-lems (P) with integrands f : Rn Rn R1 which do not depend ont. Since the turnpike theorems of Chapter 2 are of generic nature andthe subset of M which consists of all time independent integrands arenowhere dense, the results of Chapter 2 can not be applied for this sub-set. Moreover, we cannot expect to obtain the turnpike property statedabove for the general autonomous case. Indeed, if an integrand f doesnot depend on t and has a turnpike, then this turnpike should also betime independent. It means that the turnpike is a stationary trajectory(a singleton). But it is not true when a time independent integrand f isnot a convex function.

Consider the following example. Let

f(x1, x2, u1, u2) = (x21 + x22 1)2 + (u1 + x2)2 + (u2 x1)2,

(x1, x2, u1, u2) R2 R2

and consider the family of the variational problems T0

f(v1(t), v2(t), v1(t), v2(t))dt min, (P4)

(v1, v2) : [0, T ] R2 is an absolutely continuous functionsuch that (v1, v2)(0) = y, (v1, v2)(T ) = z,

xx TURNPIKE PROPERTIES

where y = (y1, y2), z = (z1, z2) R2 and T > 0. The integrand f doesnot depend on t. Since f is continuous and for each x = (x1, x2) R2 thefunction f(x, ) : R2 R1 is convex and grows superlinearly at innity,the problem (P4) has a solution for each T > 0 and each y, z R2.Clearly, if T > 0, y = (cos(0), sin(0)) and z = (cos(T ), sin(T )), then thefunction

x1(t) = cos(t), x2(t) = sin(t), t [0, T ]is a solution of the problem (P4). Thus, if the integrand f has a turnpikeproperty, then the turnpike is not a singleton.

Let T > 2, y, z R2 and let v = (v1, v2) : [0, T ] R2 be a solutionof the problem (P4). Dene a function v = (v1, v2) : [0, T ] Rn by

v(t) = y + t((cos(1), sin(1)) y), t [0, 1],v(t) = (cos(t), sin(t)), t [1, T 1],

v(t) = (cos(T 1), sin(T 1)) + (t T +1)(z (cos(T 1), sin(T 1)),t [T 1, T ].

Then T11

f(v(t), v(t))dt = 0

and T0

(v1(t)2 + v2(t)2 1)2dt T0

f(v(t), v(t))dt

T0

f(v(t), v(t))dt

= 10

f(v(t), v(t))dt + TT1

f(v(t), v(t))dt

sup{f(x1, x2, u1, u2) : x1, x2, u1, u2 R1and |xi|, |ui| 2|y|+ 2|z|+ 2, i = 1, 2}.

Thus T0

(v1(t)2 + v2(t)2 1)2dt c2(|y|, |z|)with

c2(|y|, |z|) = sup{f(x1, x2, u1, u2) : x1, x2, u1, u2 R1and |xi|, |ui| 2|y|+ 2|z|+ 2}.

Here c2(|y|, |z|) depends only on |y|, |z| and does not depend on T . Forany (0, 1) we have

mes{t [0, T ] : ||(v1(t), v2(t))| 1| > }

INTRODUCTION xxi

mes{t [0, T ] : |v1(t)2 + v2(t)2 1| > 2}

4 T0

(v1(t)2 + v22 1)2dt

4c2(|y|, |z|).It means that for most t [0, T ], v(t) belongs to the -neighborhood ofthe set {x R2 : |x| = 1}. Thus we can say that the integrand f has aweakened version of the turnpike property and the set {|x| = 1} can beconsidered as the turnpike for f .

For a general autonomous nonconvex problem (P) we also have aversion of the turnpike property in which a turnpike is a compact subsetof Rn. This subset depends only on the integrand f .

Consider the following autonomous variational problem: T0

f(z(t), z(t))dt min, z(0 = x, z(T ) = y, (Pa)

z : [0, T ] Rn is an absolutely continuous functionwhere T > 0, x, y Rn and f : R2n R1 is an integrand.

We say that a time independent integrand f = f(x, u) C(R2n)has the turnpike property if there exists a compact set H(f) Rn suchthat for each bounded set K Rn and each > 0 there exist numbersL1 > L2 > 0 such that for each T 2L1, each x, y K and an optimalsolution v : [0, T ] Rn for the variational problem (Pa), the relation

dist(H(f), {v(t) : t [, + L2]}) holds for each [L1, T L1]. (Here dist(, ) is the Hausdor metric).

We also consider a weak version of this turnpike property for a timeindependent integrand f(x, u). In this weak version, for an optimalsolution of the problem (Pa) with x, y Rn and large enough T , therelation

dist(H(f), {v(t) : t [, + L2]}) with L2, which depends on and |x|, |y| and a compact set H(f) Rndepending only on the integrand f , holds for each [0, T ] \ E whereE [0, T ] is a measurable subset such that the Lebesgue measure of Edoes not exceed a constant which depends on and on |x|, |y|.

These two turnpike properties for autonomous problems (Pa) are con-sidered in Chapters 3-5.

In Chapter 3 we consider the space A of all time independent inte-grands f M. We establish the existence of a set F A which is a

xxii TURNPIKE PROPERTIES

countable intersection of open everywhere dense sets in A such that foreach f F the weakened version of the turnpike property holds.

The turnpike property for time independent integrands is establishedin Chapter 5 for a generic element of a subset N of the space A. Thespace N is a subset of all integrands f A which satisfy some dieren-tiability assumptions.

In the other chapters of the monograph we establish a number ofturnpike results (generic and individual) for various classes of optimalcontrol problems. We study optimal control of linear periodic systemswith convex integrands (Chapter 6) and optimal solutions of linear sys-tems with convex nonperiodic integrands (Chapter 7). In Chapter 8 weestablish turnpike theorems for discrete-time control systems in Banachspaces and in complete metric spaces. Innite-dimensional continuous-time optimal control problems in a Hilbert space are studied in Chapter9. A turnpike theorem for a class of dierential inclusions arising ineconomic dynamics is proved in Chapter 10 and structure of optimaltrajectories of convex processes is studied in Chapter 11. In Chapter 12we establish a turnpike property for a dynamic discrete-time zero-sumgame.

Chapter 1

INFINITE HORIZON

VARIATIONAL PROBLEMS

In this chapter we study existence and uniform boundedness of ex-tremals of variational problems with integrands which belong to a com-plete metric space of functions. We establish that for every bounded setE Rn the C([0, T ]) norms of approximate solutions x : [0, T ] Rnfor the minimization problem on an interval [0, T ] with x(0), x(T ) Eare bounded by some constant which does not depend on T . Given anx0 Rn we study the innite horizon problem of minimizing the expres-sion

T0 f(t, x(t), x

(t))dt as T grows to innity, where x : [0,) Rnsatises the initial condition x(0) = x0. We analyse the existence andthe properties of approximate solutions for every prescribed initial valuex0.

1.1. Preliminaries

Variational and optimal control problems dened on innite intervalsare of interest in many areas of mathematics and its applications [10,11, 16, 32, 62, 63, 88, 89, 95]. These problems arise in engineering [1,3], in models of economic growth [14, 26, 27, 28, 29, 45, 46, 49-52, 60,61, 67, 68, 72, 74, 80, 86, 94], in dynamic games theory [15, 17], ininnite discrete models of solid-state physics related to dislocations inone-dimensional crystals [6, 85] and in the theory of thermodynamicalequilibrium of materials [20, 44, 53-55, 90-92, 95].

We consider the innite horizon problem of minimizing the expression T0

f(t, x(t), x(t))dt

2 TURNPIKE PROPERTIES

as T grows to innity where a function x : [0,) K is locallyabsolutely continuous (a.c.) and satises the initial condition x(0) = x0,K Rn is a closed convex set and f belongs to a complete metric spaceof functions to be described below.

We say that an a.c. function x : [0,) K is (f)-overtaking optimalif

lim supT

T0

[f(t, x(t), x(t)) f(t, y(t), y(t))]dt 0

for any a.c. function y : [0,) K satisfying y(0) = x(0).This notion, known as the overtaking optimality criterion, was in-

troduced in the economics literature by Atsumi [4], Gale [33] and vonWeizsacker [81] and has been used in control theory [3, 13, 14, 16, 39, 40].In general, overtaking optimal solutions may fail to exist. Most studiesthat are concerned with their existence assume convex integrands [13,40, 72].

Another type of optimality criterion for innite horizon problems wasintroduced by Aubry and Le Daeron [6] in their study of the discreteFrenkelKontorova model related to dislocations in one-dimensional crys-tals. More recently this optimality criterion was used in [44, 65, 66, 85].A similar notion was introduced in Halkin [34] for his proof of the max-imum principle.

Let I be either [0,) or (,). We say that an a.c. functionx : I K is an (f)-minimal solution if

T2T1

f(t, x(t), x(t))dt T2T1

f(t, y(t), y(t))dt 0

for each T1 I, T2 > T1 and each a.c. function y : [T1, T2] K whichsatises y(Ti) = x(Ti), i = 1, 2.

It is easy to see that every (f)-overtaking optimal function is an (f)-minimal solution.

In this chapter we consider a functional space of integrands M de-scribed in Section 1.1. We show that for each f M and each z Rnthere exists a bounded (f)-minimal solution Z : [0,) Rn satisfyingZ(0) = z such that any other a.c. function Y : [0,) Rn is notbetter than Z. We also establish that given f M and a bounded setE Rn the C([0, T ]) norms of approximate solutions x : [0, T ] Rnfor the minimization problem on an interval [0, T ] with x(0), x(T ) Eare bounded by some constant which depends only on f and E.

Innite horizon variational problems 3

1.2. Main results

Let a > 0 be a constant and : [0,) [0,) be an increasingfunction such that (t) as t.

Let K Rn be a closed convex set. Denote by | | the Euclidean normin Rn and denote byM the set of continuous functions f : [0,)KRn R1 which satisfy the following assumptions:

A(i) for each (t, x) [0,) K the function f(t, x, ) : Rn R1 isconvex;

A(ii) the function f is bounded on [0,) E for any bounded setE K Rn;

A(iii) for each (t, x, u) [0,)K Rn,f(t, x, u) max{(|x|), (|u|)|u|} a;

A(iv) for each M, > 0 there exist , > 0 such that

|f(t, x1, u1) f(t, x2, u2)| max{f(t, x1, u1), f(t, x2, u2)}for each t [0,), each u1, u2 Rn and each x1, x2 K which satisfy

|xi| M, |ui| , i = 1, 2, max{|x1 x2|, |u1 u2|} ;A(v) for each M, > 0 there exist > 0 such that

|f(t, x1, u1) f(t, x2, u2)| for each t [0,), each u1, u2 Rn and each x1, x2 K which satisfy

|xi|, |ui| M, i = 1, 2, max{|x1 x2|, |u1 u2|} .When K = Rn it is an elementary exercise to show that an inte-

grand f = f(t, x, u) C1([0,)Rn Rn) belongs to M if f satisesAssumptions A(i), A(iii),

sup{|f(t, 0, 0)| : t [0,)}


h2(x) |x|+ 1, x R1

and if the function h3 : R1 R1 is convex andu2 + 1 h3(u) c0(u2 + 1), |h3(u)| c0(u2 + 1)

for all u R1, where c0 is a positive constant, then the functionf(t, x, u) = h1(t) + h2(x)h3(u), (t, x, u) [0,)R1 R1

belongs to M.In Chapters 1-5 we consider variational problems with integrands be-

longing to the space M or to its subspaces. The Assumption A(i) andthe inequality f(t, x, u) (|u|)|u| a in the Assumption A(iii) guar-antee the existence of minimizers of the variational problems. Theseassumptions are common in the literature. We need the inequalityf(t, x, u) (|x|) a in A(iii) in order to show that for every boundedset E Rn the C([0, T ]) norms of approximate solutions x : [0, T ] Rnfor the variational problems on intervals [0, T ] with x(0), x(T ) E arebounded by some constant which does not depend on T . We need theAssumptions A(ii) and A(v) in order to obtain certain properties of ap-proximate solutions for variational problems on intervals [T1, T2] whichdepend on T2T1 and do not depend of T1 and T2. Note that if a func-tion f is Frechet dierentiable, then the Assumption A(v) means thatthe growth of the partial derivatives of f does not exceed the growthof f . We use it in order to establish the continuity of the function Uf

which is dened below.We equip the set M with the uniformity which is determined by the

following base:

E(N, , ) = {(f, g) MM : |f(t, x, u) g(t, x, u)| (2.1)

for each t [0,), each u Rn each x K satisfying |x|, |u| N}{(f, g) MM : (|f(t, x, u)|+ 1)(|g(t, x, u)|+ 1)1 [1, ]

for each t [0,), each u Rn and each x K satisfying |x| N}where N > 0, > 0, > 1 [37].

Clearly, the uniform space M is Hausdor and has a countable base.ThereforeM is metrizable. We will prove in Secton 1.3 that the uniformspace M is complete.

Put

If (T1, T2, x) = T2T1

f(t, x(t), x(t))dt (2.2)


where f M, 0 T1 < T2 < and x : [T1, T2] K is an a.c.function.

For f M, a, b K and numbers T1, T2 satisfying 0 T1 < T2, putUf (T1, T2, a, b) = inf{If (T1, T2, x) : x : [T1, T2] K (2.3)is an a.c. function satisfying x(T1) = a, x(T2) = b},

f (T1, T2, a) = inf{Uf (T1, T2, a, b) : b K}. (2.4)It is easy to see that < Uf (T1, T2, a, b) < for each f M, eacha, b K and each pair of numbers T1, T2 satisfying 0 T1 < T2.

Let f M. We say that an a.c. function x : [0,) K is an(f)-good function if for any a.c. function y : [0,) K,

inf{If (0, T, y) If (0, T, x) : T (0,)} > . (2.5)In this chapter we study the set of (f)-good functions and prove the

following results.

Theorem 1.2.1 For each h M and each z K there exists an (h)-good function Zh : [0,) K satisfying Zh(0) = z such that:

1. For each f M, each z K and each a.c. function y : [0,)K one of the following properties holds:

(i) If (0, T, y) If (0, T, Zf ) as T ;(ii) sup{|If (0, T, y) If (0, T, Zf )| : T (0,)} 0 such thatsup{|Zg(t)| : t [0,)} Q

for each g U and each z K satisfying |z| M .3. For each f M and each number M > inf{|u| : u K} there

exist a neighborhood U of f in M and a number Q > 0 such that foreach g U , each z K satisfying |z| M , each T1 0, T2 > T1 andeach a.c. function y : [T1, T2] K satisfying |y(T1)| M the followingrelation holds:

Ig(T1, T2, Zg) Ig(T1, T2, y) +Q.4. If K = Rn, then for each f M and each z Rn the function

Zf : [0,) Rn is an (f)-minimal solution.Corollary 1.2.1 Let f M, z K and let y : [0,) K be an a.c.function. Then y is an (f)-good function if and only if condition (ii) ofAssertion 1 of Theorem 1.2.1 holds.


Theorem 1.2.2 For each f M there exist a neighborhood U of f inM and a number M > 0 such that for each g U and each (g)-goodfunction x : [0,) K,

lim supt

|x(t)| < M.

Our next result shows that for every bounded set E K the C([0, T ])norms of approximate solutions x : [0, T ] K for the minimizationproblem on an interval [0, T ] with x(0), x(T ) E are bounded by someconstant which does not depend on T .

Theorem 1.2.3 Let f M and M1,M2, c be positive numbers. Thenthere exist a neighborhood U of f in M and a number S > 0 such thatfor each g U , each T1 [0,) and each T2 [T1 + c,) the followingproperties hold:

(i) if x, y K satisfy |x|, |y| M1 and if an a.c. function v :[T1, T2] K satises

v(T1) = x, v(T2) = y, Ig(T1, T2, v) Ug(T1, T2, x, y) + M2,

then|v(t)| S, t [T1, T2]; (2.6)

(ii) if x K satises |x| M1 and if an a.c. function v : [T1, T2] Ksatises

v(T1) = x, Ig(T1, T2, v) g(T1, T2, x) +M2,then the inequality (2.6) is valid.

Theorems 1.2.1-1.2.3 have been proved in [98].In the sequel we use the following notation:

B(x, r) = {y Rn : |y x| r}, x Rn, r > 0, (2.7)

B(r) = B(0, r), r > 0.

Chapter 1 is organized as follows. In Section 1.3 we study the spaceM and the dependence of the functionals Uf and If of f . In Section1.4 we associate with any f M a related discrete-time control systemand study its approximate solutions. Theorems 1.2.1-1.2.3 are proved inSection 1.5.


1.3. Auxiliary results

In this section we study the space M and continuity properties of thefunctionals If and Uf . The next proposition follows from AssumptionA(iv).

Proposition 1.3.1 Let f M. Then for each pair of positive numbersM and there exist , > 0 such that the following property holds:

If t [0,) and if u1, u2 Rn and x1, x2 K satisfy|xi| M, |ui| , i = 1, 2, |u1 u2|, |x1 x2| , (3.1)

then

|f(t, x1, u1) f(t, x2, u2)| min{f(t, x1, u1), f(t, x2, u2)}.Proof. Let M, > 0. Choose

0 (0, 81 inf{1, }). (3.2)It follows from Assumption A(iv) that there exist , > 0 such that thefollowing property holds:

If t [0,) and if u1, u2 Rn and x1, x2 K satisfy (3.1), then|f(t, x1, u1) f(t, x2, u2)| 0 sup{f(t, x1, u1), f(t, x2, u2)}. (3.3)

Assume that t [0,), u1, u2 Rn and x1, x2 K satisfy (3.1). Bythe denition of , , (3.2) and (3.3),

min{f(t, x1, u1), f(t, x2, u2)} (1 0)max{f(t, x1, u1), f(t, x2, u2)} (1 0)10 |f(t, x1, u1) f(t, x2, u2)| 1|f(t, x1, u1) f(t, x2, u2)|.Proposition 1.3.1 is proved.

Proposition 1.3.2 The uniform space M is complete.Proof. Assume that {fi}i=1 M is a Cauchy sequence. Clearly, for

each (t, x, u) [0,)KRn the sequence {fi(t, x, u)}i=1 is a Cauchysequence. Then there exists a function f : [0,)K Rn R1 suchthat

f(t, x, u) = limi

fi(t, x, u) (3.4)

for each (t, x, u) [0,)K Rn.In order to prove the proposition it is sucient to show that f satises

Assumption A(iv).


Let M, be positive numbers. Choose a number > 1 for which

2 1 < 81. (3.5)Since {fi}i=1 is a Cauchy sequence there exists an integer j 1 suchthat

(fi, fj) E(M, , ) for any integer i j. (3.6)By (3.5) and the properties of there exists a number 0 such that

0 > 1, (0) 2a, 2(1 + 2(0)1)2 1 < 81. (3.7)Choose 1 > 0 such that

81[(1 + 2(0)1)]2 < . (3.8)

By Proposition 1.3.1 there exist numbers , > 0 such that

> 0

and that for each t [0,), each u1, u2 Rn and each x1, x2 K whichsatisfy (3.1) the inequality

|fj(t, x1, u1) fj(t, x2, u2)| 1 min{fj(t, x1, u1), fj(t, x2, u2)} (3.9)is true.

Assume that t [0,), u1, u2 Rn, x1, x2 K satisfy (3.1). Thenthe inequality (3.9) follows from the denition of , . (2.1), (3.4), (3.6)and (3.1) imply that

(|f(t, xi, ui)|+ 1)(|fj(t, xi, ui)|+ 1)1 [1, ], i = 1, 2. (3.10)It follows from Assumption A(iii), (3.1), (3.7) and (3.9) that

min{f(t, xi, ui), fj(t, xi, ui)} 21(0), i = 1, 2. (3.11)By (3.11) and (3.10),

f(t, xi, ui)fj(t, xi, ui)1

[((1 + 2(0)1))1, (1 + 2(0)1)], i = 1, 2. (3.12)We may assume without loss of generality that

f(t, x1, u1) f(t, x2, u2). (3.13)It follows from (3.12), (3.9), (3.8) and (3.7) that

f(t, x1, u1) f(t, x2, u2) (1 + 2(0)1)fj(t, x1, u1)


((1 + 2(0)1))1fj(t, x2, u2)= (1 + 2(0)1)[fj(t, x1, u1) fj(t, x2, u2)]

+fj(t, x2, u2)[(1 + 2(0)1) ((1 + 2(0)1))1] (1 + 2(0)1)1fj(t, x2, u2) + fj(t, x2, u2)[(1 + 2(0)1)((1 + 2(0)1))1] 1[(1 + 2(0)1)]2f(t, x2, u2)

+f(t, x2, u2)[2(1 + 2(0)1)2 1] f(t, x2, u2).Therefore the function f satises Assumption A(iv). This completes theproof of the proposition.

The next auxiliary result will be used in order to establish the contin-uous dependence of the functional Uf (T1, T2, y, z) of T1, T2, y, z and thecontinuous dependence of the functional If (T1, T2, x) of f .

Proposition 1.3.3 Let M1 > 0 and let 0 < 0 < 1. Then there existsa number M2 > 0 such that the following property holds:

If f M, numbers T1, T2 satisfy0 T1, T2 [T1 + 0, T1 + 1] (3.14)

and if an a.c. function x : [T1, T2] K satisesIf (T1, T2, x) M1, (3.15)

then|x(t)| M2, t [T1, T2]. (3.16)

Proof. By Assumption A(iii) and the properties of the function there exists a number c0 > 0 such that

f(t, x, u) |u| (3.17)for each f M and each (t, x, u) [0,)K Rn satisfying |u| c0,and

f(t, x, u) 2M1(min{1, 0})1 (3.18)for each f M and each (t, x, u) [0,)K Rn satisfying |x| c0.Fix a number

M2 > 1 + M1 + a1 + c0(1 + 1) (3.19)

(recall a in Assumption A(iii)).Let f M, T1, T2 be numbers satisfying (3.14) and let x : [T1, T2]

K be an a.c. function satisfying (3.15). We will show that (3.16) holds.


Assume the contrary. Then there exists t0 [T1, T2] such that|x(t0)| > M2. (3.20)

By the denition of c0, (3.18), (3.14) and (3.15) there exists t1 [T1, T2]satisfying

|x(t1)| c0. (3.21)Set

E = [inf{t0, t1}, sup{t0, t1}], E1 = {t E : |x(t)| c0}, E2 = E \ E1.(3.22)

By the denition of c0, Assumption A(iii), (3.15), (3.22), (3.14) and(3.17),

|x(t1) x(t0)| E1|x(t)|dt +

E2|x(t)|dt

1c0 +E1|x(t)|dt 1c0 +

E1

f(t, x(t), x(t))dt

1c0 + If (T1, T2, x) + a1 1(c0 + a) + M1.It follows from this inequality, (3.20) and (3.21) that

M2 c0 1(c0 + a) +M1.This is contradictory to (3.19). The obtained contradiction proves theproposition.

The following propositon establishes an important property which willbe used in Chapter 2.

Proposition 1.3.4 Let M1, > 0 and let 0 < 0 < 1. Then thereexists a positive number such that for each f M and each pair ofnumbers T1, T2 satisfying (3.14) the following property holds:

If an a.c. function x : [T1, T2] K satises (3.15) and if t1, t2 [T1, T2] satises |t1 t2| , then |x(t1) x(t2)| .

Proof. By Assumption A(iii) and the properties of the function thereexists a number c0 > 0 such that for each f M and each (t, x, u) [0,)K Rn satisfying |u| c0 the inequality

f(t, x, u) 41(M1 + 2 + a1)|u| (3.23)is true. Choose

(0, 81(c0 + 1)1). (3.24)


Assume that f M, numbers T1, T2 satisfy (3.14), an a. c. functionx : [T1, T2] K satises (3.15) and

t1, t2 [T1, T2], 0 < |t1 t2| . (3.25)Set

E = [min{t1, t2}, max{t1, t2}],E1 = {t E : |x(t)| c0}, E2 = E \ E1.

By Assumption A(iii), the choice of c0, (3.14), (3.25) and (3.23),

|x(t2) x(t1)| E1|x(t)|dt +

E2|x(t)|dt c0 +

E1|x(t)|dt

c0 + [4(M1 + 2 + a1)]1E1

f(t, x(t), x(t))dt

c0 + [4(M1 + 2 + a1)]1(If (T1, T2, x) + a1).Combined with (3.15), (3.14) and (3.24) this inequality implies that

|x(t2) x(t1)| c0 + 41 .This completes the proof of the proposition.

We have the following result (see [9]).

Proposition 1.3.5 Assume that f M, M1 > 0, 0 T1 < T2, xi :[T1, T2] K, i = 1, 2, . . . is a sequence of a.c. functions such that

If (T1, T2, xi) M1, i = 1, 2, . . . .Then there exists a subsequence {xik}k=1 and an a.c. function x :[T1, T2] K such thatIf (T1, T2, x) M1, xik x(t) as k uniformly in [T1, T2] and

xik x as k weakly in L1(Rn; (T1, T2)).Corollary 1.3.1 For each f M, each pair of numbers T1, T2 sat-isfying 0 T1 < T2 and each z1, z2 K there exists an a.c. func-tion x : [T1, T2] K such that x(Ti) = zi, i = 1, 2, If (T1, T2, x) =Uf (T1, T2, z1, z2).

Corollary 1.3.2 For each f M, each T1, T2 satisfying 0 T1 < T2and each z K there exists an a.c. function x : [T1, T2] K such thatx(T1) = z, If (T1, T2, x) = f (T1, T2, z).


It is an elementary exercise to prove the following result.

Proposition 1.3.6 Let f M, 0 < c1 < c2 0. Thenthere exists a neighborhood U of f in M such that the set

{Ug(T1, T2, z1, z2) : g U, T1 [0,), T2 [T1 + c1, T1 + c2],z1, z2 K B(c3), i = 1, 2}

is bounded.

The next auxiliary result establishes the continuity of the functional(T1, T2, y, z) Uf (T1, T2, y, z).Proposition 1.3.7 Assume that K = Rn, f M, 0 < c1 < c2 < and M, > 0. Then there exists > 0 such that the following propertyholds:

If T1, T2 0 satisfyT2 [T1 + c1, T1 + c2] (3.26)

and if y1, y2, z1, z2 Rn satisfy|yi|, |zi| M, i = 1, 2, sup{|y1 y2|, |z1 z2|} , (3.27)

then|Uf (T1, T2, y1, z1) Uf (T1, T2, y2, z2)| . (3.28)

Proof. By Proposition 1.3.6 there exists a number

M0 > sup{|Uf (T1, T2, y, z)| : T1 [0,), T2 [T1+c1, T1+c2], (3.29)y, z B(M)}.

It follows from Proposition 1.3.3 that there exists a number M1 > 0 suchthat the following property holds:

If a pair of numbers T1, T2 0 satises (3.26) and an a.c. functionx : [T1, T2] Rn satises

If (T1, T2, x) 4M0 + 1,then

|x(t)| M1, t [T1, T2]. (3.30)Choose a number 1 > 0 such that

41(2c2 + 2a + 4ac2 + 1 +M0) < (3.31)


(see Assumption A(iii)). By Proposition 1.3.1 there exist

0 > 2 and 2 (0, 81) (3.32)such that

|f(t, x1, u1) f(t, x2, u2)| 1 inf{f(t, x1, u1), f(t, x2, u2)} (3.33)for each t [0,) and each u1, u2, x1, x2 Rn which satisfy|xi| M1 + 1, |ui| 0 1, i = 1, 2, |u1 u2|, |x1 x2| 2. (3.34)

By Assumption A(iv) there exists

3 (0, 41 inf{1, 2}) (3.35)such that

|f(t, x1, u1) f(t, x2, u2)| 1 (3.36)for each t [0,), each u1, u2, x1, x2 Rn which satisfy|xi|, |ui| 0 +M1 + 4, i = 1, 2, sup{|x1 x2|, |u1 u2|} 3. (3.37)

Choose a number > 0 for which

8(c11 + 1) < 3. (3.38)

Assume that numbers T1, T2 0 satisfy (3.26) and y1, y2, z1, z2 Rnsatisfy (3.27). By Corollary 1.3.1 there exists an a.c. function x1 :[T1, T2] Rn such that

x1(T1) = y1, x1(T2) = z1, If (T1, T2, x1) = Uf (T1, T2, y1, z1). (3.39)

Put

x2(t) = x1(t)+y2y1+(tT1)(T2T1)1(z2z1y2+y1), t [T1, T2].(3.40)

Clearlyx2(T1) = y2, x2(T2) = z2. (3.41)

It follows from (3.26), (3.27), (3.39), (3.29) and the denition of M1 that

|x1(t)| M1, t [T1, T2]. (3.42)(3.40), (3.27) and (3.26) imply that

|x1(t) x2(t)| 3, |x1(t) x2(t)| 2c11 , t [T1, T2]. (3.43)


Set

E1 = {t [T1, T2] : |x1(t)| 0}, E2 = [T1, T2] \ E1. (3.44)We have

|If (T1, T2, x2) If (T1, T2, x1)| 1 + 2 (3.45)where

j =Ej

|f(t, x1(t), x1(t)) f(t, x2(t), x2(t))|dt, j = 1, 2. (3.46)

We will estimate 1, 2 separately. By (3.42), (3.43), (3.44), (3.38),(3.35), (3.32) and the denition of 2 for each t E1,

|f(t, x1(t), x1(t)) f(t, x2(t), x2(t))| 1f(t, x1(t), x1(t))

and1 1

E1

f(t, x1(t), x1(t))dt.

It follows from this inequality, (3.39), (3.27), (3.29), (3.26) and Assump-tion A(iii) that

1 1(If (T1, T2, x1) + a(T2 T1)) 1(M0 + ac2). (3.47)By the denition of 3, (3.42), (3.43), (3.38) and (3.44),

|f(t, x1(t), x1(t)) f(t, x2(t), x2(t))| 1for each t E2 and

2 1c2. (3.48)Combining (3.45), (3.47), (3.48) and (3.31) we obtain that

|If (T1, T2, x2) If (T1, T2, x1)| 1(M0 + ac2 + c2) .Together with (3.39) and (3.41) this implies that

Uf (T1, T2, y2, z2) Uf (T1, T2, y1, z1) + .This completes the proof of the proposition.

The next proposition is an important tool which will be used in Chap-ters 1-5. It establishes that the integral functional If (T1, T2, x) dependscontinuously on f .

Proposition 1.3.8 Let f M, 0 < c1 < c2 < , D, > 0. Thenthere exists a neighborhood V of f in M such that for each g V , each


pair of numbers T1, T2 0 satisfying T2 T1 [c1, c2] the followingproperty holds:

If an a. c. function x : [T1, T2] K satisesinf{If (T1, T2, x), Ig(T1, T2, x)} D, (3.49)

then|If (T1, T2, x) Ig(T1, T2, x)| .

Proof. It follows from Proposition 1.3.3 there exists a number S > 0such that

|x(t)| S, t [T1, T2] (3.50)for each g M, each T1, T2 0 satisfying T2 T1 [c1, c2] and eacha.c. function x : [T1, T2] K which satises Ig(T1, T2, x) D + 1.

Choose (0, 1), N > S and > 1 such that(c2+1) 41, (N)N > 4a, (1)(c2+D+ac2+1) 41 (3.51)

and putV = {g M : (f, g) E(N, ,)}

(see (2.1)). Assume that g V ,T1, T2 0, T2 T1 [c1, c2] (3.52)

and x : [T1, T2] K is an a.c. function satisfying (3.49). By the choiceof S the inequality (3.50) is true. Put

E1 = {t [T1, T2] : |x(t)| N}, E2 = [T1, T2] \ E1.It follows from (3.50) and the choice of V and N that

|f(t, x(t), x(t)) g(t, x(t), x(t))| , t E1. (3.53)Dene

h(t) = inf{f(t, x(t), x(t)), g(t, x(t), x(t))}, t [T1, T2]. (3.54)It follows from (3.50), (3.51), Assumption A(iii) and the denition of V ,N that for t E2

(f(t, x(t), x(t)) + 1)(g(t, x(t), x(t)) + 1)1 [1,], (3.55)|f(t, x(t), x(t)) g(t, x(t), x(t))| ( 1)(h(t) + 1).

By (3.53), (3.52), (3.55), (3.49), (3.54), Assumption A(iii) and (3.51),

|If (T1, T2, x) Ig(T1, T2, x)| E1|f(t, x(t), x(t)) g(t, x(t), x(t))|dt


+E2|f(t, x(t), x(t)) g(t, x(t), x(t))|dt c2 + ( 1)

E2

(h(t) + 1)dt

c2 + ( 1)c2 + ( 1)(D + ac2) .The proposition is proved.

The next result establishes that the functional Uf (T1, T2, y, z) dependscontinuously on f . It is also an important tool which will be used inChapters 1-5.

Proposition 1.3.9 Let f M, 0 < c1 < c2 < , c3, > 0. Thenthere exists a neighborhood V of f in M such that

|Uf (T1, T2, y, z) Ug(T1, T2, y, z)|

for each g V , each T1, T2 0 satisfying T2 T1 [c1, c2] and eachy, z K B(c3).

Proof. By Proposition 1.3.6 there exist a neighborhood V1 of f in Mand a positive number D0 such that

|Ug(T1, T2, z1, z2)|+ 1 < D0for each g V1, each T1 [0,), T2 [T1 + c1, T1 + c2] and eachz1, z2 K B(c3), i = 1, 2. It follows from Proposition 1.3.8 that thereis a neighborhood V of f in M such that V V1 and that

|If (T1, T2, x) Ig(T1, T2, x)| inf{1, }

for each g V , each T1, T2 0 satisfying T2T1 [c1, c2] and each a.c.function x : [T1, T2] K which satisfy

min{If (T1, T2, x), Ig(T1, T2, x)} D0 + 2.

The validity of the proposition now follows from the equality

Ug(T1, T2, y, z) = inf{Ig(T1, T2, x) :

x : [T1, T2] K is an a.c. functionsatisfying x(T1) = y, x(T2) = z, Ig(T1, T2, x) D0 + 1}

which holds for g V , T1 0, T2 [T1 + c1, T1 + c2] and y, z Ksatisfying |y|, |z| c3.


1.4. Discrete-time control systems

In this section we associate with f M a related discrete-time controlsystem. We establish a boundedness of approximate solutions of thissystem (see Proposition 1.4.2). This result plays a crucial role in theproof of Theorem 1.2.3.

Let f M, z K and let 0 < c1 < c2 < . It follows from Propo-sition 1.3.6 that there exist a positive number M0 and a neighborhoodU0 of f in M such that

|Ug(T1, T2, y, z)| M0 for each g U0,each T1 [0,), T2 [T1 + c1, T1 + c2] (4.1)

and each y, z K B(2|z|+ 1).Proposition 1.3.3 implies that there is a number M1 > 0 such that

2M0 + 2 Ug(T1, T2, y, z) for each g M,each T1 [0,), T2 [T1 + c1, T1 + c2], (4.2)

and each y, z K satisfying |y|+ |z| M1.Proposition 1.4.1 Let a number M1 > 0 satisfy (4.2) and let M2 > 0.Then there are an integer N > 2 and a neighborhood U of f in M suchthat for each g U , each [0,), each T [c1, c2] and each pair ofintegers q1, q2 satisfying 0 q1 < q2, q2q1 N the following assertionshold:

1. If {zi}q2i=q1 K satises{i {q1, . . . , q2} : |zi| M1} = {q1, q2}

and if yi = zi, i = q1, q2, yi = z, i = q1 + 1, . . . , q2 1, thenq21i=q1

[Ug( + iT,+ (i+ 1)T, zi, zi+1)

Ug( + iT,+ (i + 1)T, yi, yi+1)] M2; (4.3)2. If {zi}q2i=q1 K satises

{i {q1, . . . , q2} : |zi| M1} = {q1}and if yq1 = zq1, yi = z, i = q1 + 1, . . . , q2, then the inequality (4.3) isvalid.


Proof. It follows from Proposition 1.3.6 that there exist a positivenumber M3 and a neighborhood U of f in M such that

U U0,|Ug(T1, T2, y, z)| M3 for each g U, each T1 [0,),

T2 [T1 + c1, T1 + c2] and each y, z K B(2|z|+ 1 + 2M1).Fix an integer N M2 + 4M3 + 4. The validity of the proposition nowfollows from the denition of U , M3, N , (4.1) and (4.2).

Proposition 1.4.2 Assume that a positive number M1 satises (4.2)and M3 > 0. Then there exist a neighborhood V of f in M and anumber M4 > M1 such that for each g V , each [0,), eachT [c1, c2] and each pair of integers q1, q2 satisfying 0 q1 < q2 thefollowing assertions hold:

1. If a sequence {zi}q2i=q1 K satises|zq1 |, |zq2 | M1, max{|zi| : i = q1, . . . , q2} > M4, (4.4)

then there is a sequence {yi}q2i=q1 K such that yqj = zqj , j = 1, 2 andq21i=q1

[Ug( + iT,+ (i + 1)T, zi, zi+1)

Ug( + iT,+ (i+ 1)T, yi, yi+1)] M3. (4.5)2. If a sequence {zi}q2i=q1 K satises

|zq1 | M1, max{|zi| : i = q1, . . . , q2} > M4, (4.6)then there is a sequence {yi}q2i=q1 K such that yq1 = zq1 and the in-equality (4.5) is true.

Proof. There exist a neighborhood U U0 of f in M and an integerN > 2 such that Proposition 1.4.1 holds with M2 = 4(M3 + 1). Itfollows from Proposition 1.3.6 that there exist a positive number r1 anda neighborhood V of f in M such that

V U, |Ug(T1, T2, y, z)|+ 1 < r1 for each g V, each T1 [0,),(4.7)

T2 [T1 + c1, T1 + c2] and each y, z K B(|z|+ 1 +M1).By Proposition 1.3.3 there exists M4 > M1 such that

inf{Ug(T1, T2, y, z) : g M, T1 [0,), T2 [T1 + c1, T1 + c2], (4.8)


y, z K, |y|+ |z| M4} > 3r1N + 4 + 4M3 + 3ac2N(recall a in Assumption A(iii)).

Let g V , [0,), T [c1, c2], 0 q1 < q2, {zi}q2i=q1 K.We prove Assertion 1. Assume that (4.4) holds. Then there is j {q1, . . . , q2} such that |zj | > M4. Set

i1 = max{i {q1, . . . , j} : |zi| M1},

i2 = min{i {j, . . . , q2} : |zi| M1}.If i2 i1 N , then by the denition of V , U , N and Proposition 1.4.1there exists a sequence {yi}q2i=q1 K which satises (4.5) and yqi = zqi ,i = 1, 2.

Now assume that i2 i1 < N . Putyi = zi, i {q1, . . . , i1}{i2, . . . , q2}, yi = z, i = i1+1, . . . , i21. (4.9)

It follows from (4.9), (4.7), Assumption A(iii) and the denition ofi1, i2, j that

q21i=q1

[Ug(+ iT,+(i+1)T, zi, zi+1)Ug(+ iT,+(i+1)T, yi, yi+1)]

(4.10)

=i21i=i1

[Ug(+iT,+(i+1)T, zi, zi+1)Ug(+iT,+(i+1)T, yi, yi+1)]

Ug( + (j 1)T,+ jT, zj1, zj) a(i2 i1 1)c2 (i2 i1)r1.By this relation and the denition of j, M4 (see (4.8))

q21i=q1

[Ug( + iT,+ (i+ 1)T, zi, zi+1) (4.11)

Ug( + iT,+ (i + 1)T, yi, yi+1)] 4M3 + 4.This completes the proof of Assertion 1.

We prove Assertion 2. Assume that (4.6) holds. Then there is j {q1, . . . , q2} such that |zj | > M4. Set i1 = sup{i {q1, . . . , j} : |zi| M1}.

There are two cases: 1) |zi| > M1, i = j, . . . , q2; 2) inf{|zi| : i =j, . . . , q2} M1. Consider the rst case. We set

yi = zi, i = q1, . . . , i1, yi = z, i = i1 + 1, . . . , q2.


If q2 i1 N , then (4.5) follows from the denition of V , U , N andProposition 1.4.1. If q2 i1 < N , then (4.5) follows from the denitionof {yi}q2i=q1 , i1, j,M4, (4.7) (see (4.10), (4.11) with i2 = q2).

Consider the second case. Set i2 = inf{i {j, . . . , q2} : |zi| M1}.If i2 i1 N , then by the denition of V,U,N and Proposition 1.4.1there exists a sequence {yi}q2i=q1 K which satises (4.5) and yqi = zqi ,i = 1, 2. If i2i1 < N we dene a sequence {yi}q2i=q1 K by (4.9). Then(4.10) and (4.11) follows from (4.9), the denition of i1, i2, j,M4, (4.7).Assertion 2 is proved. This completes the proof of the proposition.

1.5. Proofs of Theorems 1.1-1.3

Construction of a neighborhood U . Let f M, z K, M > 2|z|.It follows from Proposition 1.3.6 that there exist a positive number M0and a neighborhood U0 of f in M such

|Ug(T1, T2, y, z)| M0 for each g U0, each T1 [0,), (5.1)T2 [T1 + 41, T1 + 4] and each y, z K B(2|z|+ 1).

It follows from Proposition 1.3.3 that there exists a number M1 > Mfor which

2M0 + 1 < inf{Ug(T1, T2, y, z) : g M,T1 [0,), T2 [T1 + 41, T1 + 4],

y, z K, |y|+ |z| M1}. (5.2)By (5.1), (5.2) there exists a neighborhood U1 of f in M and a numberM2 such that

U1 U0, M2 > M1 and Proposition 1.4.2 holds with M3 = 1, (5.3)c1 = 41, c2 = 4, V = U1, M4 = M2.

Proposition 1.3.6 implies that there exist a positive number Q0 and aneighborhood U2 of f in M such thatU2 U1, |Ug(T1, T2, y, z)|+ 1 Q0 for each g U2, each T1 [0,),

(5.4)T2 [T1 + 41, T1 + 4] and each y, z K B(M2 + 1).

By Proposition 1.3.3 there exists a number

Q1 > Q0 + M2 + 1 (5.5)

such that the following property holds:


If g M, T1, T2 satisfy0 T1 < T2, T2 T1 [41, 4]

and if an a.c. function x : [T1, T2] K satises Ig(T1, T2, x) 2Q0 + 2,then

|x(t)| Q1, t [T1, T2]. (5.6)It follows from Proposition 1.3.6 that there exist a positive number

Q2 and a neighborhood U of f in M such thatU U2, Q2 > Q1,

|Ug(T1, T2, y, z)|+ 1 < Q2 for each g U, (5.7)each T1 [0,), T2 [T1+41, T1+4] and each y, z KB(2Q1+4).We may assume without loss of generality that there exists a positivenumber Q3 such that

|g(t, y, u)|+ 1 < Q3 for each g U, each t [0,) (5.8)

and each y K B(2M2 + 2), u B(2M2 + 2).

Construction of a function Zg : [0,) K. Let g U , z K,|z| M . By Corollary 1.3.2 for any integer q 1 there exists an a. c.function Zgq : [0, q] K such that

Zgq (0) = z, Ig(0, q, Zgq ) =

g(0, q, z). (5.9)

It follows from Proposition 1.4.2 and the denition of Zgq , U1, M2 that

|Zgq (i)| M2, i = 0, . . . , q, q = 1, 2, . . . . (5.10)There exists a subsequence {Zggj}j=1 such that for any integer i 0there exists

zgi = limjZgqj (i). (5.11)

By Corollary 1.3.1 there exists an a.c. function Zg : [0,) K suchthat for each integer i 0,

Zg(i) = zgi , Ig(i, i+ 1, Zg) = Ug(i, i+ 1, zgi , z

gi+1). (5.12)

It follows from (5.9), (5.10) and (5.4) that

Ig(i, i+ 1, Zgq ) < Q0, i = 0, . . . , q 1, q = 1, 2, . . . . (5.13)


(5.10), (5.11), (5.12) and (5.4) imply that

Ig(i, i+ 1, Zg) < Q0, i = 0, 1, . . . . (5.14)

By (5.13), (5.14) and the denition of Q1 (see (5.5), (5.6))

|Zgq (t)| Q1, t [0, q], q = 1, 2, . . . , |Zg(t)| Q1, t [0,). (5.15)Therefore for each g U and each z K satisfying |z| M we denea.c. functions Zgq : [0, q] K, q = 1, 2, . . . and Zg : [0,) Ksatisfying (5.9)-(5.15).

The next auxiliary result shows that the sequence {zqi }i=0 is (g)-goodfor each g U .Lemma 1.5.1 Let g U , z K, |z| M and let a pair of integersq1, q2 satisfy 0 q1 < q2. Then if a sequence {yi}q2i=q1 K satises|yq1 | M1, then

q21i=q1

[Ug(i, i+ 1, zgi , zgi+1) Ug(i, i+ 1, yi, yi+1)] 4 + 4Q2. (5.16)

Proof. Assume that a sequence {yi}q2i=q1 K satises |yq1 | M1. Wewill show that (5.16) holds.

Let us assume the converse. Thenq21i=q1

[Ug(i, i+ 1, zgi , zgi+1) Ug(i, i+ 1, yi, yi+1)] > 4 + 4Q2. (5.17)

By Corollaries 1.3.1 and 1.3.2 we may assume without loss of generalitythat if a sequence {yi}q2i=q1 K satises yq1 = yq1 , then

q21i=q1

[Ug(i, i+ 1, yi, yi+1) Ug(i, i+ 1, yi, yi+1)] 0.

(5.3) and (5.5) imply that

|yi| M2 < Q1, i = q1, . . . , q2. (5.18)By Proposition 1.3.5, (5.9), (5.11) and (5.13) for any integer i 0,

Ug(i, i+ 1, zgi , zgi+1) lim infj U

g(i, i+ 1, Zgqj (i), Zgqj (i+ 1)).

Therefore there exists an integer q > q2 + 1 such thatq2

i=q1

[Ug(i, i+ 1, zgi , zgi+1) Ug(i, i+ 1, Zgq (i), Zgq (i+ 1))] 1. (5.19)


We dene a sequence {hi}qi=0 K as follows:hi = Zgq (i), i {0, . . . , q1} {q2 + 1, . . . , q}, hi = yi, i = q1 + 1, . . . , q2.

(5.20)It follows from (5.20), (5.9), Corollary 1.3.1, (5.19) and (5.17) that

0 q1i=0

[Ug(i, i+ 1, Zgq (i), Zgq (i+ 1)) Ug(i, i+ 1, hi, hi+1)]

=q2

i=q1

[Ug(i, i+ 1, Zgq (i), Zgq (i+ 1)) Ug(i, i+ 1, hi, hi+1)]

=q2

i=q1

[Ug(i, i+ 1, Zgq (i), Zgq (i+ 1)) Ug(i, i+ 1, zgi , zgi+1)]

+q2

i=q1

Ug(i, i+ 1, zgi , zgi+1)

q21i=q1

Ug(i, i+ 1, yi, yi+1)

+Ug(q1, q1 + 1, yq1 , yq1+1)

Ug(q1, q1 + 1, hq1 , hq1+1) Ug(q2, q2 + 1, hq2 , hq2+1) 3 + 4Q2+Ug(q2, q2 + 1, zgq2 , z

gq2+1

) + Ug(q1, q1 + 1, yq1 , yq1+1)

Ug(q1, q1 + 1, hq1 , hq1+1) Ug(q2, q2 + 1, hq2 , hq2+1).Combined with (5.20), (5.18), (5.10), (5.11), (5.5) and (5.7) this relationimplies that

0 3 + 4Q + Ug(q2, q2 + 1, zgq2 , zgq2+1) + Ug(q1, q1 + 1, yq1 , yq1+1)

Ug(q1, q1 + 1, Zgq (q1), yq1+1)Ug(q2, q2 + 1, yq2 , Zgq (q2 + 1)) 3 + 4Q2 4Q2.

The contradiction we have reached proves the lemma.

Lemma 1.5.1 implies the following result.

Lemma 1.5.2 Let g U , z K, |z| M , an integer q 0 and letT (q,). Then

Ig(q, T, Zg) Ig(q, T, x) + 4 + 4Q2 + Q0 + 2a (5.21)for each a.c. function x : [q, T ] K such that |x(q)| M1 (recall a inAssumption A(iii)).


Proof. Assume that an a.c. function x : [q, T ] K satises |x(q)| M1. There exists an integer q1 q such that q1 < T q1 +1. It followsfrom Lemma 1.5.1 and (5.12) that

Ig(q, q1, Zg) Ig(q, q1, x) + 4 + 4Q2. (5.22)By Assumption A(iii) and (5.14)

Ig(q1, T, x) a, Ig(q1, T, Zg) Q0 + a. (5.23)(5.22) and (5.23) imply (5.21). The lemma is proved.

The next lemma, which follows from Lemma 1.5.2, establishes thatthe function Zg is (g)-good for each g U and each z K satisfying|z| M .Lemma 1.5.3 Let g U , z K, |z| M and 0 T1 < T2. Then

Ig(T1, T2, Zg) Ig(T1, T2, x) + 4 + 4Q2 + Q0 +Q3 + 3a (5.24)for each a.c. function x : [T1, T2] K such that |x(T1)| M1.

Proof. Assume that an a.c. function x : [T1, T2] K satises|x(T1)| M1.

There exists an integer q 0 such that q T1 < q + 1. Setx1(t) = x(T1), t [q, T1], x1(t) = x(t), t [T1, T2]. (5.25)

By Lemma 1.5.2,

Ig(q, T2, Zg) Ig(q, T2, x1) + 4 + 4Q2 + Q0 + 2a. (5.26)By Assumption A(iii) and (5.26),

Ig(T1, T2, Zg) = Ig(q, T2, Zg) Ig(q, T1, Zg) Ig(q, T2, Zg) + a (5.27) Ig(q, T2, x1) + 4 + 4Q2 + Q0 + 3a.

It follows from (5.25) and (5.8) that |Ig(q, T1, x1)| Q3. (5.24) nowfollows from this relation and (5.27), (5.25). The lemma is proved.

The following auxiliary result shows that a sequence {yi}i=0 K isnot good if lim supi |yi| > M2. It plays an important role in the proofof Theorem 1.2.2.

Lemma 1.5.4 Let g U , z K, |z| M , {yi}i=0 K,lim supi

|yi| > M2. (5.28)


Then

N1i=0

[Ug(i, i+1, yi, yi+1)Ug(i, i+1, zgi , zgi+1)] as N . (5.29)

Proof. There are two cases:

a) lim infi

|yi| > 21M1; b) lim infi

|yi| 21M1.

Consider the case a). Set hi = z for i = 0, 1, . . .. It follows from (5.1),(5.2) that

Ug(i, i+ 1, yi, yi+1) Ug(i, i+ 1, hi, hi+1) M0 + 1for all large i. (5.29) now follows from this relation and Lemma 1.5.1.

Consider the case b). By (5.28) there exists a subsequence {yik}k=1such that

0 < i1, |yik | < M1, sup{|yj | : j = ik, . . . , ik+1} > M2, k = 1, 2, . . . .(5.30)

It follows from (5.3), (5.30) and Proposition 1.4.2 that for any integerk 1 there exists a sequence {hj}ik+1j=ik K such that hj = yj , j {ik, ik+1},

ik+11j=ik

[Ug(j, j + 1, yj , yj+1) Ug(j, j + 1, hj , hj+1)] 1. (5.31)

Fix an integer q 4. By (5.30), Lemma 1.5.1 and (5.31) for an integerN > iq

N1j=iq

[Ug(j, j + 1, zgj , zgj+1) Ug(j, j + 1, yj , yj+1)] 4 + 4Q2,

iq1j=i1

[Ug(j, j + 1, zgj , zgj+1) Ug(j, j + 1, hj , hj+1)] 4 + 4Q2,

N1j=0

[Ug(j, j + 1, zgj , zgj+1) Ug(j, j + 1, yj , yj+1)]

=i11j=0



+iq1j=i1

[Ug(j, j + 1, zgj , zgj+1) Ug(j, j + 1, hj , hj+1)]

+iq1j=i1

[Ug(j, j + 1, hj , hj+1) Ug(j, j + 1, yj , yj+1)]

+N1j=iq


i11j=0

[Ug(j, j+1, zgj , zgj+1)Ug(j, j+1, yj , yj+1)]+2(4+4Q2) (q1).

This completes the proof of the lemma.The next lemma implies Theorem 1.2.2. Its proof is based on Lemma

1.5.4.

Lemma 1.5.5 Assume that g U , z K, |z| M and y : [0,) Kis an a.c. function which satises

lim supt

|y(t)| > Q1. (5.32)

ThenIg(0, T, y) Ig(0, T, Zg) as T . (5.33)

Proof. There are two cases:

a) lim supi

|y(i)| > M2; b) lim supi

|y(i)| M2

where i is an integer. Consider the case a). It follows from Lemma 1.5.4,(5.12) that

Ig(0, q, y) Ig(0, q, Zg) as an integer q . (5.34)Let T > 0. There exists an integer q(T ) 0 such that

q(T ) < T q(T ) + 1. (5.35)By Assumption A(iii) and (5.14),

Ig(q(T ), T, y) a, Ig(q(T ), T, Zg) = Ig(q(T ), q(T ) + 1, Zg) (5.36)Ig(T, q(T ) + 1, Zg) Q0 + a.

Together with (5.34) these relations imply that

Ig(0, T, y) Ig(0, T, Zg) Ig(0, q(T ), y)


Ig(0, q(T ), Zg)Q0 2a as T .Consider the case b). There exists an integer i0 2 such that

|y(i)| M2 + 21 for all integers i i0. (5.37)By (5.37), (5.32), (5.4) and the denition of Q1 (see (5.5)),

Ni=0

[Ig(i, i+ 1, y) Ug(i, i+ 1, y(i), y(i+ 1))] as N . (5.38)

Dene a sequence {di}i=i0 K asdi0 = z, di = y(i) for all integers i > i0.

By Lemma 1.5.1 and the denition of {di}i=i0 for any integer N i0+1N

i=i0+1

[Ug(i, i+ 1, y(i), y(i + 1)) Ug(i, i+ 1, zgi , zgi+1)]

=N

i=i0

[Ug(i, i+ 1, di, di+1) Ug(i, i+ 1, zgi , zgi+1)]

+Ug(i0, i0 + 1, zgi0, zgi0+1) Ug(i0, i0 + 1, z, y(i0 + 1))

4 4Q2 + Ug(i0, i0 + 1, zgi0,zgi0+1

) Ug(i0, i0 + 1, z, y(i0 + 1)).Together with (5.28), (5.12) this implies that

Ni=0

[Ig(i, i+ 1, y) Ig(i, i+ 1, Zg)] + as N . (5.39)

Let T > 0. There exists an integer q(T ) 0 satisfying (5.35). Clearly(5.36) holds. (5.33) now follows from (5.36) and (5.39). The lemma isproved.

The following lemma implies Assertion 1 of Theorem 1.2.1.

Lemma 1.5.6 Let g U , z K, |z| M and let y : [0,) K be ana.c. function. Then one of the relations below holds:

(i) Ig(0, T, y) Ig(0, T, Zg) as T ;(ii) sup{|Ig(0, T, y) Ig(0, T, Zg)| : T (0,)} 0 such that

|y(t)| Q1 + 21, t [i0,). (5.40)


Fix an integer i > i0. By Corollary 1.3.1 there exists an a.c. functiony : [i 1,) K such thaty(i 1) = z, y(t) = y(t), t [i,), Ig(i 1, i, y) = Ug(i 1, i, z, y(i)).

(5.41)(5.7), (5.41), (5.40), (5.5) imply that |Ug(i1, i, z, y(i))| Q2. It followsfrom this relation, (5.41), Lemma 1.5.2 and Assumption A(iii) that foreach T > i,

Ig(i, T, y) Ig(i, T, Zg) = Ig(i 1, T, y) Ig(i 1, T, Zg) (5.42)Ig(i 1, i, y) + Ig(i 1, i, Zg) 4 4Q2 Q0 2aIg(i 1, i, y) + Ig(i 1, i, Zg) 4 5Q2 Q0 3a.

(5.42) holds for each integer i > i0 and each T > i.Let S > i0+1, T > S+1. There exists an integer i > i0+1 such that

i 1 S < i. Clearly (5.42) holds. By Assumption A(iii) and (5.14),Ig(S, i, y) a, Ig(S, i, Zg) = Ig(i1, i, Zg)Ig(i1, S, Zg) Q0+a.Together with (5.42) this implies that

Ig(S, T, y) Ig(S, T, Zg)= Ig(i, T, y) Ig(i, T, Zg) + Ig(S, i, y) Ig(S, i, Zg) (5.43)

4 5Q2 2Q0 5a.We established (5.43) for each S > i0 + 1 and each T > S + 1.

Assume that (ii) does not hold. It follows from (5.14), AssumptionA(iii) and (5.43) which holds for each S > i0 + 1, T > S + 1 that

inf{Ig(0, T, y) Ig(0, T, Zg) : T (0,)} > .Therefore sup{Ig(0, T, y)Ig(0, T, Zg) : T (0,)} =. By Assump-tion A(iii) and (5.14) sup{Ig(0, i, y) Ig(0, i, Zg) : i = 1, 2, . . .} = .Together with (5.43) which holds for each S > i0 + 1, T > S + 1 thisimplies (i). The lemma is proved.

The next auxiliary result implies Assertion 4 of Theorem 1.2.1.

Lemma 1.5.7 Let K = Rn, g U and let z K satisfy |z| M . Thenthe function Zg is a (g)-minimal solution.

Proof. Let us assume the converse. Then there exist T1 0 andT2 > T1 such that

Ig(T1, T2, Zg) > Ug(T1, T2, Zg(T1), Zg(T2)).


Choose a number

(0, 81[Ig(T1, T2, Zg) Ug(T1, T2, Zg(T1), Zg(T2))] (5.44)and an integer q0 > T2 + 5. By Corollary 1.3.1 there exists an a.c.function y : [T1, T2] K such thaty(Ti) = Zg(Ti), i = 1, 2, Ig(T1, T2, y) = Ug(T1, T2, Zg(T1), Zg(T2)).

(5.45)It follows from (5.10), (5.11), (5.12) and Proposition 1.3.7 that thereexists an integer k > 2q0 + 4 for which

|Ug(i, i+ 1, Zg(i), Zg(i+ 1)) Ug(i, i+ 1, Zgk(i), Zgk(i+ 1))| (5.46)(2q0 + 1)1, i = 0, . . . , 2q0 + 1,

|Ug(q0, q0 + 1, Zg(q0), Zgk(q0 + 1)) (5.47)Ug(q0, q0 + 1, Zgk(q0), Zgk(q0 + 1))| (2q0 + 1)1.

By Corollary 1.3.1 and (5.45) there exists an a.c. function x : [0, k] Ksuch that

x(t) = Zg(t), t [0, T1] [T2, q0], x(t) = y(t), t [T1, T2], (5.48)x(t) = Zgk(t), t [q0 + 1, k],

Ig(q0, q0 + 1, x) = Ug(q0, q0 + 1, x(q0), x(q0 + 1)).

It follows from (5.48), (5.9) that

Ig(0, k, x) Ig(0, k, Zgk). (5.49)By (5.48), (5.9), (5.12), (5.46), (5.47) and (5.44),

Ig(0, k, x) Ig(0, k, Zgk) = Ig(0, q0 + 1, x) Ig(0, q0 + 1, Zgk)= (Ig(0, q0, x) Ig(0, q0, Zg)) + (Ig(0, q0, Zg) Ig(0, q0, Zgk))

+Ig(q0, q0 + 1, x) Ig(q0, q0 + 1, Zgk) Ig(T1, T2, y) Ig(T1, T2, Zg)

+q01i=0

[Ug(i, i+ 1, Zg(i), Zg(i+ 1)) Ug(i, i+ 1, Zgk(i), Zgk(i+ 1))]

+Ug(q0, q0 + 1, Zg(q0), Zgk(q0 + 1)) Ug(q0, q0 + 1, Zgk(q0), Zgk(q0 + 1))

Ig(T1, T2, y) Ig(T1, T2, Zg) + .It follows from this relation, (5.44), (5.45) that

Ig(0, k, x) Ig(0, k, Zgk) Ig(T1, T2, y) Ig(T1, T2, Zg) + < .


This is contradictory to (5.49). The obtained contradiction proves thelemma.

Proof of Theorem 1.2.1. At the begining of Section 1.5 for each f Mand each M > 2|z| we constructed a neighborhood U of f in M and foreach g U and each z K satisfying |z| M we dened a.c. functionsZg : [0,) K, Zgq : [0, q] K, q = 1, 2, . . . satisfying (5.9)-(5.15).Clearly an a.c. function Zf : [0,) K was dened for every f Mand every z K. By Lemmas 1.5.5, 1.5.6 for each f M and eachz K the function Zf is (f)-good and Assertion 1 of Theorem 1.2.1holds.

Assertion 2 of Theorem 1.2.1 follows from (5.15) which holds for everyg U (U is a neighborhood of f in M) and each z K satisfying|z| M .

Assertion 3 of Theorem 1.2.1 follows from Lemma 1.5.3. Lemma 1.5.7implies Assertion 4 of Theorem 1.2.1. Theorem 1.2.1 is proved.

Theorem 1.2.2 follows from Lemma 1.5.5.

Proof of Theorem 1.2.3 Fix z K. It follows from Proposition 1.3.6that there exists a positive number M0 and a neighborhood U0 of f inM such that

|Ug(T1, T2, y, z)| M0 for each g U0,each T1 [0,), T2 [T1 + c, T1 + 2c+ 2], (5.50)

and each y, z K B(2|z|+ 1).By Proposition 1.3.3 we may assume without loss of generality that

inf{Ug(T1, T2, y, z) : g M, T1 [0,), T2 [T1 + c, T1 + 2c+ 2],(5.51)

y, z K, |y|+ |z| M1} > 2M0 + 1.There exist a positive number S1 and a neighborhood U1 of f inM suchthat

U1 U0, S1 > M1 and Proposition 1.4.2 holds with (5.52)M3 = M2 + 2, M4 = S1, V = U1, c1 = c, c2 = 2c+ 2.

It follows from Proposition 1.3.6 that there exist a positive number M3and a neighborhood U of f in M such thatU U1, |Ug(T1, T2, y, z)|+ 1 < M3 for each g U, each T1 [0,),

(5.53)


T2 [T1 + c, T1 + 2c+ 2] and y, z K B(S1).It follows from Proposition 1.3.3 that there exists S > S1 + 1 such thatthe following property holds:

If g M, T1 [0,), T2 [T1 + c, T1 +2c+2] and if an a.c. functionv : [T1, T2] K satises Ig(T1, T2, v) 2M3 + 2M2 + 2, then

|v(t)| S, t [T1, T2].Assume that g U , T1 [0,), T2 c + T1. We will show that

property (i) holds.Let x, y K, |x|, |y| M1 and let v : [T1, T2] K be an a.c. function

which satises

v(T1) = x, v(T2) = y, Ig(T1, T2, v) Ug(T1, T2, x, y) +M2. (5.54)There is a natural number p such that pc T2 T1 < (p + 1)c. SetT = p1(T2 T1). Clearly T [c, 2c]. By (5.54) and Corollary 1.3.1,

p1i=0

[Ug(T1 + iT, T1 + (i+ 1)T, v(T1 + iT ), v(T1 + (i+ 1)T ))

Ug(T1 + iT, T1 + (i+ 1)T, yi, yi+1)] M2for each sequence {yi}pi=0 K satisfying y0 = v(T1), yp = v(T2). Itfollows from this, (5.52), (5.54) and Proposition 1.4.2 that

|v(T1 + iT )| S1, i = 0, . . . , p.By this relation and (5.54), (5.53) for i = 0, . . . , p 1,

Ig(T1 + iT, T1 + (i+ 1)T, v) Ug(T1 + iT, T1 +(i+1)T, v(T1 + iT ), v(T1 +(i+1)T ))+M2 < M3 +M2.

It follows from this relation and the denition of S that

|v(t)| S, t [T1, T2].Therefore property (i) holds. Analogously to this we can show thatproperty (ii) holds. The theorem is proved.

Chapter 2

EXTREMALS

OF NONAUTONOMOUS PROBLEMS

In this chapter we show that the turnpike property is a general phe-nomenon which holds for a large class of nonautonomous variationalproblems with nonconvex integrands. We consider the complete met-ric space of integrands M introduced in Section 1.1 and establish theexistence of a set F M which is a countable intersection of open ev-erywhere dense sets in M such that for each f F and each z Rn thefollowing properties hold:

(i) there exists an (f)-overtaking optimal function Zf : [0,) Rnsatisfying Zf (0) = z;

(ii) the integrand f has the turnpike property with the trajectory{Zf (t) : t [0,)} being the turnpike.

Moreover we show that the turnpike property holds for approximatesolutions of variational problems with a generic integrand f and thatthe turnpike phenomenon is stable under small pertubations of a genericintegrand f .

2.1. Main results

Let a > 0 be a constant and let : [0,) [0,) be an increasingfunction such that

(t) + as t.Denote by | | the Euclidean norm in Rn. We consider the space

of integrands M introduced in Section 1.1. This space consists of allcontinuous functions f : [0,) Rn Rn R1 which satisfy thefollowing assumptions:


A(i) for each (t, x) [0,)Rn the function f(t, x, ) : Rn R1 isconvex;

A(ii) the function f is bounded on [0,) E for any bounded setE Rn Rn;

A(iii)f(t, x, u) max{(|x|), (|u|)|u|} a

for each (t, x, u) [0,)Rn Rn;A(iv) for each pair of positive numbers M, there exist , > 0 such

that if t [0,) and if u1, u2, x1, x2 Rn satisfy|xi| M, |ui| , i = 1, 2, max{|x1 x2|, |u1 u2|} ,

then

|f(t, x1, u1) f(t, x2, u2)| max{f(t, x1, u1), f(t, x2, u2)};A(v) for each pair of positive numbers M, there is a positive number

such that if t [0,) and if u1, u2, x1, x2 Rn satisfy|xi|, |ui| M, i = 1, 2, max{|x1 x2|, |u1 u2|} ,

then|f(t, x1, u1) f(t, x2, u2)| .

We equip the set M with two topologies where one is weaker thanthe other. We refer to them as the weak and the strong topologies,respectively. For the set M we consider the uniformity determined bythe following base:

Es() = {(f, g) MM : |f(t, x, u) g(t, x, u)| for each t [0,) and each x, u Rn},

where > 0. It is not dicult to see that the uniform space M withthis uniformity is metrizable and complete. This uniformity generatesin M the strong topology.

We also equip the set M with the uniformity which is determined bythe following base:

E(N, , ) = {(f, g) MM : |f(t, x, u) g(t, x, u)| for each t [0,) and each x, u Rn satisfying |x|, |u| N,

(|f(t, x, u)|+ 1)(|g(t, x, u)|+ 1)1 [1, ]for each t [0,) and each x, u Rn satisfying |x| N},

Extremals of nonautonomous problems 35

where N > 0, > 0, > 1. This uniformity which was introduced inSection 1.2, generates in M the weak topology. By Proposition 1.3.2the space M with this uniformity is complete.

We consider functionals of the form

If (T1, T2, x) = T2T1

f(t, x(t), x(t))dt (1.1)

where f M, 0 T1 < T2 < + and x : [T1, T2] Rn is an a.c.function.

For each f M, each pair of vectors y, z Rn, each T1 0 and eachT2 > T1 we set

Uf (T1, T2, y, z) = inf{If (T1, T2, x) : x : [T1, T2] Rn (1.2)is an a.c. function satisfying x(T1) = y, x(T2) = z},

f (T1, T2, y) = inf{Uf (T1, T2, y, u) : u Rn}. (1.3)It is not dicult to see that Uf (T1, T2, y, z) is nite for each f M,each y, z Rn and all numbers T1, T2 satisfying 0 T1 < T2.

Recall the denition of an overtaking optimal function given in Section1.1 and the denition of a good function introduced in Section 1.2.

Let f M. An a.c. function x : [0,) Rn is called (f)-overtakingoptimal if for any a.c. function y : [0,) Rn satisfying y(0) = x(0),

lim supT

T0

[f(t, x(t), x(t)) f(t, y(t), y(t))]dt 0.

Let f M. We say that an a.c. function x : [0,) Rn is an(f)-good function if for any a.c. function y : [0,) Rn, the function

T If (0, T, y) If (0, T, x), T (0,)is bounded from below.

In this chapter we establish the existence of a set F M which is acountable intersection of open (in the weak topology) everywhere dense(in the strong topology) subsets of M such that the following theoremsare valid.

Theorem 2.1.1 1. For each g F and each pair of (g)-good functionsvi : [0,) Rn, i = 1, 2,

|v2(t) v1(t)| 0 as t.2. For each g F and each y Rn there exists a (g)-overtaking

optimal function Y : [0,) Rn satisfying Y (0) = y.


3. Let g F , > 0 and Y : [0,) Rn be a (g)-overtaking optimalfunction. Then there exists a neighborhood U of g in M with the weaktopology such that the following property holds:

If h U and if v : [0,) Rn is an (h)-good function, then|v(t) Y (t)| for all large t.

Theorem 2.1.2 Let g F , M, > 0 and let Y : [0,) Rn be a (g)-overtaking optimal function. Then there exists a neighborhood U of g inM with the weak topology and a number > 0 such that for each h Uand each (h)-overtaking optimal function v : [0,) Rn satisfying|v(0)| M ,

|v(t) Y (t)| for all t [,).Theorems 2.1.1 and 2.1.2 establish the existence of (g)-overtaking op-

timal functions and describe the asymptotic behavior of (g)-good func-tions for g F .Theorem 2.1.3 Let g F , S1, S2, > 0 and let Y : [0,) Rn be a(g)-overtaking optimal function. Then there exists a neighborhood U of gin M with the weak topology, a number L > 0 and an integer Q 1 suchthat if h U , T1 [0,), T2 [T1 + LQ,) and if an a.c. functionv : [T1, T2] Rn satises one of the following relations below:(a) |v(Ti)| S1, i = 1, 2, Ih(T1, T2, v) Uh(T1, T2, v(T1), v(T2)) + S2;

(b) |v(T1)| S1, Ih(T1, T2, v) h(T1, T2, v(T1)) + S2,then the following property holds:

There exist sequences of numbers {di}qi=1, {bi}qi=1 [T1, T2] such thatq Q, bi < di bi + L, i = 1, . . . , q,

|v(t) Y (t)| for each t [T1, T2] \ qi=1[bi, di].Theorem 2.1.4 Let g F , S, > 0 and let Y : [0,) Rn be a(g)-overtaking optimal function. Then there exist a neighborhood U ofg in M with the weak topology and numbers , L > 0 such that for eachh U , each pair of numbers T1 [0,), T2 [T1+2L,) and each a.c.function v : [T1, T2] Rn which satises one of the following relationsbelow:

(a) |v(Ti)| S, i = 1, 2, Ih(T1, T2, v) Uh(T1, T2, v(T1), v(T2)) + ;(b) |v(T1)| S, Ih(T1, T2, v) h(T1, T2, v(T1)) +


the inequality |v(t) Y (t)| is valid for all t [T1 + L, T2 L].Theorem 2.1.4 establishes the turnpike property for any g F .The results of this chapter have been established in [103].In the sequel we use the notation

B(x, r) = {y Rn : |y x| r}, x Rn, r > 0, (1.4)B(r) = B(0, r), r > 0.

Chapter 2 is organized as follows. In Section 2.2 for a given f Mand a given neighborhood of f in M with the strong topology we con-struct an integrand f which belongs to this neighborhood and estab-lishes the turnpike property for f. We also study the structure of ap-proximate solutions of variational problems with integrands belonging toa small neighborhood of f in the weak topology. Theorems 2.1.1-2.1.4are proved in Section 2.3. In Section 2.4 we discuss analogs of Theorems2.1.1-2.1.4 for a class of periodic variational problems. In Section 2.5we show that Theorems 2.1.1-2.1.4 also hold for certain subspaces of Mwhich consist of smooth integrands. In Section 2.6 we consider an ex-ample of an integrand which has the turnpike property and an exampleof an integrand which does not have the turnpike property.

2.2. Preliminary lemmas

Fix f M and z Rn. Let > 0, M > |z| and let an a.c. functionZf : [0,) Rn be as guaranteed by Theorem 1.2.1. We have that Zfis an (f)-good function, Zf (0) = z and for each T1 0, T2 > T1,

Uf (T1, T2, Zf (T1), Zf (T2)) = I

f (T1, T2, Zf ). (2.1)

First we dene functions fMr for r > 0 such that fMr f as r 0+

in the strong topology and such that each fMr has the turnpike property.Fix a continuous bounded function M : [0,)Rn [0,) which

satises the following assumptions:

B(i) {(t, x) [0,)Rn : M (t, x) = 0} = {(t, Zf (t)) :t [0,)} {(t, x) [0,)Rn : |x| M + 2};

B(ii) for any > 0 there is a positive number such that if t [0,)and if x1, x2 Rn satisfy |x1 x2| , then

|M (t, x1) M (t, x2)| ;


B(iii) for any positive number there is > 0 such that if t [0,)and if x Rn satises

|x Zf (t)| and |x| M + 1,then M (t, x) .Remark 2.2.1 Consider a continuous function : R1 [0, 1] for which

(t) = 1, t (,M + 1], (t) = 0, t [M + 2,),(t) > 0, t (M + 1,M + 2).

Let q be a natural number.Dene a bounded continuous function M : [0,)Rn R1 by

M (t, x) = |x Zf (t)|q(|x|), t [0,), x Rn. (2.2).It is easy to verify that the function M satises assumption (B).

Dene a function fM : [0,)Rn Rn R1 byfM (t, x, u) = f(t, x, u) +

M (t, x), t [0,), x, u Rn. (2.3)It is easy to verify that fM M and to prove the following result.Lemma 2.2.1 Let M > |z| and V be a neighborhood of f in M with thestrong topology. Then there exists a number r0 > 0 such that fMr Vfor every number r (0, r0).

Fix a natural number p. It follows from Theorem 1.2.1 and Theorem1.2.2 that there exist a number Mf > 0 and a neighborhood W f of f inM with the weak topology such that

|z|, sup{|Zf (t)| : t [0,)} < Mf (2.4)and

lim supt

|x(t)| < Mf (2.5)

for each g W f and each (g)-good function x : [0,) Rn. Thereexist a positive number M0(f, p) and an open neighborhood W0(f, p) off in M with the weak topology such that

W0(f, p) W f , M0(f, p) > 2Mf + 2p + 2 (2.6)and Theorem 1.2.3 holds with

M1,M2 = 2Mf + 2p + 2, c = 41, S = M0(f, p), U = W0(f, p). (2.7)


There exists a neighborhood W (f, p) of f in M with the weak topologyand a number M(f, p) such that

W (f, p) W0(f, p), M(f, p) > 2M0(f, p) + 2 (2.8)and Theorem 1.2.3 holds with

M1,M2 = 2M0(f, p) + 2, c = 41, S = M(f, p), U = W (f, p). (2.9)It follows from Lemma 2.2.1 that there is a positive number r(f, p) suchthat

fM(f,p)+1r W (f, p) for each r (0, r(f, p)). (2.10)Fix r (0, r(f, p)) and set

f = fM(f,p)+1r . (2.11)

We study the structure of approximate solutions of variational prob-lems with integrands belonging to a small neighborhood of f in thestrong topology. We show that the integrand f has the turnpike prop-erty and the function Zf is its turnpike. The next lemma establishesthat each approximate solution dened on an interval [T1, T2] is closeenough to the turnpike Zf at a certain point of [T1, T2] if the integrandis close enough to f and T2 T1 is large enough.Lemma 2.2.2 Let 0 (0, 1). Then there exist a neighborhood U of fin M with the weak topology and an integer N 8 such that if g U ,T 0 and if an a.c. function v : [T, T + N ] Rn satises

max{|v(T )|, |v(T +N)|} 2M0(f, p) + 2, (2.12)

Ig(T, T + N, v) Ug(T, T + N, v(T ), v(T + N)) + 2M0(f, p) + 2,then there is an integer i0 [0, N 6] such that

|v(t) Zf (t)| 0, t [i0 + T, i0 + T + 6]. (2.13)Proof. It follows from Proposition 1.3.6 that there exist a positive

number S0 and an open neighborhood U0 of f in M with the weaktopology such that

U0 W (f, p), |Ug(T1, T2, y1, y2)|+ 1 < S0 for each g U0,

each T1 [0,), T2 [T1 + 41, T1 + 8] (2.14)and each y1, y2 B(M(f, p) + 1), i = 1, 2.


By Theorem 1.2.1 there is a positive number S1 such that the inequality

If (T1, T2, Zf ) If (T1, T2, v) + S1holds for each T1 0, T2 > T1 and each a.c. function v : [T1, T2] Rnwhich satises |v(T1)| M(f, p) + 1.

(2.4), Proposition 1.3.6 and Assertion 4 of Theorem 1.2.1 imply thatthere exists

S2 > sup{|If (T1, T2, Zf )| :T1 [0,), T2 [T1 + 41, T1 + 8]}+ 2M(f, p). (2.15)

By Proposition 1.3.4 there exists (0, 81) such that for each g M,each T1, T2 [0,) satisfying 41 T2 T1 8 and each a.c. functionv : [T1, T2] Rn satisfying

Ig(T1, T2, v) 2S0 + 2S1 + 2S2 + 2, (2.16)the following property holds:

If t1, t2 [T1, T2] and if |t1 t2| , then|v(t1) v(t2)| 1610. (2.17)

There exists a number 1 (0, 410) such that Assumption B(iii) holdswith M = M(f, p)+1, = 410, = 1. Fix a natural number N > 48for which

41(61N 8)1r > 2M(f, p) + 2S0 + 6a + 4 + S1 (2.18)(recall a in Assumption A(iii)). By Proposition 1.3.6 there exist a num-ber S3 > 0 and a neighborhood U1 of f in M with the weak topologysuch that

U1 U0,|Ug(T1, T2, y1, y2)|+ 1 < S3 for each g U1 each T1 [0,), (2.19)

T2 [T1 + 41, T1 + N + 4] and each y1, y2 B(M(f, p) + 2), i = 1, 2.By Propositions 1.3.8 and 1.3.9 there is an open neighborhood U of fin M with the weak topology such that U U1 and that for each g Uand each T1 [0,), T2 [T1+41, T1+N +4] the following propertieshold:

a) if an a.c. function v : [T1, T2] Rn satisesmin{If(T1, T2, v), Ig(T1, T2, v)} S3 + 2M(f, p) + 2,

then |If(T1, T2, v) Ig(T1, T2, v)| 41;


b)|Uf(T1, T2, y1, y2) Ug(T1, T2, y1, y2)| 41

for each y1, y2 B(M(f, p) + 2), i = 1, 2.Assume that g U , T [0,) and v : [T, T + N ] Rn is an

a.c. function which satises (2.12). We show that (2.13) holds with aninteger i0 [0, N 6].

Let us assume the converse. Then for any integer i [0, N 6],sup{|v(t) Zf (t)| : t [i + T, i+ T + 6]} > 0. (2.20)

By the inequalities (2.12), Theorem 1.2.3 and the choice of W (f, p),M(f, p) (see (2.8) and (2.9)),

|v(t)| M(f, p), t [T, T + N ]. (2.21)(2

[Alexander J. Zaslavski] Turnpike Properties in Th(BookFi.org)

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