7AD-A141 698 DIRECTIDNAL CONVEXIT AND FINITE OPTIMALT CONDITIONS /A(U) WSCONSIN UNIV MAO SON MATHEMATICS RESEARCH CENTERA BRESSAN MAR 84 MRC-TSR-2664 DAAG29-80-C-0041

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MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU Of STANDARDS 196 A

DIRECTIONAL CONVEXITY AND FINITE

U Alberto Bressan

Mathematics Research CenterUniversity of Wisconsin-Madison610 Walnut StreetMadison, Wisconsin 53705

March 1984C-,

-_ (Received September 26, 1983)

Approved for pubiq releaseDistribution unlimited D I

<p\ELEC T Eft

Sponsored by JU 11984

U. S. Army Research OfficeP. 0. Box 12211Research Triangle ParkNorth Carolina 27709

84 0.5 01 07

UNIVERSITY OF WISCONSIN-MADISONMATHD4ATICS RESEARCH CENTER

DIRECTIONAL CONVEXITY AND FINITE

OPTIMALITY CONDITIONS

Alberto Bressan*

Technical Summary Report #2664March 1984

ABSTRACT

Summary. For A C Rd , we say that a set A C Rd is A-convex if the

segment pq is contained in A whenever p, q e A and p - q e A. For the

control system x(t) - G(x(t))ou(t), x(o) - 0 e Rd , u(t) e QC sm for every

t e [0,T], assuming that the reachable set R(T) is A-convex, an extension

of the Pontryagin Maximum Principle is proved. If x(u,T) lies on the

boundary of R(T), conditions on the first order tangent cone at x (u,T) can

indeed be combined with restrictions placed on finite difference vectors of

the form y - x(u,T), with y e R(T), y - x(u,T) e A. This result is

complemented by conditions insuring the directional convexity of the reachable

set. They rely on a uniqueness assumption for solutions of the Pontryagin

equations and are proven by means of a generalized Mountain Pass Theorem.

AMS (MOS) Subject Classifications: 49E15, 58E25, 93C10

Key Words: Directional convexity, Critical point, Nonlinear control system,Necessary Conditions for optimality.

Work Unit Number 5 (Optimization and Large Scale Systems)

*Istituto di Matematica Applicata, Universita di Padova, 35100 ITALY.

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.

- ..- --

Ac ceosion For

SIGNIFICANCE AND EXPLANATION - I IC B.A&

Just i ff .'..

Consider a control system of the form By

Distr:iAv~i' i

1 - os

(t) (x(t))ui(t) , x(o) - 0 e Rd,i= 1 Dist p ci:.,

u(t) = (u1(t), ... , u e(t)) e a t e t[O,T].

Assume that an admissible control u(*) steers the system to a point x(u,t)

on the boundary of the reachable set R(t). Then, by studying the variations

x(u,t) - x(u,t) for controls u which are infinitesimally close to u, one

concludes that u must satisfy the Pontryagin Maximum Principle.

This paper establishes further necessary conditions for optimality, by

considering finite difference vectors of the type y - x(u,t), y being any

point in R(t), not necessarily close to x (u,t). These conditions are

significant whenever the reachable set is a priori known to be directionally

convex. More precisely, we say that R(t) is A-convex if a segment pq is

entirely contained in R(t) whenever its end points p,q lie in R(t) and

the direction of p-q falls inside a preassigned cone of directions A. An

efficient method for determining the A-convexl ty of a reachable set is

developed.

The major application of the present results appears in (3], where they

are used to determine the local time-optimal stablizing controls in a generic

3-dimensional problem.

The responsibility for the wording and views expressed in this descriptivesummary lies with MRC, and not with the author of this report.

64

DIRECTIONAL CONVEXITY AND PINITE OPTIMALITY CONDITIONS

Alberto Bressan*

1. Introduction.

Consider a control system of the form

x(t) - G(x(t))u(t), x(o) - 0 e R , (1.1)

where G is a dxm matrix with continuously differentiable entries and the control u

lies in the admissible set

U ( (u - (u1,...,u m ) e 2([0,T],RP)i u(t) e 0 a.e.}, (1.2)

0 being a fixed compact convex subset of Tm. Given a control u e u, let t + x(u,t)

be the corresponding solution of (1.1) and call R(t) the reachable set at time t.

Several necessary conditions are known in order for a trajectory x(u,-) to reach a

boundary point of R(T) at time T [2,7,8]. All of these conditions are obtained from a

local analysis: to test the optimality of a control u(-), a one-parameter family of

control functions u() is constructed, which generates at time T a tangent vector

v - lim [X(U ,T) - X(U,T)]/E.E 0 (1.3)

If. by choosing different families of controls u (*), one can generate tangent vectors

dv 1, .,v n whose positive span is all of R , then it can usually be shown that

x(u,T) lies in the interior of R(T). In the present paper we consider not only

infinitesimal tangent vectors of the form (1.3), but also finite difference vectors:

w . y - x(u,T) (1.4)

where y is any point in R(T). Assume that there exist first order tangent vectors

v1,...,v n and finite difference vectors wi, ..., wV such that the positive span of

{v1 , .I., vn , w1 , ..., wV }I is all of Rd . The a-priori assumption that R(T) is convex

would then imply x(u,T) e int R(T).

Cletituto di Matematica Applicata, Universita di Padova, 35100 ITALY.

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.

Unfortunately, in a nonlinear setting such an assumption in far too restrictive for most

of the relevant applications. A less stringent oondition is the following kind of

directional convexity:

oefinition. Let A be a subset of Rd- A set A S Rd is A-convex if p, q e A and

p - q e A imply Ep + (1-C)q e A for all P, e [0,11.

Using the notion of A-convexity we obtain an extension of the Pontryagin Maximum

Principle for the system (1.1).

Theorem 1. Assume that the reachable set R (T) in A-convex for some A S Rd. Let

u e U be a control such that x(u,T) lies on the boundary of (T). Then there exists

a nontrivial adjoint vector X(o) such that

i(t) -A(t) * G (x(u,t)) * u(t), (1.5)X

( A(t), G(x(u,t))u(t) > = max {[<(t),G(x(u,t)).u> ; u e n}

a.e. in 10,T], (1.6)

< X(T), y - x(u,T) > < 0 (1.7)

for all y e R(T) such that y - x(u,T) lies in the interior of A.

Here (1.5) and (1.6) are a restatement of the Maximum Principle, while the non-positivity

of the inner product in (1.7) poses an additional requirement whenever the A-convexity

of the reachable set is a-priori known. Of course, Theorem I would have little

significance unless we provide some efficient way to determine the A-convexity of (T)

for some set A. This is indeed the major concern of the present paper. Let the

sets R(t) be bounded for 0 ( t ( T. Then for all nontrivial n e Rd there exists at

least one control u C. ) for which the Pontryagin equations

;(t) = G(x(t)) u(t)

- -X(t) - G (x(t)).u(t) (1.8)

x(o) - 0, ACT) -

< A(t), G(x(t))u(t)> - max [<A(t),G(x(t))u >s u e Q)

are a.e. satisfied. Indeed R (T) is compact, hence the problem

max < n, x(u,T) > (1.9)

ueU

-2-

has at least one solution. The control un which attains the maximum clearly yields a

solution to (1.8). We will infer the directional convexity of R(T) from the uniqueness

of the solution of (1.8) for certain n, relying on a version of the Mountain Pass

Theorem [1,4]. In the following we call V the hyperplane orthogonal to the vector

w e Rd . Our main result is

Theorem 2. Assume that the reachable sets R(t), 0 ( t 4 T, for the control system

(1.1) are uniformly bounded. Then R(T ) is A-convex, A being the set of all w e Rd

such that for every nontrivial n e

a) the equations (1.8) have a unique solution, say (u(.), x(o), )(*)),

b) for a.e. t e [0,T], u(t) is the unique point in A where the map

u + < A(t), G(x(t))-u > attains its maximum.

Notice that, for the set A defined in Theorem 2, w e A always implies

Ew e A for every C e R, C # 0. A thus represents a family of directions. This

motivates the term "directional convexity" used throughout the paper.

The major application of the present results appears in [2), where the above theorems rule

out the optimality of certain bang-bang controls for which Pontryagin's test is

inconclusive. This leads to the solution of the generic local time-optimal stabilization

problem for systems of the type X X(x) + Y(x)u, in dimension 3.

2. Preliminaries

Consider a mapping # from a Hilbert space H into Rd . Its Frechet differential at a

point u e H is denoted by D#(u). We say that * is CI if the map u + O#(u)

from H into the space of continuous linear operators L(H;R d ) from H into Rd is

continuous. For the definition of the operator norm on L(H,Rd ) and for the basic

properties of differential calculus in abstract spaces our general reference is Dieudonne

-3-

II i

__________-- - - -. -

[6]. If f H R U is TApschitz continuous, the generalized directional derivative of

f at u in the Airection v is

0-f (ulv) - li Z [f(u+gv) - f(u)],

u~u

and the generalized gradient of f at u, denoted 3f(u), in the subdifferential of the

convex function v + f°(U,v) at the origin (5). Thus w e 3f(u) if, for all v e H

< w'v > 4 f (u,v). If U is a closed convex subset of H, we denote by

r(u) and rl(u) the tangent and the normal cone to U at a point

u e U. If M ) 0 and du(v) denotes the distance from a point v e H to U, the

generalized gradient of the map v + M * dU(v) at u e U is

3 (M-du) (u) - (w e r (u); IVi I MI. (2.1)

We write P(x,r) for the closed ball centered at x with radius r, int A and co A

for the interior and the convex closure of the set A. Consider now the special case

where N = 2 ([o,T];3), U is the convex set defined at (1.2) and L 2 + Rd is the

map u(-) x(u,T) generated by (1.1). Then # is continuously differentiable (see [21

for details) and D#(;) is the linear mapT

u(C) + I M(T,s) Gx(u,s))u(s)ds (2.2)0

where s + M(T,s) is the dxd matrix fundamental solution of

;(t) - -z(t) * G (x(ut)) (t)

with M(T,T) - I. Here Gx denotes the differential of the map G :R d Rd in

(1.1). For n e Rd, the differential of the map u(*) < n, f(u) > at u is the

linear mapT

u(') J X(s) Gx(u,s))u(s)ds, (2.3)0

X(.) being the unique solution to

(t - -A(t) G (x(u,t))u(t) , X(T) -(2.4)

In this case, if e() EU, r (u) is the set of L2 functions z(.) such that

-4-

T

%(s) (u( ) - u(s))ds ( 0 Vue U. (2.5)0

3. Proof of Theorem 1.

Consider a control u e U, and suppose that the conclusion of Theorem 1 does not hold for

du. Then for every nontrivial vector T1 4 R one of the following alternatives holds:

i) if X(*) denotes the unique solution of (1.5) for which X(T) r1, then

)(t) G(x(ut))u(t) < max ({(t)G(x(u't))u1 u C 0}

for t in a subset J S [0,T] of positive measure, or

ii) there exists some y e (T) with

w - y - x(u,T) e int A and < n, y - x(u,T) > > 0.

In the firset case, let u. I-) be a control in U such that

Xt)*G(x(u,t))uQ(t) - max (Xlt)*G(x(u,t))u1 u e 0)

for a.e. t e (0,T]. For C e (0,11, the convex combination Cu (.) + (1-E)u(.) is

again an admissible control. It was shown in (21 that

v - lim [x(Cu + (1-t)u,T) - x( ,T)]/C

is a first order tangent vector whose inner product with n is strictly positive. A

standard argument now implies that 10 is the positive span of d + I vectors : vI , ... ,

vn, w1 , ... , wv with the following property%

For i - 1, ... , n there exists a control us(.) e U such that

v, - lim [x(Cu +(l-C)u,T) - xlu,T))/g D #(u) - (uC+ xi,)/ *u u

with D #(u) given at (2.2), and for j v1, ... , v there exists a point yj e R(T)

such that wj " y- - x(u,T).

Moreover, wi lies in the interior of A. By constructing suitable convex combinationsirof the vectors vi, wj we will show that the above assumptions imply

x(u,T) e int RMT). Define

An+V" ((c 1 ... n, Cn 1 .... C ) Rn+ ci o, n+Vc , 1).

i-I

. . .. . . . .T d e - * i"|t --5 -

dE P i i4 q4

For c e An", c e (0,1] set

po(c,) x(C-O)u + C ). ci(ui.l) ' T).

By induction on J - 1, ..., v define

Pj(c 1 C) = PJ-1 (c,A) + cCn+ j (yj-pj_ (c,))

and set p(c,)} p v (c,(). Let p > 0 be such that D(wjip) S A for all

j-1, ... , V. Let M - max I1 i-. .... v). Choose j > 0 so small that

P' [2v(M+p)l -J and (p o(c, - x(u,T)l 4 0/2 for all (c,r) e An xfFJ. An

easy inductive argument now yields

Ijp(c,) - x(uT)l 4 P/2 + P p

for all c, E, J. hence IYJ j Pj-1 - wji I p. If p,-l(cC) e R(T), then also

p (cC) e R(T), due to the A-convexity of the reachable set. By induction on j it

follows that p(c,) a R(T) for all (cE) e 6 n+ V x 10,C]. The continuoum Frechet

differentiability of the map * u(-) 4 x(u,t) implies that

i-1 inI

S• (u) ) •n c(Ui -)i I

+ J D (U+C i ci(ui-u)) - D(u)j" Il ci(ui -u)d1 i.1

= I }n civi + o(C) (3.1)i=1

with lin o(Z)/ 4 0 uniformly w.r.t. c e AnD+V.E+0

Moreoverltm ( -pjllC,)) "Cn ~ (j3.2)

lim (PJ(cC) - p.l(C,)]/t " Cn+j lim (y-j 1 (c 1 )) cn+j j

E+0 C+O

n+valso holds uniformly w.r.t. c e A Together, (3.1) and (3.2) yield

li !p(c,E) - x(,T)]/E - In civi + ).Cvi-i .- I

n+V

uniformly w.r.t. c e A The argument that completes the proof is now well

established. Let 6 be the distance of 0 e Rtd from the boundary of

co {v1,...,vnw 1,... ,wV) and choose Co > 0 so small that co 4 and

Ip(c,Ao ) - xC' ,T) - to ( In c vj + V W <Cto, 8/2i=1 i )I

for all c e An+ V . Consider the injective map o : A + Rd defined by

0(c) - X(uT) + to(I n i + IV C+j

i-1

For z e B(x(u,T), Co5) define F(x) = p(O W:), CO). By (3.3),

iF(z) - zi < to. 6/2. For each so e B(x(u,T), to , 8/2), an application of Brouwer's

theorem ([7] pg. 251) now implies the existence of some z e B(x(u,T), E06 ) for which

F(z) = zO . Hence B(x(u,T), 06/2) S R(T), Q.E.D.

4. An abstract result on directional convexity

The additional condition (1.5) stated in Theorem 1 is useful only if one can prove that

the reachable set R(T) is A-convex for some (hopefully large) set A C R . Recasting

the problem in a more general setting, we now examine the directional convexity of the

image #(U) of a convex set U under an arbitrary differentiable map 4.

Theorem 3. Let U be a closed, convex, bounded set in a Hilbert space H and let * be

a C1 map from a neighborhood of U into R d , with *(U) compact and 1D*1 uniformly

bounded on U. Then #(U) is A-convex, A being the set of vectors w e Rd such that,

for every nontrivial 11 e w

a') there exists a unique u e U for which

< q, D#(u) - y > C 0 for all y e r(u),

b') If un e u, zn e r (U) , n - 1, 2, ... and II-D*(un ) - Zn| I 0, then the

sequence (Un)n)1 has a convergent subsequence.

-7-

Notice that in condition b') both z n and the map v + < n, D*(u n ) v > are continuous

linear functional. on H, hence the norm of their difference is well defined. To prove

the theorem let p' - *(u'), p" - *(u") e Rd, u', u" e u, p"-p, = we A. If the segment

joining p' to p* is not entirely contained in *(U), the compactness of #(U)

implies

B(1p' + (l- )p", p) n #(U) - P (4.1)

for some 1 e (0m1), p > 0. Let ID$(u)l 4 NO for every u in a neighborhood U. of

U, and extend # from Uo to a function, still called #, defined and globally

Lipschitz continuous on the whole space H, so that, say

IuIY - *(u 2 )1 4 M lu1-u21 V u1,u2 e H. (4.2)

Let di(p) be the distance of the point p e H from the line X through p', p" and

denote by du(v) the distance of v a H from the convex set U. Define the scalar

functional f on H by

f(v) - dI(#(v)) + 2Mdu(V). (4.3.

Notice that f is globally Lipschitz continuous, with Lipschitz constant 3M. By setting

m(v) - inf (lyl; y e of(v)},

the elementary properties of generalized gradients [51 imply

m(v) ) M > 0 vv e u. (4.4)

Let E be the set of all continuous paths Y [0,1] H with y(0) - p', y(l) - p".

Set

c - inf sup f (y( ))SeZ eCo, 1]

and obeerve that c ) P. Indeed, if y e E, by (4.1) to (4.3)

sup f(y(&)) ) sup f(WU(Y())) ) p,C~eto w O1] e[O, 1 ]

VU being the orthogonal projection on U. our next goal is to apply the deformation

lemma, proved in [4] for Lipschitz continuous functionals, and conclude that c is a

critical value for f. We first check that the Palais-Smale condition is satisfied.

-8-

Lenmma 1 Any sequence (un) n~o in H such that flu) n p/2 and m~u n) 0 possesses a

convergent subsequence.

Proof. If Une U, by (4.4) m~u n) M. We can thus assume that

d8n eu for all n > 1. Let I I(x) be the orthogonal projection of x e R on the line

I through p', p". For each n > 1, define the unit vector

n1n W1 W£Du) n V(un) d I dW$u)n(45

The denominator in (4.5) is no less then p/2, hence nn is well defined.

$)I. ±iMoreover ni n8 (p"-p B y possibly taking a subsequence, we can assume that n nn n

converges to some unit vector ni e w1 Relying on the assumption b'), the lemma will be

proved by exhibiting a sequence (Zn) n;, such that In * (u) -znE converges to

zero. The generalized gradient of f at un is

'f(u) n *D(u)+ S

with s n (y e r 1(u n) yl 4 2M). Hence m(u n) 0 implies that the distance between

Ti n* OD*(u n) and the set S. tends to zero. Let zn be the projection of Tn. * D#(u n

on S . Then zn e r (u ) and

lim In-*(u n Z n~ 1 li1m ( 1 iD*(u n )rin.D*(un )I+IT n0*fu n )-z nI 0.

By b') some subsequence of (u n) 011 converges.

To complete the proof of Theorem 3, for c e Ri define the sets

AC = u e Hl NOu < c I

K { u e H; f(u) -c, 0 e 3f(u) }

c

existence of a homeomorphism h H + H such that

h(u) =u for u 8 A_ A_ , (4.6)c+p/2 c-p/2

h(A --A_ for some > 0. (4.7)

if Y eE is apath for which

max fVY()) e. c + C,

-9-

then the path y'(.) - h(y(.)) lies in E and

Max f(y'(E)) ( c -Ce (0, 11

contrary to the definition of c This proves the existence of some u e H for which

o e 3f(u) and f(u) - c B . By (4.4), u e U. Setting

'- [Nit(u) - (u)]/d t(#'(u)) one has

This means n.D#(u) e r (u), hence < D#(u)*y, n > 4 0 for all y e F(u). On the other

hand, the compactness of the reachable set #(U) implies the existence of some u e U

for which

S*(U), n> - max ( < (u), n > u e u).

Since # is continuously differentiable on a neighborhood of U, u satisfies the

necessary conditions for optimality, i.e. < D*(u)-y, iT > C 0 for all y e r(u).

However, u 0 u because

This contradicts the uniqueness assumption a') and proves the theorem.

5. Proof of Theorem 2

Since G is C I and the sets R(t) are uniformly bounded for 0 4 t ( T, it is

known that the map # : u(-) + x(u.T) is continuously Frechet differentiable. By (2.2),

|D#(u)I is uniformly bounded as u ranges on a suitable small neighborhood of U.

oreover, (U) - R(T) is compact. Let w e A and n e w, w 1 0. It will be shown

that # satisfies the assumptions a'), b') stated for 4 in Theorem 3. Let u(-) e u

be an admissible control for which

< D#(U).y, " > • 0 vy e r(;) (5.1)

Let u(-) be any control in U and take y() - u(-) - u(,) e r(u). Using (2.3). (5.1)

yieldT

A(s) G(x(u,-)) (us) u(s))ds • 0 (5.2)0

-10-

MN1

with A(-) defined at (2.4). Since u(-) was an arbitrary control in U, u(-) yields

a solution of the Pontryagin equations (1.8). The u lueness assumption a) thus implies

Nov consider a sequence of controls C' () e u and linear functionals on

L (O~lil~),say zn(-), with z6 r I(u ), n t , 2, .. , such that

n n

Setting A C.) to be the unique solution ofn

n(t) -A n Ct) G x(x(un ,t))u n (t) A nCT) -n (5.4)

condition (5.3) can be written as

him [ JT AX (t)G(x)(un ,t))*v ntWdtn- (5.5)

- fT. z~t nW v nt) dtl - 0

for every bounded sequence (vn ) n;Iin L 2 (10,ri 3 ).

Since u n t) e 1 for all n and a... t e (0,T), the trajectories x C. x~u ,*

are uniformly Lipschitz continuous, and the same is true for the adjoint vectors

I ). By possibly taking a subsequence and relabeling, we can assume that

n a n X ~ in the norm topology of C(,TId) whln C)*x) a n A()*( ,TR) whl

un(' u(-) weakly, for some x(-), A(-), u(-). Por every t e 10,T]

Xn (u ,t) tGx()u()snn 0 n Gxs unsds

I(t) = 1+ fT X (a) G (x (8)) un sds.

Letting n *- in the above equalities, we obtain

;(t) f'G(;(s))u(s)ds, (5.6)

+(t ft X~+ (s) G1 (x(s))u~s)ds. (5.7)

Therefore x(-) is actually the trajectory of (1.1) corresponding to the control u(-)

and IC') solves the correct adjoint equation in (1.8). The maximality condition in

(1.8) also holds. indeed, let u(-) e U. Since U is bounded, such is the sequence

v(') -U(') -un C.). Thus (5.5) implies

o i.eo(% t)).cuct) u n(t,)dt 0 o n(* (u(t, - u (t,,dt

I i (t) G(x(t)) (u(t)-ul(t))dt (5.8)

because, by (2.5), IT zn(t) (u(t) - un(t))dt 4 0 for all n ) 1. Since u(- ) e U was

arbitrary, (5.8), together with (5.6) and (5.7), shows that u(.), x(.), A(.) afford a

solution to (1.8). Therefore the strong uniqueness condition b) can now be invoked.

Intuitively, b) states that u(.) is an exposed point of U, hence the convergence

Un(-) + u(-) takes actually place in the norm topology of L2 ([0,T], R") as well. A

precise argument runs as follows. Suppose that un(*) did not converge to u in the

L2 -norm. Then, because of the boundedness of n, there exist £ > 0, a subsequence

Uv and a set J C (O.TJ with positive Lebesque measure such that 1u(t) - u(t)I N C

for t e J and all V ; 1. For t e j, set

8(t) - X(t), G(x(t))u(t) > -max {<X(t), G(x(t))u > i u e S1, lu - u(t)l ; e I

8(*) is measurable and strictly positive a.e. on J, hence

-J 6(t)dt > 0.

For each V ) 1, define vvCt) - u(t) - u,(t) if t e J, v,(t) = 0 if t e J. Clearly

vC(.) e r(u ). Recalling that in (5.3) z e r (u ), one has

IT Cs() G (%x C))vCa)ds - z (s)VV()ds

I fj A V(s) GC(xV(s))v,(s)d

; Ij T(s)G(x(s)) vV(s)ds - JjIXv(s)-T(s)I-IG(x(s)) t-Ivv(s)lds

- J I)V(s)I'lG(x,(s)) - G(x(a))I.vv (s)lds (5.9)

Since the sequence vv is bounded, by (5.5) the left-hand side of (5.9) tends to zero as

v + f. However, the first integral on the right-hand side of (5.9) is ) F for all

V, while the other two integrals tend to zero. This yields a contradiction and

establishes the strong convergence of the sequence un to u. Theorem 3 can thus be i

applied, completing the proof.

-12-

REFRAUNCES

1. A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory

and applications, 3. Funct. Anal. 14 (1973), 349-381.

2. A. Bressan, A high-order test for optimality of bang-banq controls, SIAN J. Control,

to appear.

3. A. Bressan, The generic local time-optimal stabilizing control@ in dimension 3, to

appear.

4. K. C. Chang, Variational methods for non-differentiable functionals and application.

to partial differential equations, preprint.

5. F. H. Clarke, Generalized gradients of Lipschitz functionals, Advances in Math. 40

(1981) 52-67.

5. 3. Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1960.

7. A. 3. Irener, The high order maximal principle and its applications to singular

extrenals, SIAN 3. Control 15 (1977), 256-293.

8. R. B. Lee and L. Markus, Foundations of optimal control theory, Wiley 1967.

AS/jb

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Directi cal Ccrexity and Finte Optimlity Summary Report - no specificreporting period

CcsKitions S. PERFORMING ORG. REPORT NUMBER

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Alberto Bressan DAAG9-80-C-0041

S. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASKAREA & WORK UNIT NUMBERS

Mathematics Research Center, University of Work Unit Nunber 5 -610 Walnut Street Wisconsin Optimzation and LargeMadison, Wisconsin 53706 Scale Systes

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Research Triangle Park, North Carolina 27709 1314. MONITORING AGENCY MANE & AODRESS(II d4//teint boa Comtrlling Office) II. SECURITY CLASS. (of Afil report)

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17. DISTRIBUTION STATEMENT (a1 the abearct enteed In Blook 20, it different brn Report)

IS. SUPPLEMENTARY NOTES

IS. KEY WORDS (Cnlm on reverse side it necesaey and IdentiIty by block nuiber)

Directional ocwexaty, critical point, nonlinear control system, necessaryconditions for optiniality

20. ABSTRACT (Contine on revers side If nocessay and idbftify by bleck number)

summary. For A C Rd , we say that a set A E Rd is A-convex if the

segment pq is Tontained in A whenever p, q edA and p -cq R A. For the

control system x(t) - G(x(t))Ou(t), x(o) - 0 e R , u(t) e n - A for every

t I (0,T), assuming that the reachable set R(T) is _A-convex, an extension

of the Pontryagin Maximum Principle is proved. If x(u,T) lies on the

boundary of R(T), conditions on the first order tangent cone at x (uT) can

DD '-- 1473 ,OITION OF I NOV 5IS OBSOLETEUC SACURIY . UNCLASSIFIEDil SECURITY CL.ASSIFICATION OF THIS PAGE (Mm Date kerN0

ABSTRACT (Omtinued)

indeed be combined with restrictions placed on finite difference vectors ofthe form y - x(u,T), with y e R(T), y - x(u,T) e A. This result iscomplemented by conditions insuring the directional convexity of the reachableset. They rely on a uniqueness assumption for solutions of the Pontryaginecuations and are proven by means of a generalized Mountain Pass Theorem.