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A Hamilton-Jacobi approach to the differential inclusion problem Offin, Daniel C. 1979

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A HAMILTON - JACOBI APPROACH TO THE DIFFERENTIAL INCLUSION PROBLEM Daniel B.Sc, University of by C. O f f i n B r i t i s h Columbia, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF A p r i l , © Daniel C. BRITISH COLUMBIA 1979 O f f i n , 1979 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Mft-TH E f t i R T / C ^ The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D E - 6 B P 7 5 - 5 1 ! E ( i i ) ABSTRACT In the c l a s s i c a l calculus of v a r i a t i o n s , the Hamilton - Jacobi theory leads, under general hypotheses, to s u f f i c i e n t conditions f o r a l o c a l minimum. The optimal control problem as well has i t s own Hamilton -Jacobi approach to s u f f i c i e n t conditions f o r optimality. In t h i s thesis we extend t h i s approach to the d i f f e r e n t i a l i n c l u s i o n problem; a general, nonconvex, non d i f f e r e n t i a b l e control problem. In p a r t i c u l a r , the f a m i l i a r Hamilton - Jacobi equation i s generalized and a corresponding necessary condition (chapter 2) i s obtained. The s u f f i c i e n c y condition (chapter 3) i s derived and an example i s presented where i t i s shown how :this r e s u l t may lead to considerable s i m p l i f i c a t i o n . F i n a l l y , we show (chapter 4) how the c l a s s i c a l theory of canonical transformations may be brought to bear on c e r t a i n Hamiltonian inclusions associated with the d i f f e r e n t i a l i n c l u s i o n problem. Our main to o l w i l l be the generalized gradient, a set valued d e r i v a t i v e f o r L i p s c h i t z functions which reduces to the s u b d i f f e r e n t i a l of convex analysis i n the convex case and the f a m i l i a r d e r i v a t i v e i n the C"*" case. ( i i i ) TABLE OF CONTENTS Chapter 0 - Introduction page 1 Chapter 1 - Preliminaries (A) Generalized Gradients page 6 (B) The D i f f e r e n t i a l Inclusion Problem page 9 Chapter 2 - Necessary Conditions page 13 Chapter 3 - S u f f i c i e n t Conditions page 35 Chapter 4 - Canonical Transformations page 47 Bibliography page 62 (iv) Acknowledgements I would l i k e to express my appreciation to the University of B r i t i s h Columbia f or t h e i r generous f i n a n c i a l assistance and the use of t h e i r resources during my stay at UBC. Special thanks are due to Frank Clarke f o r proposing the topic and providing my i n i t i a l impetus and a sense of d i r e c t i o n . His c r i t i c i s m s and comments, made throughout the duration of the work, provided a constant source of stimulation. Many thanks to C o l i n Clark and Frank Clarke f o r reading the f i n a l manuscript. F i n a l l y , I would l i k e to thank P a t r i c i a Bott who c a r e f u l l y typed the manuscript, and who generously added her enthusiasm to my depleted reserves. (1) 0. Introduction The basic problem i n the calculus of va r i a t i o n s consists of mini-mizing the fu n t i o n a l / J L ( x ( t ) , x ( t ) ) d t over the functions x(') i n a s p e c i f i e d class which s a t i s f y the endpoint constraints x(0) = x Q and x ( l ) = x.^  . Under ce r t a i n conditions i t i s possible to associate with the function L(x,v) the Hamiltonian H(",Q defined by H(x,p) = <p,v> - L(x,v), where v = v(x,p) i s obtained by sol v i n g f o r v i n the equation P = L v U , v ) . A well known necessary condition i n order that the arc x(') furnish a minimum f o r the problem i s that the arc x(«), together with the arc p(«) defined by p(t) = L v ( x ( t ) , x ( t ) ) , s a t i s f y Hamilton's canonical equations: (0.1) x(t) = H ( x ( t ) , p ( t ) ) , -P(t) = H (x ( t ) , p ( t ) ) . In the nineteenth century the central importance to t h i s problem of a rel a t e d f i r s t - o r d e r p a r t i a l d i f f e r e n t i a l equation, the Hamilton -Jacobi equation defined below i n (0.2), became apparent. In p a r t i c u l a r (2) the fundamental s u f f i c i e n c y theorem of the calculus of v a r i a t i o n s depends upon the existence of a s o l u t i o n W(t,x), i n an appropriate domain, of the equation (0.2) 3W(t,x) + H ( x , 3 W j - ^ x ) ) = 0 (see H. Rund [ 1] f o r d e t a i l s ) . Furthermore a complete s o l u t i o n of (0.2), depending on n independent parameters W(t,x,X^,...,X ), may be used to generate a transformation of coordinates (x,p) + (X,P) f o r which the transformed version of equations (0.1) i s X(t) = 0 , P(t) = 0 . This transformation i s a s p e c i a l case of a "canonical transformation"; see Courant and H i l b e r t [ 1] f o r d e t a i l s . The standard problem i n optimal control theory consists of minimizing the functional / j L ( x ( t ) , u ( t ) ) d t subject to the t r a j e c t o r y x(«) s a t i s f y i n g (0.3) x(t) = f ( x ( t ) , u ( t ) ) (0.4) u(t) ^ U almost everywhere where U i s a compact set i n R m and x(0) = x Q , x ( l ) = x1 . Naturally associated with t h i s problem i s the value function, (3) defined f o r (t,y) e [0,1] X R n by S(t,y) = infimum /^L(x(s),u(s))ds where the infimum i s taken over the class of t r a j e c t o r i e s x : [ t , l ] -> R n which s a t i s f y (0.3), (0.4) and the endpoint constraints x(t) = y , x( l ) = x1 . This function, when d i f f e r e n t i a b l e , can be shown to be a so l u t i o n of the following p a r t i a l d i f f e r e n t i a l equation (see W.H. Fleming and R.W. Rishel [ 1 ] ) , (0.5) max { S (s,x(s)) + <S (s,x(s)) , f(x(s),u(s))> - L(x(s),u(s)) } = 0 u e u Z X which i s , as we s h a l l see, the natural analogue of the Hamilton - Jacobi equation f o r th i s problem. Related to equation (0.5) i s a s u f f i c i e n t condition f o r optimality. Suppose there exists a s o l u t i o n of (0.5) S(t,x), such that f o r some control u*(t) with corresponding t r a j e c t o r y x * ( t ) , (0.6) S t ( t , x * ( t ) ) + < S x ( t , x * ( t ) ) , f ( x * ( t ) , u * ( t ) ) > _ L ( x* ( t ) j U* ( t ) ) = o . then u*(«) i s an optimal control with x*(«) the optimal t r a j e c t o r y . To demonstrate t h i s we rewrite equation (0.5) as follows: / j L ( x ( t ) , u ( t ) ) d t > / J { S t ( t , x ( t ) ) + <S x ( t , x ( t ) ) , f ( x ( t ) , u ( t ) ) > } d t = S ( l , X l ) - S(0,x 0) . We need only notice that equation (0.6) implies that equality holds when x = x* and u = u*. An equivalent reformulation of the optimal control problem i s as (4) follows: minimize <j>(x(l)) subject to (0.7) x £ E(x) and (0.8) x(0) = x, 0 where the multifunction E(') i s given by E(x) = f(x,U) When we consider a larger class of multifunctions, the d i f f e r e n t i a l  i n c l u s i o n problem, as the l a t t e r formulation i s c a l l e d , i s more general than the optimal control problem. This can be seen f o r example from the case when the constraint set U i s i t s e l f a multifunction dependent upon the state: U = U(x). In t h i s case we simply define Following F.H. Clarke [ 1], we introduce the Hamiltonian H(•, •) fo r a multifunction E ( - ) : i n the case i n which we are minimizing a function <(>(x(lj) of the endpoint. I f we are minimizing an i n t e g r a l f u n c t i o n a l f ^h(x,x)dt, then we define H(x,p) = max { <p,v> - L(x,v):v S E(x) } . For the optimal control problem f o r example E(x) = f(x,U) and H(x,p) = max { <p,f(x,u)> - L(x,u):u 6 U } E(x) = f(x,U(x)) (0.9) H(x,p) = max { <p,v>: v £ E(x) } (5) so that equation (0.5) may be written S.(t,x) + max { <S (t,x),f(x,u)> - L(x,u) } = 0 or Z u G U X S t(t,x) + H(x,S x(t,x)) = 0 . This formalism suggests that a Hamilton - Jacobi theory might be developed f o r the d i f f e r e n t i a l i n c l u s i o n problem. However the d i f f i c u l t i e s of applying Hamilton - Jacobi techniques i n optimal control theory are compounded by the problem of characterizing the d i f f e r e n t i a l properties of the value function. W.H. Fleming and R.W. Rishel [ l] show that under sui t a b l e smoothness assumptions, the value function i s l o c a l l y L i p s c h i t z . In f a c t , as we s h a l l see, f o r the more general d i f f e r e n t i a l i n c l u s i o n problem there are examples to show that the value function i s not n e c e s s a r i l y d i f f e r e n t i a b l e . To circumvent t h i s d i f f i c u l t y we w i l l develop a Hamilton -Jacobi theory f o r the d i f f e r e n t i a l i n c l u s i o n problem using a calculus f o r nonsmooth functions introduced and developed by F.H.Clarke [ 3 ] . In so doing, the Hamilton - Jacobi equation f o r the d i f f e r e n t i a l i n c l u s i o n problem w i l l be stated i n terms of "generalized gradients", set valued derivatives which reduce to the f a m i l i a r d e r i v a t i v e i n the case and the s u b d i f f e r e n t i a l of convex analysis i n the convex case. In Chapter 2, we w i l l derive the Hamilton - Jacobi equation from the properties of the value function and subsequently obtain a necessary condition f o r optimality. In Chapter 3 we develop the c h a r a c t e r i s t i c s u f f i c i e n c y condition of Hamilton - Jacobi theory and f i n a l l y i n Chapter 4, we show that the c l a s s i c a l theory of canonical transformations can be brought to bear upon the "Hamiltonian i n c l u s i o n s " that correspond'to d i f f e r e n t i a l i n c l u s i o n problems. (6) 1. Preliminaries We begin by r e c a l l i n g some elements of the calculus of generalized gradients and subsequently s t a t i n g pertinent information and d e f i n i t i o n s concerning the d i f f e r e n t i a l i n c l u s i o n problem. (A) Generalized Gradients. Throughout t h i s paper we w i l l be dealing with r e a l valued functions f:R n -> R. The function f i s s a i d to be l o c a l l y L i p s c h i t z on R n i f given an a r b i t r a r y compact set K of Rn, there exists a constant k < °°, perhaps depending on K, such that |f(x) - f ( y ) | < k|y - x| whenever x,y belong to K. A theorem of Rademacher, (see H. Federer [ 1]) asserts that a l o c a l l y L i p s c h i t z function i s d i f f e r e n t i a b l e everywhere except p o s s i b l y on a set of n-dimensional Lebesque measure 0. The convex h u l l of a set E contained i n R n i s denoted by coE. D e f i n i t i o n : I f f i s l o c a l l y L i p s c h i t z on Rn, the generalized gradient of f at x, denoted 9f(x), i s given by co { lim V f ( y . ) : Vf(y.) e x i s t s , lim Vf(y.) exists and y. -> x } i-x» The generalized gradient i s an example of a set valued mapping. A n n R n mapping from R to the subsets of R E:R -> 2 , i s c a l l e d a multifunction. D e f i n i t i o n : The multifunction E(.) i s said to be upper semicontinuous provided the following holds: given any sequence of points x^ converging to x and a sequence of elements v^ of E(x n) converging to a point v i n Rn, (7) then v belongs to E(x). Some properties of the generalized gradient as a set valued mapping include: (1.1) 3f(-) ^ $ (1.2) 3f(") i s convex and compact. (1.3) 3f(«) i s an upper semicontinuous multifuntion. We s h a l l now state a proposition g i v i n g the r e l a t i o n between generalized gradients and the usual gradient. The proof i s supplied i n F.H. Clarke [ 2]. Proposition 1. The following are equivalent: (1.4) Sf(x) = iO a singleton; (1.5) Vf(x) e x i s t s , Vf(x) = ? and Vf i s continuous at x r e l a t i v e to the set upon which i t e x i s t s . Example: Consider the l o c a l l y L i p s c h i t z function f:R -> R defined by 2 f(x) = x i f x < 1 = 1 i f x > 1 x We attempt to evaluate 9 f ( l ) . We consider an a r b i t r a r y sequence x -*- 1 f o r which f'(x^) e x i s t s f o r a l l n and notice that the sequence f'(x^) has at most two convergent subsequences with l i m i t s 2 and -1 r e s p e c t i v e l y . Taking the convex h u l l , we f i n d that 3 f ( l ) = [-1,2], D e f i n i t i o n : I f f i s l o c a l l y L i p s c h i t z on Rn, the generalized d i r e c t i o n a l d e r i v a t i v e of f at x i n the d i r e c t i o n v, denoted f°(x;v) i s given by (8) f°(x;v) = lim sup f ( y + Av) - f(y) y+x;UO A The usual one-sided d i r e c t i o n a l d e r i v a t i v e of a function f:R n -> R, denoted f' ( x ; v ) , i s given by f'(x;v) = lim f ( x + Av) - f(x) A+0 A whenever t h i s l i m i t e x i s t s . A l o c a l l y L i p s c h i t z function f:R n -> R i s c a l l e d regular at x i n case f°(x;v) = f 1 ( x ; v ) f o r every v i n Rn. Example: The function f(x) considered i n the previous example i s not regular. However the function g(x) = -f(x) does have the property of r e g u l a r i t y . To show t h i s we observe that f o r y < 1, g(7 + A) - g(y) < 0 < g ( l + A) - g(l) . A A Therefore, g°(l;l) = lim sup g(y+A) - g(y) = lim g(l+A) - g(l) = g' . . y+l;X4-0 A A+0 A An i d e n t i c a l argument shows that g°(l;-l) = g'.(l;.-l)> a n d we may deduce that g i s regular at 1. For other values of x, g i s continuously d i f f e r -entiable and hence regular. Proposition 2. f°(x;v) = maximum { <£,v> : ? £ 3f(x) } The proof may be found i n F.H. Clarke [ 2 ] . From proposition 2, we may deduce that (1.6) 3f(x) =.{ ?: <?,v> <^  f°(x;v) f o r every v i n R° } The chain r u l e f o r composition of mappings as well as the mean value (9) property have been adapted f o r generalized gradients. Proposition 3. Let g be a continuously d i f f e r e n t i a b l e mapping g:Rn Rm. Let h:R m -> R be L i p s c h i t z . Then i f f:R n -* R i s given by f = h o g we have 3f(x) C 3h(g(x)) o Dg(x) . The proof may be found i n F.H. Clarke [ 3 ] . Proposition 4. I f x and y are d i s t i n c t points of R n then there i s a point z i n the open l i n e segment between x and y such that f(y) - f(x) € <y-x,3f(z)> . The proof may be found i n G. Lebourg [ 1]. (B) The D i f f e r e n t i a l Inclusion Problem. When considering the problem of. existence of solutions f o r the optimal control problem, A. F i l i p p o v found i t convenient to study the set valued mapping x•+ f(x,U) = {f(x,u) : u e U} and was able to show that i f the t r a j e c t o r y x:[0,l] -> R n s a t i s f i e d the r e l a t i o n (0.7), x ? E(x), with E(x) defined as E(x) = f(x,U) then x(.) also s a t i s f i e d the r e l a t i o n s (0.3) and (0.4), f o r an appropriate u, x(t) = f ( x ( t ) , u ( t ) ) and u(t) G U res p e c t i v e l y . As mentioned i n the introduction, we w i l l undertake a study of the more general d i f f e r e n t i a l i n c l u s i o n problem. (10) In general, we w i l l consider a multifunction E ( t , x ) , (t,x) £ [0,1] X R n possessing the following p r o p e r t i e s : (1.7) f o r each t i n [0,1] and x i n Rn, E(t,x) w i l l be a nonempty compact set. (1.8) f o r each x i n Rn, the multifunction t •+ E(t,x) i s measureable; that i s ' { t e [ o , i ] : E(t,x) n s / o l i s Lebesque measureable f o r every closed set S. (1.9) E(t,x) i s L i p s c h i t z i n x; that i s , there e x i s t s a k £ L*(0,1) with the following property: given any two points x 2 > x 2 ^ n ^ and v^ i n E ( t , x ^ ) , there i s some i n E ( t , X 2 ) such that l v l " V2I 1 kC t) l x i " X2I ' In the event that E i s independent of t, the L i p s c h i t z constant k(t) w i l l be replaced by a number k. (1.10) E(t,x) i s integrably bounded; that i s , there e x i s t s b £ L^(0,1) such that f o r a l l t i n [0,1] x i n R n and v i n E ( t , x ) , |v| 1 b(t) . D e f i n i t i o n : An absolutely continuous mapping x:[0,l] R n i s s a i d to be a t r a j e c t o r y f o r the multifunction E(t,x) provided that x(t) G E ( t , x ( t ) ) f o r almost every t i n [0,1] . „ The d i f f e r e n t i a l i n c l u s i o n problem can then be formulated: given a l o c a l l y (11) L i p s c h i t z function <t>:Rn -»• R and a multifunction s a t i s f y i n g properties (1.7) through (1.10) we attempt to minimize <}>(x(l)) subject to (1.11) x(t) S E ( t , x ( t ) ) f o r almost every t i n [0,1] (1.12) x(0) = X Q . D e f i n i t i o n : A relaxed t r a j e c t o r y f o r the multifunction E(t,x) i s an absolutely continuous mapping x:[0,l] -»- Rn which s a t i s f i e s x(t) S coE(t,x(t)) a. e. The relaxed d i f f e r e n t i a l i n c l u s i o n problem consists of minimizing the func t i o n a l 4>(x(l)) subject to x . £ coE(t,x) and x(0) = X Q . I f a multifunction i s compact, convex, L i p s c h i t z and integrably bounded then i t s t r a j e c t o r i e s are sequentially compact i n the topology of uniform convergence. The following proposition from F.H. Clarke [4] allows us to conclude that (1.13) i n f { o>(x(l)): x e E(t,x) , x(0) = X Q } =--inf { <J>(x(l)): x G coE(t,x) , x(0) = X Q } . Proposition 5. Let E(t,x) be measureable i n t L i p s c h i t z i n x and integrably bounded. Then f o r every relaxed t r a j e c t o r y y ( * ) j given any p o s i t i v e 6 there i s a tr a j e c t o r y x(«) f o r E with x(0) = y(0) and IIx - yll < 6 . J CO -The proof i s supplied i n F.H. Clarke [4] . (12) Definition: Given t ^  [0,1] and y Rn the value function for the differential inclusion problem denoted V(t,y), is given by V(t,y) = inf Kx(l)) where the infimum is taken over a l l trajectories of E which satisfy (1.14) x(t) = y. Given that the multifunction E is integrably bounded we may deduce that for any trajectory x(«) of E |x(l) - x(0) | < llbllj so that in particular V(t,«) is bounded from below on compact sets. The existence of a solution z(») for the relaxed problem is ensured and equation (1.13) tells us (Kz(l)) = V(t,y). Definition: The Hamiltonian function, denoted H(t,x,p) is defined by H(t,x,p) = max { <p,v> : v £ E(t,x) } . The assumptions (1.7) through (1.10) imply that H is locally Lipschitz in the (x,p) argument. (13) 2. Necessary Conditions. In Chapter 1 we introduced the value function; V(t,y) = i n f { ( K x ( l ) ) : x e E(x), x(t) = y } . Our immediate goal w i l l be to demonstrate that under s u i t a b l e r e s t r i c t i o n s on the choice of multifunction E, the value function w i l l be l o c a l l y L i p s c h i t z on [0,1] X Rn. Subsequently we w i l l derive the generalized Hamilton - Jacobi equation and present a necessary condition associated with t h i s equation. We s h a l l assume that the multifunction E(t,x) s a t i s f i e s conditions (1.7) through (1.10), and (2.1) E(t,x) i s L i p s c h i t z i n t; that i s , there exists a number k' < » with the following property: given any two points t ^ and t ^ i n [0,1] and v^ i n E ( t ^ , x ) , there i s some i n E ( t 2 , x ) such that \ \ ~ v 2 | l k ' | t 1 , t 2 | . We w i l l need the following d e f i n i t i o n s (see F.H. Clarke [ 4 ] ) : D e f i n i t i o n : The function p:[0,l] X R n X R n [0,«>) i s defined by, p(t,x,v) = d[v, E(t,x)] . D e f i n i t i o n : I f x(«) i s an absolutely continuous arc defined on the i n t e r v a l [ t , l ] with t G [0,1), we define 1 d t(x) = ft p ( s , x ( s ) , x(s))ds . ( 1 4 ) The following proposition i s taken from the same source. Proposition 1: There e x i s t s a p o s i t i v e number K with the following property: given any arc x(*) defined on [0,1] there exists a t r a j e c t o r y y(«) f o r E defined on [0,1] such that x(0) = y(0) and IIx - yll < Kd_(x) . J co — 0 Corollary 1: Proposition 1 remains v a l i d i f we are considering arcs defined on a smaller i n t e r v a l [ t , l ] where t belongs to [0,1). frroof: Define the change of parameter 6 :[ t , l ] -+[0,1] by v s ) = f ^ r ; Given an arc x(«) defined on [ t , l ] we define the arc u(«) on [0,1] by u(p) = x ( 6 ^ ( p ) ) = x ( ( l - t ) p + t) . Set E(p,u) = ( l - t ) E ( ( l - t ) p + t, u) and P(P,u,v) = d[v, E(p,u)] f o r (p,u,v) £ [0,1] X R n X Rn. It follows that E has properties i d e n t i c a l to E. Let v(«) be the t r a j e c t o r y f o r E, whose existence i s asserted i n Proposition 1, which s a t i s f i e s " v " u " [ 0 1] - K / 0 P CP.u-CpD» u(p))dp . Define the arc y(-) over [ t , l ] by (15) y(s) = v(e t(s)) Notice that y(s) = J _ v(8 (s)) E _1_ E(8. (s),v(6 (s))) = E(s,y(s)) 1-t 1 1-t and p(p,u(p), u(p)) = d[u(p), E(p,u(p))] = d[ ( l - t ) x ( s ) , ( l - t ) E ( s , x ( s ) ) ] = ( l - t ) d [ x ( s ) , E(s,x(s))] where s = ( l - t ) p + t. Therefore we may deduce that 1 1 K/ Qp(p,u(p),u(p))dp = K/ tp(s,x(s),x(s))ds and h - x l l [ t j l ] = l l v " u l l [ 0 , l ] - K / t P ( s ' x ( s ) ^ C s ) ) d s • F i n a l l y we need only notice that y(t) = v(0) = u(0) = x(t) . Q.E.D. Lemma 1. Suppose there exists a t r a j e c t o r y x(«) f o r E defined on [ t , l ] , t G [0,1), with i n i t i a l condition x(t) = y. Given a compact set C containing y there exists a constant n < °°, perhaps depending on C, with the following property: i f y^ S C there exists a t r a j e c t o r y x^(*) f o r E defined on [ t , l ] with i n i t i a l condition x^(t) = y^ which s a t i s f i e s | K x ( i ) ) - K x ^ l ) ) | < n|y - y±\ • Recall <f):Rn -+ R i s assumed to be l o c a l l y L i p s c h i t z . Proof: Define the arc x(-) over [ t , l ] by (16) x(s) = x(s) + y 1 - y Notice that X' (s) = x(s) e E(s,x(s)) . Since E i s L i p s c h i t z i n x, there i s some v £ E(s,x(s)) which s a t i s f i e s |x'(s) - v | < k(s)|x(s) - X(s)| = k ( s ) | y - y^ . Therefore d[X»(s), E ( s , X ( s ) ) H k(s) |y - y | and /Jp(s,X(s), X'(s))ds < /Jk(s) |y - y j d s < I l k H J y - . C o r o l l a r y 1 applies and allows us to assert the existence of a t r a j e c t o r y x^(«) f o r E which s a t i s f i e s x ^ t ) = X(t) = y 1 and IIx n - xll < KlIkIL |y - yJ . 1 «> — 1 1 1 ' Therefore, »x 1 - x l l ^ < ( K l l k l ^ + l ) | y - y j . * There exists a compact set C with the following property: f o r any tr a j e c t o r y x^f*) of E with i n i t i a l point y^ contained i n the compact * set C, x ^ s ) i s contained i n C for a l l s i n [ t , l ] . To see th i s we notice that f o r any such t r a j e c t o r y ; x 1 ( s ) = x ^ t ) + / ^ ( ^ d u so that | x 1 C s ) I < Ix 1 Ct) I + /^k(u)du < | x 1 C t D I + B k l l j . (17) Hence the class of such t r a j e c t o r i e s i s uniformly bounded i n the supremum norm. To complete the proof of Lemma 1 we l e t be a L i p s c h i t z constant * f o r the function $(•) v a l i d on the compact set C . Then UCxjCl)) - 4»(x(l)) | •< K j x ^ l ) - x ( l ) | < K 1CKllklt 1 + 1)|y - yx\ . We set n = KjCKllkll + 1 ) . Q.E.D. Proposition 2: V(t,y) i s l o c a l l y L i p s c h i t z i n y. Proof: Given an a r b i t r a r y compact set C we w i l l show that |v(t,7l) - v(t,y 2)| < nl/j - y 2 | whenever y and y 2 belong to the set C. The constant n i s the same as that i n Lemma 1. If x^(') i s a t r a j e c t o r y f o r E with i n i t i a l data ( t , y ^ ) , Lemma 1 asserts the existence of a t r a j e c t o r y x 2(*) with i n i t i a l data ( t , y 2 ) which s a t i s f i e s (2.2) V ( t , y 2 ) - V ( t , 7 l ) < <Kx 2(l)) - V(t,y x) < <))Cx1Ci)) - V ( t , y p + n|y : - y 2 | . Let x™ denote a minimizing sequence of t r a j e c t o r i e s with i n i t i a l data (t,y^) and x^ the corresponding t r a j e c t o r i e s with i n i t i a l data ( t , y 2 ) which s a t i s f y | K x " ( l ) ) - Kx2(D)| i nlxj - y 2 | • (18) We deduce from the i n e q u a l i t i e s (2.2) that V(t,y 2) - V{t,Yl) < lim - V ( t , 7 l ) + n|y 1 - y 2 | = n|y x - y 2 | Reversing t h i s argument, that i s deducing the existence of a t r a j e c t o r y with i n i t i a l data (t,y^) from a given t r a j e c t o r y with the i n i t i a l data ( t , y 2 ) , allows us to deduce that v(t, y i) - v(t,y2) < n|y 1 - y 2 | which concludes the proof. Q.E.D. Lemma 2: Given e > 0 there e x i s t s a constant g(e) < °° with the following property: i f t ^ and t 2 belong to [0,1 - e] and x^-) i s a t r a j e c t o r y f o r E defined on [ t ^ , l ] with i n i t i a l conditions x ^ ( t p = Y> there exists a t r a j e c t o r y x 2 ( - ) defined on [ t 2 , l ] which s a t i s f i e s x 2 ( t 2 ) = y and | K x 2 ( l ) ) - K x ^ l ) ) | £ S(e) | t 2 - t 1 | . Proof: Consider the mapping x:[t , 1] [ t ,1] defined by x(r) = 1 - t x ( r - t 2 ) + t1 . T-rr-2 It follows that T _ 1 ( S ) = Us - t ^ + t 2 where m = 1 - t 1 . m l _ t " 2 Since x^fs) belongs to E ( s , x 1 ( s ) ) i t follows from (2.1) that there i s some v s belonging to E(x 1 ( s ) , x 1 ( s ) ) such that |v s - x ^ s ) ] < k'|s - x _ 1 ( s ) | = 2sk'|t 1 - t 2 | . 1 - h (19) Therefore, d l x j f s ) , l E ( x " 1 ( s ) , x 1 ( s ) ) ] < | l y - X l ( s ) | ~ m < I l v - V I + I v - x 1 (s) — '— s s' ' s 1 ^  ' mm < | . v s | | l - 1| + |v - x ( s ) | m Since v g belongs to E(x 1 ( s ) , x 1 ( s ) ) we may deduce from (1.10) that dlxjCs) , 1 E ( T " 1 ( S ) , X 1 ( S ) ) ] < b ( x _ 1 ( s ) ) \ t ± - t | + 2sk'|t - t | m 1 - ^ 1 - t , I f E\s,y) = l E ( x " 1 ( s ) , y ) and m p(s,y,v) = dlv , E(s,y)] i t follows that p ( s , x 1 ( s ) , x 1 ( s ) ) < b ( T 1 ( s ) ) + 2sk'|t 1 - t 2 | . e From Coro l l a r y 1 we may deduce the existence of a t r a j e c t o r y v(») f o r E, defined on the i n t e r v a l [ t ^ , l ] , which s a t i s f i e s v ( t p = x 1 ( t 1 ) = y Iv - x, II < K ( l l b l L + 2k 1) I t , - t j 1 ° ° — 1 '1 2' and Define the arc * 2 ( ' ) o v e r t n e i n t e r v a l [ t 2 , l ] by x 2 ( r ) = v(x(r)) It follows that x 2 ( r ) = mv 2(x(r» £ mE(x(r) ,v(x (r))) = E ( r , x 2 ( r ) ) (20) so that x 9(«) i s a t r a j e c t o r y f o r E which s a t i s f i e s x 2 ( t 2 ) = v ( t x ) = y and x2(D = v ( l ) Therefore | x 2 ( l ) - x ^ l ) | < KCllbllj. + 2k') | t 2 - t j £ and |4,(x 2(l)) - ( K X j U ) ) ! i M K C l l b l ^ +.2k')|t 2 - t x f o r an appropriate L i p s c h i t z constant M. Proposition 3: Given E > 0, V(t,y) i s L i p s c h i t z i n t on the i n t e r v a l [0,1-E]. The proof i s i d e n t i c a l to that given f o r Proposition 1 and w i l l be omitted. Here i s an example f o r which the optimal arc z(«) has the following property: the value function V(t,y) i s not d i f f e r e n t i a b l e anywhere on the point set Our example takes place i n R and we w i l l denote a generic point i n R' by ( x>y)• Define the multifunction E(x,y) by {(s,z(s)) : s belongs to [0,1]} . 2 2 E(x,y) = {(0,0)} U {(-1,-1)} and the objective functional <j> by <j>(x,y) = | (x,y) | . (21) We consider the problem of minimizing <Kx(l),y(l)) over all trajectories (x(•),y(•)) of E, defined on [t,l] and satisfying the ini t i a l condition (x(t),y(t)) = (x 0,y 0) . For this particular problem the optimal trajectory for the ini t i a l data (t,XQ, XQ) can be shown to satisfy (x(s),y(s)) = ' (0,0) i f t < s < (l-t)-x Q+t (-1,-1) i f 1-x < s < l provided (1-t) > x^ and (x(s),y(s)) = (-1,-1) i f (1-t) < xf In order to consider arbitrary in i t i a l data (t,x,y) with 0 _< t <_ 1 and 2 (x,y) in the first quadrant of the plane, let D be the diagonal in R : D = {(x,y) : y = x, 0 <_ x <_ 1} . Given (x Q,y 0) let PQ= (pQ,p0) be the point of D closest to (x Q,y 0) Figure 1 (22) Case 1: 1-t <_ . The optimal t r a j e c t o r y s a t i s f i e s (0,0) i f t < s < 1-p (x(s),y(s))= 0 L(-l,-l) i f l - p Q < s < 1 In t h i s case V ( t , x Q , y Q ) = |(x 0,y 0) - ( P 0 , P 0 ) | = d [ ( x 0 , y 0 ) , D] . Case 2: 1-t < p Q . The optimal t r a j e c t o r y s a t i s f i e s (x(s),y(s))-= (-1,-1) and V ( t , x Q , y 0 ) = |(x 0,y 0) - (1-t,1-t)| . A convenient way to summarize t h i s information i s as follows: l e t h be the vector (-1,-1). Then V(t,x,y) = d[ (x,y),D] i f h-(x,y) > h-(1-t,1-t) = |(x,y) - ( l - t , l - t ) | i f h-(x,y) < h - ( l - t , l - t ) I f i s the hyperplane indicated i n Figure 2 then V(t,x,y) = d[(x,y),D] provided (x,y) belongs to the halfspace below L t = |(x,y) - (1-t,1-t) | provided (x,y) belongs to the halfspace above ( 2 3 ) Figure 2 This representation i s convenient when deriving d i f f e r e n t i a l properties of V . For example, i t can be seen that (see Figure 3 ) f o r e > 0 V ( S , 1 - S , 1 - S + E ) = £ and V ( s , 1 - S , 1 - S - E ) = £ J2 Figure 3 I t may be concluded therefore that V i s nowhere d i f f e r e n t i a b l e on the set { ( s , l - s , l - s ) : 0 £ s <_ 1 } . (24) For the i n i t i a l data (0,1,1) the optimal arc i s given by Z(s) = ( l - s , l - s ) . In order that we may compute V ( s , l - s , l - s ) we consider sequences s -»- s; x -»• 1-s; y -> 1-s; and VV(s ,x ,y ) -> v as n -»• °°. There are n n 'n ' n n'-'n' two separate cases to consider when computing v V ( s n > x n > v n ) . Case 1: (x ,y )*h < h * ( l - s ,1-s ) i n which case v n JnJ — v n' n^ VV(s ,x ,y ) = (x +y - 2 ( l - s ),x -(1-s ),y -(1-s )) ^ n' n"n^ v n Jn *• nJ' n ^ n^'^n v n " (x ,y ) - ( l - s ,1-s ) v n Jn K n n Case 2: (x ,y )*h > h - ( l - s ,1-s ) i n which case VV(s ,x ,y ) = ( 0' xn-Pn' yn-Pn ) n n n f , ^ •> r-r (x ,Y ) - (p ,p ) 1 n J n r n r n ' 1 where (P n>P n) i - s the closest point i n D to ( x n > y n ) ' I t i - s easy to see that i n Case 2, lim VV(s ,x ,y ) = ± 1 (0,-1,1). n n n —pr In Case 1 we notice that i t i s s u f f i c i e n t to consider lim VV(s ,x ,y ) ^ n n JnJ x - (1-s 1 when the r a t i o n n^ remains constant as n •> ». The quantities y -(1-s ) •'n n^ lim v^Csn'xn'>rn) m a y then be parameterized with t h i s r a t i o . In p a r t i c u l a r : fo r k belonging to [-1,0], (l+k,k,l) belongs to 8V(s,1-s,1-s); 717k2 f o r p belonging to [0,°°), (l+p,l,p) belongs to 3V(s,l-s,l-s); TITp^ (25) f o r m belonging to [-1,0] (1+m,l,m) belongs to 3V( s , l - s , l - s ) . TT+m2 In a l l cases the (x,y) components of 3V l i e on the un i t c i r c l e of the (x,y) plane while the t component varies from 0 when (x,y) = _1_ (-1,1), J2 to a maximum value of -Jl when (x,y) = _1_ (1,1) and back to 0 when 72 (x,y) = 1 (1,-1). Taking the convex h u l l of these points gives us 71 the two-dimensional set shown i n Figure 4. Figure 4 To begin the discussion of necessary: c 6 h a i t i o n s w e::prove a'-result which has been adapted from W.H. Fleming and R.W. Rishel [ l ] . Proposition 4. I f x(») i s a t r a j e c t o r y f o r E defined on [ t , l ] then V(s,x(s)) i s a nondecreasing function of s on [ t , l ] . I f z(«) i s an optimal t r a j e c t o r y fo r i n i t i a l conditions z(t) = y then V(s,z(s)) i s constant on [ t , l ] . Proof: For any S-^ S2 which s a t i s f y t <^  s^ <_ s^ ^ _ 1 ; V ( s 1 , x ( s 1 ) ) < V ( s 2 , x ( s 2 ) ) . I f z(-) i s optimal, V ( t , z ( t ) ) = cj>(z(l)). Therefore f o r any s i n [ t , l ] , (26) <j>(z(l)) = V ( t , z ( t ) ) < V(s,z(s)) < 4>(z(l)) so that equality must hold and V(s,z(s)) i s constant on [ t , l ] Proposition 5: For every (t,y) belonging to [0,1] X Rn; 0 < i n f { lim sup V(t+ ,y+Av) - V(t,y) } v E E(t,y) A+0 A Proof: We consider the difference quotient V(t+ ,y+Av) - V(t,y) A fo r A > 0 and v £ E ( t , y ) . Define the arc x(-) on the i n t e r v a l [t,t+A] by x(t+s) = y + sv where A > 0. An argument s i m i l a r to that presented i n Co r o l l a r y 1 may be used to show the following: there exists a t r a j e c t o r y x (•) f o r E A defined f o r s E [t,t+A] which s a t i s f i e s x^(t) = y and (2.3) Hx - x x l l [ t ^ t + x ] < K / J p(t+s,x(t+s),x' (t+s))ds . It follows e a s i l y from our assumptions on the multifunction E that p (*,•,•) i s continuous. Therefore i f we consider a sequence of A tending to 0, we may conclude that 1 P(t+s,x(t+s),x'(t+s))ds -> p ( t , x ( t ) , x ' (t)) = 0 . A It follows from (2.3) therefore, that (27) Now f o r A > 0, i f x (•) denotes the t r a j e c t o r y of equation (2.3) and n A i s an appropriate L i p s c h i t z constant f o r V then |V(t+A,y+Av) - V(t+A,x rt+A)) | <_n|x(t+X) - x,(t+A) I = o(A) . A A Therefore f o r a l l A s u f f i c i e n t l y small, -o(A) + V(t.+A,x x(t+A)) - V(t,y) < V(t+A,y+Av) - V(t,y) __ __ _ Since x,(•) i s a t r a j e c t o r y f o r E, Proposition 4 implies V(t+A,x A(t+A)) - V(t,y) > 0 . It follows that -o(A) < V(t+A,y+Av) - V(t,y) and A A 0 <_ lim sup V(t+A,y+Av) - V(t,y) A*0 A Proposition 5 follows since t h i s holds f o r a l l v £ E ( t , y ) . Q.E.D. The following Lemmas w i l l be needed i n the derivation of the generalized Hamilton - Jacobi equation. Lemma 3: I f V(*,») i s d i f f e r e n t i a b l e at (t,y) and z(«) i s an optimal relaxed t r a j e c t o r y with i n i t i a l condition z(t) = y, then there exists a vector £ belonging to 3z(t) n coE(t,y) such that V t(t,y) + V ( t , y ) - ? = 0 (28) Proof: We w i l l show that i n f a c t 8z(t) C coE(t,y) and that there i s an element c. of 9z(t) which s a t i s f i e s V t ( t , y ) + V y ( t , y ) - c = 0 . Given a sequence A^ tending to 0, the mean value property implies the existence of A' G (0.A ") such that z(t+A ) = z(t) + A c v n v n n fo r some t,^ G 9z(t+A^). By considering an appropriate subsequence A.. , we may assume that £ -> It follows from uppersemi-continuity of 8z(») that ? G 3z(t). We may conclude therefore z(t+A.) = z(t) + A.£ + o(A.) and J J • J V(t+A. 3z(t+A.)> - V ( t , z ( t ) ) = V(t+A.,z(t)+A . c : ) - V ( t , z ( t ) ) + o(\.) V t ( t , z ( t ) ) + V y ( t , z ( t ) ) - C + o(X.) A. However i t follows from Proposition 4, since z(») i s optimal, that V(t+X , z(t+A )) - V ( t , z ( t ) ) = 0 . Therefore; V t ( t , z ( t ) ) + V ( t , z C t ) D - c = 0 . To demonstrate that 8z(t) C coE(t,y) we l e t B denote the set (29) {s:t<s<l and z(s) does not exist} U {s:t<_s<l and z(s) £ coE(s,z(s))} . Then B has measure 0 and as i s shown i n F.H. Clarke [ 2], 9z(t) = 3 B z ( t ) where 9_z(t) i s the convex h u l l of a l l l i m i t s z(s ) with s -> t such B ^ n n that s n belongs to [ t , l ] \ B . For any such sequence s , z ( s n ) belongs • to coE(s n J ) z ( s n ) ) . Therefore lim z ( s n ) e coE(t,z(t)) by uppersemicontinuity of the multifunction E(«,«). Taking the convex h u l l s of a l l such l i m i t s we f i n d that 9 D z ( t ) C coE(t,z(t)) . Q.E.D. D Lemma 4: For every (t,y) belonging to [0,1]X Rn, min min { a + b«v }^_0 . 9V(t,y) E(t,y) Proof: We consider (a,b) £ 9V(t,y) as represented by a convex combination of l i m i t s of gradients of V; that i s , m (a,b) = lim y a.[ a. , b. ] . , l i i n-x» 1=1 where a n , b n = VV(t n,y n) : t n -»- t ; y\ •+ y for i = l,...,m. Given i i v l y i l l v G E ( t , y ) , since E(«,«) i s L i p s c h i t z , we may deduce the existence of u^ €E E(t,y") and v^ E E ( t " , y n ) which s a t i s f y , r e s p e c t i v e l y , l u i ~ v l 1 k ( t ) | y ? - y| and i n n i . . j n . v. - u. < k' t. - t . 1 I I 1 — 1 I 1 (30) It follows that v1} -+ v since \ \ ~ v| < k ' | t ? - t | + k ( t ) | y j - y| Therefore, m . i • v r n , n n-, a + b«v = lim £ a..{a. + b.*v.} . 1 l l i i m lim Z a . V ( t n , y n ; l , v n ) n-x» i = l We may conclude from Proposition 5 that V ' ( t n , y n ; l , v n ) > 0 and consequently a + b-v >_ 0 f o r a r b i t r a r y (a,b) £ 9V(t,y). It follows that min min { a + b - v } > _ 0 . Q.E.D. 3V(t,y) E(t,y) Theorem 1: For every (t,y) belonging to [0,1] X Rn, min { a - H(t,y,-b) } = 0 9V(t,y) Proof: We suppose temporarily that V i s d i f f e r e n t i a b l e at ( t , y ) . The optimal relaxed t r a j e c t o r y with i n i t i a l data ( t , y ) , which i s known to e x i s t (see discussion i n Chapter 1(B)), w i l l be denoted z ( * ) . From (31) Lemma 3 we deduce the existence of a vector t, belonging to coE(t,y) ( i f z(t) exists we may take c=z(t)) such that V t ( t , y ) + V (t,y)-? = 0 . Since z, £ coE(t,y) there i s some convex combination {a^} and v^ E(t,y) such that m Z a [V (t,y) + V (t,y)-v ] = 0 i = l y However, V t ( t , y ) + V y ( t , y ) - v . = V ( t , y ; l,v.) > 0 and consequently f o r some i V t(t,y) + V y ( t , y ) - v . = 0 . We may conclude from t h i s and Proposition 5 that whenever V i s d i f f e r e n t i a b l e (2.4) i n f V'(t,y ; l,v) = 0. E(t,y) Given (a,b) G 3V(t,y) such that (a,b) = lim VV(t n,y n) n-*=° fo r sequences t -*"t and Y^*y> l e t v n e E(t ,y ) s a t i s f y (see equation (2.4)) V 1 (t ,y ; l , v ) = 0 . By considering a s u i t a b l y chosen subsequence, v^, we may assume that v^ ->v. It follows by uppersemicontinuity that v G E ( t , y ) . Consequently (32) (2.5) a + b«v = lim V ft.,y.) + V (t.,y.)-v. = 0 . From (2.5) and Lemma 4 we may conclude that (2.6) min min { a + b - v } = 0 . 3V(t,y) E(t,y) Recall that H(t,y,-b) = max {(-b)«v} so that equation (2.6) may be v€E(t,y) rephrased as min { a - H(t,y,-b) } = 0 . Q.E.D. 3V(t,y) We w i l l say a L i p s c h i t z function W(t,y) i s a so l u t i o n of the generalized Hamilton - Jacobi equation i f f o r every (t,y) belonging to [0,1] X R n (2.7) max { a + H(t,y,b) = 0 . 3W(t,y) Theorem 1 could be rephrased then by s t a t i n g that -V(t,y) i s a so l u t i o n of the generalized Hamilton - Jacobi equation. The following r e s u l t i s a necessary condition i n order that z(») be an optimal t r a j e c t o r y . Theorem 2: If z(«) i s an optimal t r a j e c t o r y f o r i n i t i a l data ( t , y ) , then f o r every s £ [ t , l ] , the following holds: there exists £(s) £ 3z(s)i~E(s,z(s)) and (a(s),b(s)) G 3V(s,z(s)) such that (2.8) a(s) + b(s ) - s(s) = 0 . Furthermore, (33) (2.9) -b(s)-C(s) = H(z(s),-b(s)) and we may take £(s) = z(s) whenever z i s d i f f e r e n t i a b l e at s. Proof: Suppose z(*) i s d i f f e r e n t i a b l e at an i n t e r i o r point s of the i n t e r v a l [ t , l ] . Consider f o r A £ [0,1-s], the arc z(s+A) = z(s) + Az(s) . We define the L i p s c h i t z function f(A) by f(A) = V(s+A,z(s+A)) and note that since z i s continuously d i f f e r e n t i a b l e , the chain r u l e f o r generalized gradients (Proposition 1.3) states that 9f(A) C 3V(s+A,z(s+A)) • [ l , z ( s ) ] . We w i l l now demonstrate that 0 3f(0) which w i l l imply (2.10) 0 G 3V(s,z(s)) • [ l , z ( s ) ] . Since z(s) e x i s t s , z(s+A) = z(s) + Az(s) + o(A) as A4-0 . It thereby follows that V(s+A,z(s+A)) - V(s+A,z(s+A)) = o(A) as A+0 and consequently f(A) - f(0) = V(s+A,z(s+A)) - V(s,z(s)) + o(A) . From Proposition 4 we notice that V(s+A,z(s+A)) - V(s,z(s)) = 0 ( 3 4 ) and c o n c l u d e t h a t f ( A ) - f ( 0 ) = o(A) . An i d e n t i c a l argument shows t h a t f ( - A ) - £(0) = o(A) so t h a t f ( 0 ) e x i s t s and f 1 ( 0 ) = 0. T h e r e f o r e , 0 = - f 1 ( 0 ) < f°(0;-l) and 0 = f ' ( 0 ) < f°(0;l) w h i c h i m p l i e s ( P r o p o s i t i o n 1.2) t h a t 0 G 3 f ( 0 ) . E q u a t i o n (2.10) f o l l o w s f rom t h i s f a c t . We w i l l now c o n s i d e r a sequence o f p o i n t s s^ £ ( t , l ) and s^ -»-s such t h a t Z(S^)-H;(S). i t f o l l o w s by u p p e r s e m i c o n t i n u i t y t h a t £(s) G 3 z ( s ) r i E ( s , z ( s ) ) . From e q u a t i o n (2.10) we deduce t h e e x i s t e n c e o f p o i n t s (a^,b^) b e l o n g i n g t o 3 V ( s ^ , z ( s ^ ) ) w h i c h s a t i s f y , f o r e v e r y i , a. + b . • z ( s . ) = 0 . 1 1 v xJ By c o n s i d e r i n g a p p r o p r i a t e subsequences, we may assume t h a t (a^,b^)->(a(s) , b ( s ) ) and a g a i n by u p p e r s e m i c o n t i n u i t y i t f o l l o w s t h a t ( a ( s ) , b ( s ) ) e 3 V ( s , z ( s ) ) . T h e r e f o r e , a ( s ) + b ( s ) • S ( s ) = 0 . To complete t h e s t a t e m e n t o f t h i s theorem, we n o t i c e f r o m Theorem 1 t h a t a ( s ) - ( - b ( s ) ) ' C ( s ) > a ( s ) - H ( s , z ( s ) , - b ( s ) ) > 0 . E q u a l i t y t h e r e b y h o l d s and - b ( s ) - ? ( s ) = H ( s , z ( s ) , - b ( s ) ) . Q.E.D. (35) 3. S u f f i c i e n t Conditions. In the c l a s s i c a l calculus of v a r i a t i o n s , s u f f i c i e n t conditions f o r an optimal s o l u t i o n were tackled with the construction of " f i e l d s of extremals".[ R. Courant [ l ] ] . This method was elegant but the technical d i f f i c u l t i e s were awesome and few problems could be completely analysed i n t h i s manner. Hence, to a l l intents and purposes, the method was not p r a c t i c a l i n the same sense as the necessary conditions. For the optimal control problem, the s i t u a t i o n i s somewhat better owing to a more e x p l i c i t use of the value function. However, technical d i f f i c u l t i e s increased i n d i r e c t proportion to the p r e c i s i o n with which one wanted to specify the d i f f e r e n t i a l properties of t h i s function. In t h i s chapter, we present a s u f f i c i e n c y theorem, or a v e r i f i c a t i o n theorem as i t i s sometimes c a l l e d , which may serve to eliminate some of the computational problems encountered i n the analysis of optimal control problems. We begin by s t a t i n g a converse to Proposition 4 of Chapter 2 which w i l l lay the groundwork f o r the main theorem. Recall the objective functional cf>:Rn ->- R i s by assumption a l o c a l l y L i p s c h i t z function. Proposition 1. Let W(s,y) be a r e a l valued, l o c a l l y L i p s c h i t z function defined on [0,1] X R n such that W(l,y) = Ky) f o r every y belonging to Rn. Let (t ,y ) be given i n i t i a l conditions and suppose f o r any t r a j e c t o r y x(«) of E, defined on [ t Q , l ] and s a t i s f y i n g x(t ) = y , W(s,x(s)) i s nondecreasing on [ t ,1]. I f z(-) i s a t r a j e c t o r y (36) f o r E defined on [ t Q , l ] , s a t i s f y i n g z ( t Q ) = YQ> a n d such that W(s,z(s)) i s constant on [ t ,1]; then z(«) i s optimal f o r the given i n i t i a l conditions and W(t ,y ) = V(t ,y ) v o o o o where V(-,«) i s the value function. Proof: For any t r a j e c t o r y x(-) such that x ( t Q ) = y , with equality f o r x = z. That i s , 4>(z(l)) 1 cf»(x(l)) and z(>) i s thereby optimal for the i n i t i a l conditions (t -,y ). Therefore W(t Q,y o) = 4>(z(l)) = v ( t o ^ o ) Q > E - D -Theorem 1: Let -W(s,y) be a l o c a l l y L i p s c h i t z s o l u t i o n of the generalized Hamilton - Jacobi equation; max { a + H(t,y,b)} = 0 . 3W(t,y) Suppose W s a t i s f i e s the boundary conditions W(l,y) = <Ky) fo r every y belonging to Rn. Let (t ,^ ) be given i n i t i a l conditions and suppose the t r a j e c t o r y z(«) s a t i s f i e s z ( t Q ) = YQ and (37) 3W(s,z(s)) • [ l , z ( s ) ] = 0 a.e.; then z(-) i s an optimal t r a j e c t o r y f o r (t »y ) and W(t ,Y ) = V(t ,y ) . ^ o o o Jo Proof: We begin by noting that i f -W(s,y) i s a s o l u t i o n of (2.7) then W(s,y) i s a s o l u t i o n of the following three equations: (3.1) max { a + H(t,y,b) } = 0 ; -3W(t,y) min { a - H(t,y 3-b) } = 0 ; 3W(t,y) min min { a + v b } = 0 3W(t,y) E(t,y) This serves to show more c l e a r l y the connection between the s u f f i c i e n c y condition and the Hamilton - Jacobi equation (2.7). We w i l l f i r s t demonstrate that W(s,x(s)) i s a nondecreasing function of s f o r every t r a j e c t o r y x(*) of E. We observe that W(s,x(s)) i s L i p s c h i t z as a function of s. It follows that the set G, defined by G = {s:d_ W(s,x(s)) exists} n {s:x(s) exists} ds has Lebesgue measure 1-t i f *(•) i s defined over [ t Q , l ] . For a point s belonging to G we define f (T) = W ( S + T , X ( S ) + T X ( S ) ) f o r T G [0,e] f o r some e > 0. Then the conditions f o r the chain r u l e property are met (see Proposition 1.3) and we may deduce that (38) 3f (0) C 9W(s,x(s)) - [ l , x ( s ) ] . Since s G G i t f o l l o w s that x(s+i) = x(s) + xx(s) + o(x) and W(S+T,X(S+T)) - W(s,x(s)) = W(S+T,X(S)+TX(S)) - W(s,x(s)) + O(T) . Therefore, d_ W(s,x(s)) ds = l i m f ( r ) - f (0) e 3£ (0) C 9W(s,x(s)) • [ l , x ( s ) ] . T+0 ^ s T However 9W(s,x(s)* [ l , x ( s ) ] >^  0 from equation (3.1) so that we may conclude: d W(s,x(s)) _> 0 almost everywhere and W(s,x(s)) i s i n c r e a s i n g on the ds i n t e r v a l [ t ,1] . o For the t r a j e c t o r y z(«) i t f o l l o w s t h a t d _ W ( s , z ( s ) ) G 9W(s,z(s)) • [ l , z ( s ) l a.e.; ds however by hypothesis 9W(s,z(s)) • [ l , z ( s ) ] = 0 a.e. so that W(s,z(s)) i s constant f o r s G [ t ,1]. The t r a j e c t o r y z(-) and the f u n c t i o n W(»,«) thereby s a t i s f y the c o n d i t i o n s o f P r o p o s i t i o n 1 and we may conclude that z(») i s optimal. Q.E.D. The derived necessary c o n d i t i o n s f o r the d i f f e r e n t i a l i n c l u s i o n problem could be s t a t e d as 0 G 9W • [ l , z ] and 9W • [ l, z ] >_> 0 . (39) The s u f f i c i e n c y condition requires that aw • [ 1,'z] = 0 . Could the s u f f i c i e n c y theorem be sharpened so as to decrease the above discrepancy? To show that i n f a c t t h i s i s not po s s i b l e , we w i l l demon-stra t e that when the value function i s regular the necessary condition may be strengthened to coincide with the s u f f i c i e n t condition. Recall that r e g u l a r i t y requires V°(t,y;i,v) = V'(t,y;i,v) f o r i = ±1 and v €E Rn. This condition could be shown to always be the case when the problem i s "convex" i n a c e r t a i n sense. Theorem 2: I f z(») i s an optimal t r a j e c t o r y f o r the problem with i n i t i a l data (t,y) and i f the value function i s regular then 8V(s.,z(s)) •• [ l , z ( s ) ] = 0 a.e. In order to prove t h i s theorem i t w i l l be necessary to introduce the notions of generalized normal and tangent cones to an a r b i t r a r y closed set Q. For a more d e t a i l e d discussion see F.H. Clarke [ 2 ] . We define the function d:Rn -+ R by d(x) = min {|x-e| : e £ Q} . It i s e a s i l y v e r i f i e d that d(x) i s uniformly L i p s c h i t z with constant 1. D e f i n i t i o n : The cone of normal .vectors to Q at e, denoted N^fe), i s the closure of the set (40) { p £ Rn : sp belongs to 9d(e) for some s belonging to (0,°°) } . Definition: The cone of tangent vectors to Q at e, denoted T^(e), is the cone dual to Ng(e). T Q O ) = { v : <5,v><£ 0 for a l l £ in N (e) } An alternate characterization is: (3.2) d belongs to T^(e) i f and only i f d°(e;d) <_ 0 . We consider an arbitrary locally Lipschitz function F(x) on Rn. We set Q = {x : F(x)r 0} and assume for convenience that 0 £ Q. Lemma 1: If F is regular at 0 then 3F(0) C N (0) Proof: Consider an element d of T_(0). For A > 0, let e G Q satisfy V A It follows that Ad - e = d(Ad) . A F(Ad) = F(Ad) - F(e.) < Kd(Ad) A where K is an appropriate Lipschitz constant for F. Therefore, lim sup F(Ad) < K lim sup d(Ad) < d°(0;d) <_ 0 . A4-0 A A4-0 A If F happens to be regular at 0 then (41) F°(0;d) = lim sup F(Ad) <_ 0 AIO d f o r every d G TQ(0)• That i s <£,d> £ 0 f o r a l l ? G 3F(0) and d G T (0). Therefore, 3F(0) C [T Q(0)]° = N Q(0) . Q.E.D. Proof of Theorem 2: We r e c a l l that a necessary condition f o r the t r a j e c t o r y z(«) to be optimal f o r the i n i t i a l data (t,y) i s that V(s,z(s)) be constant on [ t , l ] (see Proposition 2.4). We could restate t h i s by saying f o r a l l s belonging to the i n t e r v a l [ t , l ] , (s,z(s)) belongs to the set { (s,x) : V(s,x) = V(t,y) } . I f we denote t h i s set by Q, we may conclude that the arc z(*) defined by (s,z(s)) = z(s) belongs to the closed set Q f o r every s belonging to [ t , l ] . At t h i s point we r e c a l l the following theorem from F.H. Clarke [ 2]. Theorem: The t r a j e c t o r y z(s) belongs to the closed set $ f o r s > t i f and only i f z'(s) i s tangent to $ at z(s) almost everywhere; that i s , z'(s) belongs to T $ ( z ( s ) ) a.e. . It follows from t h i s theorem that z'(s) = [ l , z ( s ) ] belongs to T n ( s , z ( s ) ) a.e. . (42) Therefore, f o r every (a,b) belonging to N Q ( S , Z ( S ) ) , [ l , z ( s ) ] • (a,b) < 0 . In p a r t i c u l a r , Lemma 1 asserts that (3.3) [ l , z ( s ) ] • 9V(s,z(s)) <0 a.e. . Now z(«) and V(«,») s a t i s f y the generalized Hamilton - Jacobi equation, Theorem (2.1), min { a + <b,z(s)> } = 0 a.e. . 9V(s,z(s)) It follows that (3.4) [ l , z ( s ) ] • 3V(s,z(s)) ^ 0 a.e. . Inequalities (3.3) and (3.4) imply that [ l , z ( s ) J • 8V(s,z(s)) = 0 a.e. . Q.E.D. •Example: 2 We consider the problem i n R of minimizing {%x(l) + X2(l)} over a l l t r a j e c t o r i e s (x(s),X2(s)) which s a t i s f y (x^(t),X2(t))= (y^,y2) and ( x 1 ( s ) , x 2 ( s ) ) .£ {(v.-lx^s) |) : -1 < v < 1} . This problem could be reformulated as minimize {%x(l) + y 2 - /^|x|ds} I * I over a l l arcs x(s) such that x(t) = y, and |x| <_ 1. ( 4 3 ) A reasonable guess as to the nature of the optimal arc x(s) i s that i t w i l l look, for some switching time T, as follows: Figure 5 X* To v e r i f y t h i s supposition we attempt to construct a s o l u t i o n W of the generalized Hamilton - Jacobi equation; min { a - H(y ,y ,-b) } = 0 3W(t,y ry 2) which s a t i s f i e s the boundary conditions W(l,y 1,y 2) = <$>(y1,y2) = h&1 + y2 . and for which the arc i n question ( x ^ ( s ) , x 2 ( s ) ) s a t i s f i e s SWfSjX^s) ,x 2 ( s ) ) • [ l j X j f s ) ,x 2(s)] = 0 a.e. . To begin with, we must decide which switching time w i l l give an optimal t r a j e c t o r y . We consider arcs x(s) = y + (s-t) on [t,T] = y1 + (T-t) - (s-T) on [T,l] and an associated performance function (44) f(T) = %(y 1 +2T-t-l) - /J|x(s)|ds + y 2 . We will restrict attention to y^ > 0 since the optimal policy for y^ <_ 0 is clearly x = -1. After routine computations, we find f -3T2+[ 3-4(y rt)] T+fy^t) (3/2+2t-y1)+%t2-l+y2 f(T) =< i f T < hi 1 - (y r t ) ] T 2 _ T + ( t_y ( y r t ) + ht2 + y 2 i f T > hi 1 - (y^t)] : Using calculus we can decide for which values of T f(T) will have a global minimum over the interval [ t , l ] . It turns out that i f we partition the strip { (t,y x) : 0 < t < 1 , y± > 0 } as in the following diagram; Figure 6 c B A > then the minimum for f(T) is achieved when T = t in region A; T = t in region B; T = h in region C. Substituting these values of T into the expression for f(T) we obtain a function W(t,y^,y2); W(t,y ry 2) = -J2t2-y1t+3/2y1-y2-l+y2 in A, (45) = ht^ - ht + y^t - y1 + y 2 i n B, 2~ = -ht2 + t/2 + y t - y1 - h + y 2 i n C. 2~" It i s e a s i l y v e r i f i e d that W(l,y 1,y 2) = hy1 + y r To show that W(l,y^,y 2) s a t i s f i e s the Hamilton - Jacobi equation, we note that the Hamiltonian f o r the multifunction E ( y r y 2 ) = UvHyJ) : - l < v < 1} i s given by H ( y 1 , y 2 , p 1 > p 2 ) = p 1 - P 2 1 I i f P X > o = - P X - P 2 l y i l i £ P i < 0 • We derive the function VW; vW(t,y 1,y 2) = [-t - y x + 3/2,-t + 3/2 - 2 y r l ] i n A = [ t - h + y 1 , t - h,l] i n B = [-t + y1 + h,t - % , l ] i n C. The set 9W(t,y^,y 2) i s e a s i l y computed on the boundary of these regions. To demonstrate that W(t,y^,y 2) i s a s o l u t i o n of the generalized Hamilton -Jacobi equation, one l a s t computation i s necessary i n each of the regions A, B, C of which we give one, the other two being s i m i l a r . In region A, - t + 3/2 - 2y 1 > 0 so that (46) = (-t - y± + 3/2) - [ - t + 3/2 - 2y± + = 0 . F i n a l l y , to prove optimality of the t r a j e c t o r y with the chosen switching time, suppose f o r example the i n i t i a l data i s {t,y^,y^){ = (%,%,y 2). Then the optimal arc would be x 1 ( s ) = 1/2 + (s -• 1/4) f o r 1/4 < s < 1/2 =. 3/4 - (s - 1/2) f o r 1/2 < s < 1 x 2 ( s ) = y 2 - / J / 4 | X l | d s . For a l l s such that 1/4 < s < 1/2 , 3W(s,x 1(s),x 2(s)) • [ l , x r x 2 ] = (-S + X l ( s ) + 1/2) + (s - 1/2) - | x 1 ( s ) | = 0 . f o r 1/2 < s < 1 , 8W(s,x 1(s),x 2(s)) • [ l , ^ , ^ ] = (s - 1/2 + x x ( s ) ) - (s - 1/2) - Ix^s)! = 0 which proves optimality by Theorem 1. (47) 4. Canonical Transformations. Let U be a given open subset of Rn. A more general version of the differential inclusion problem, consists of minimizing <))(x(l)) over a l l trajectories x(*) of the multifunction E, which lie in Q and satisfy the boundary conditions x(0) e CQ and x(l) e C1 . When E(',«) satisfies conditions (1.7) through (1.10), <)>(•) is locally Lipschitz and C^ , C^  are closed sets in Rn the following theorem from F.H. Clarke [5], gives necessary conditions for optimality: Theorem 2: (F.H. Clarke [5]) If z(«) solves the above differential inclusion problem then there is an arc p(*) and a number X equal to 0 or 1 such that: (4.1) (-p(t),z(t)) e 3H(t,z(t),p(t)) a.e., p(0) e Nc (z(0)), o -p(l) £ N „ (z(l)) + X9<J.(z(l)) and L l X + |p| is never 0. The differential inclusion of (4.1) is called a Hamiltonian inclusion in analogy with the classical Hamiltonian equations of mechanics and optics. The solution of such Hamiltonian inclusions thereby assumes a position of some importance. As a contribution to the development of techniques suited to the solution of these inclusions, we give a theorem on the transformation properties of "extremal trajectories" and show (48) how t h i s r e s u l t may be used, with the concept of canonical transformations, i n the analysis of Hamiltonian systems. We f i r s t develop some terminology and notation. We consider the problem which consists of minimizing / j F ( t , y ( t ) , y ( t ) ) d t subject to the arc y(*) s a t i s f y i n g various endpoint constraints (which s h a l l be suppressed i n our discussion) . The class of admissable arcs i s the class of absolutely continuous functions y:[0,l]->R n which l i e i n the open subset ft of Rn. The integrand F(t,«,-) w i l l be assumed to be l o c a l l y L i p s c h i t z and Lebesgue measureable i n the t argument. The arc y(«) i s c a l l e d an extremal f o r the integrand F provided i t s a t i s f i e s the Euler - Lagrange d i f f e r e n t i a l i n c l u s i o n , that i s , i f there exists an absolutely continuous arc £(•) which s a t i s f i e s (|(t),C(t)) E 8 F ( t , y ( t ) , y ( t ) ) a.e. (see F.H. Clarke [6] f o r d e t a i l s ) . 2 n n We s h a l l consider a C transformation of coordinates f:R X R -+R , f(t,Y) = y > with (t,Y) representing the new coordinates. The term "transformation of coordinates" i s intended to imply that the matrix fy(t,Y) i s nonsingular everywhere so that, l o c a l l y , we can obtain the inverse transformation h(t,y) = Y . We w i l l also assume that the range of the mapping f includes the open subset ft (49) Q C range £ . Then given an arc y(') which l i e s i n fi, we define the inverse image Y'(«) by the i m p l i c i t equation f ( t , Y ( t ) ) = y(t) . Within the set tt, an equivalent representation could be given by the inverse transformation Y(t) = h ( t , y ( t ) ) . We s h a l l have occasion to consider an appropriate mapping from the tangent space to the tangent space Ry. We notice that whenever the d e r i v a t i v e Y(t) exists f t ( t , Y ( t ) ) + f y ( t , Y ( t ) ) o Y ( t ) = y(t) and we define the transformation j : R X R n X R n -»• R n by (4.2) j(t,Y,V) = f t ( t , Y ) + f Y(t,Y)oV . In equation (4.2) f y i s an nXn matrix and f and V are column vectors of length n. F i n a l l y , we consider a l o c a l l y L i p s c h i t z integrand F(t,y,v) and the associated mapping F*:R X R n X R n -> R defined by F*(t,Y,V) = F(t,y,v) where (y,v) = (f(t,Y),j(t,Y,V)) . Theorem 1: The extremals f o r the integrand F(t,y,v) are i n one to one correspondence, v i a the change of coordinates f(t,Y) = y, with the (50) extremals of the integrand F*(t,Y,V) . Proof: I f we set g(t,Y,V) = [ f ( t , y ) , j ( t , Y , V ) ] = (y,v) then g i s at least continuously d i f f e r e n t i a b l e ; whenever VF e x i s t s , VF* exists and VF*(t,Y,V) = VF(t,y,v)oDg(t,Y,V) where Dg i s the Jacobian of the transformation g. It follows from the d e f i n i t i o n of generalized gradients that (4.3) 9F*(t,Y,V) D 9F(t,y,v)oDg(t,Y,V) In p a r t i c u l a r , i f y(*) i s an extremal t r a j e c t o r y which l i e s i n the set ft, there i s an arc £(•) which s a t i s f i e s (|(t),?(t)) G 9 F ( t , y ( t ) , y ( t ) ) a.e. . From the i n c l u s i o n (4.3) we may deduce that (4.4) (5(t),C(t))oDg(t,Y(t),Y(t)) G 9F*(t,Y(t),Y(t)) a.e.. Now Dg(t,Y,V) i s a l i n e a r transformation from R^n into R^n which has the matrix representation Dg(t,Y,V) f y ( t , Y ) 0 j y(t,Y,V) f Y ( t , Y ) We may, therefore, rewrite i n c l u s i o n (4.4) as (51) (4.5) U(t)of Y(t,Y(t)) + 5(t)oJ Y(t,Y(t),Y(t)) , ? ( t ) o f y ( t , Y ( t ) ) ] belongs to 9F*(t,Y(t),Y(t)) a.e. . From equation (4.2), we may derive j y(t,Y,V) = f t Y ( t , Y ) + £ y Y(t,Y)oV . It follows that.the i n c l u s i o n (4.5) may be written as [| 0 f y ( t , Y ) + 5°f t Y(t,Y) + 5 o f Y Y ( t , Y ) u Y , 5o.£Y(t,Y)] belongs to 9F*(t,Y,Y) a.e. To demonstrate that Y(*) i s an extremal, i t s u f f i c e s to show that d_£(t) f y ( t , Y ( t ) ) = 5°f Y(t,Y) + Cof t y(t,Y) + 5 o f y Y ( t , Y ) o Y . dt ' • For notational convenience, we set G(t,Y,£) = ?of Y(t,Y) Then d J ( t ) o f (t,Y(t)) = G (t,Y, 5) + G (t,Y,£>Y + G (t,Y,£)u f . dt 1 1 ^ We make the following observations: G t(t,Y,0 = Co£ Y t(t,Y) = 5 o f t y C t , Y ) since f is C 2; Gy(t,Y,g)= Cof y Y(t,Y) so that GY(t,Y,g)«Y = &»fyY(t,Y)oY ; G (t,Y,£)(•) = G(t,Y,«) which is just the dual linear mapping [f y(t,Y)] so that G c(t,Y,5)o| = G(t,Y,|) = 5 o f y ( t , Y ) . (52) It follows from t h i s and our previous remark that Y(«) i s an extremal t r a j e c t o r y . The conclusion of the theorem follows provided that given an extremal Y(*) f o r the integrand F*(t,Y,V), the image t r a j e c t o r y y(«) i s an extremal f o r the integrand F(t,Y,V). This demonstration i s i d e n t i c a l to the one given when we use the inverse transformation g(t,y) = Y instead of f(t,Y) = y. Q.E.D. Let y(-)> P(*) be t r a j e c t o r i e s f o r the Hamiltonian i n c l u s i o n (-P(t) , y ( t ) ) e 3H(t,y(t),p(t)) . We are interes t e d i n those transformations of coordinates (t,y,p) -> (Y,P) f o r which the image t r a j e c t o r i e s Y(-)> P(') i n turn s a t i s f y a Hamiltonian i n c l u s i o n (-P(t) , Y(t)) G 9H*(t,Y(t),P(t)) f o r some Hamiltonian function H*(t,Y,P). For our purposes, we s h a l l c a l l the transformation canonical i f t h i s condition i s met. For a more general discussion of canonical transformations i n a c l a s s i c a l s e t t i n g , see Caratheodory [ l ] . An important class of canonical transformations, used widely i n c l a s s i c a l mechanics, are those derivable from a generating function. 2 We are given a C , r e a l valued function <f>(t,y,Y). The transformation i s to be induced through the following r e l a t i o n s : (4.6) p = cfy(t,y,Y) (53) (4.7) . -P = <|>Y(t,y,Y) (4.8) H*(t,Y,P) = H(t,y,p) + <j>t(t,y,Y) . It w i l l be assumed that the matrix i s nonsingular everywhere so that (4.6) may be solved ( l o c a l l y ) f o r Y Y = f(t,y,p) . Equation (4.7) then implies P = g(t,y,p) and the desired transformation i s [Y,P] = [f( t , y , p ) , g(t,y,p)] Notice that nonsingularity implies that equation (4.7) may be solved l o c a l l y f o r y so that the inverse transformations are r e a d i l y a v a i l a b l e . Equation (4.8) gives us a rule f o r forming the new Hamiltonian. Since H* can be obtained by a straightforward c a l c u l a t i o n , canonical transformat-. ions generated i n t h i s fashion are e s p e c i a l l y u s e f u l . The function <j> i s c a l l e d a generating function f o r the canonical transformation. The following Lemma w i l l be used to show that i n f a c t , such transformations are canonical. Lemma 1: 2 Suppose that f o r some C function cj>, the two l o c a l l y L i p s c h i t z functions F(t,y,y) and G(t,y,v) are re l a t e d as follows: F(t,y,v) = G(t,y,v) + (j>t(t,y) + <j>y(t,y)0v . Then F and G generate the same extremals. Proof: (54) Clearly, i f VF exists then VG exists and conversely. Furthermore, VF = VG + (<|>ty+<|> o v , <j> ), and from the definition of generalized gradient 9F(t,y,v) = 3G(t,y,v) + (* (t,y) + * (t.y)ov , <fr (t,y)) The extremals generated by F are those arcs y(*) for which there exists an arc £(•) such that (1,5) G 9F(t,y,y) . This condition is satisfied i f and only i f (|,C) e 8G(t,y,y) + (<|>ty + <fr o y , <jy) which in turn is satisfied i f and only i f [d_U(t)-<j> (t,y(t)) , S(t)-<|. (t,y(t))] G 8G(t,y(t),y(t)) a.e. . dt y y Therefore, y(*) is an extremal for the integrand G. Q.E.D. Proposition 1: 2 The transformation induced by a C generating function is canonical and the image trajectories Y(«)j P(*) satisfy . (-P(t),Y(t)) G 9Ht(t,Y(.t),P(-tj with H* given in (4.8). The proof is adapted from a similar argument given in Gelfand and Fomin [ 1] . Proof: We consider an integrand F of the form (55) F(t,y,p,v,u) = p-v - H(t,y,p) . Our state space is the 2n-dimensional space of pairs (y,p). We make the following observations: VF(t,y,p,v,u) exists i f and only i f VH(t,y,p) exists, and in that case VF(t,y,p,v,u) = [(0,v) - VH(t,y,p) , p , 0] . It follows from the definition of generalized gradients that 9F(t,y,p,v,u) = [(0,v) - 9H(t,y,p) , p , 0] . The Euler - Lagrange inclusion for F is (|(t),?(t),5(t),?(t)) G 9F(t,y(t),p(t),y(t),p(t)) from which we may deduce the following relations: (4.9) ?(t) = p(t) ?(t) = 0 (p(t),0) G (0,y(t)) - 9H(t,y(t),p(t)) . The relations (4.9) are equivalent to (4.10) (-P(t),y(t)) G 3H(t,y(t),p(t)) . Therefore, extremals for the integrand F are identical with trajectories of the Hamiltonian inclusion (4.10). Let (y(*)>P(*)) denote an extremal for the integrand F. We will represent the transformation induced by equations (4.6) and (4.7) as Y=f(t,y,p) , P=g(t,y,p) . Then the image trajectories are Y(t) = f(t,y(t),p(t)) , P(t) = g(t,y(t),p(t)) . (56) We consider the mappings V(t,y,p,v,u) = f (t,y,p) + £ (t,y,p) o v + f (t,y,p ) o u y p and U(t,y,p,v,u) = g (t,y,p) + g (t,y,p) o v + g (t,y,p ) o u . u / p It may be verified that Y(t) = V(t,y(t),p(t),y(t),p(t)) and P(t) = U(t,y(t),p(t),y(t),p(t)) . With H* given by (4.8) we consider the integrand G(t,y,p,v,u) = g(t,y,p)-V(t,y,p,v,u) - H*(t , f(t,y,p) , g(t,y,p)) . The trajectories generated by the integrand G*(t,Y,P,V,U) = G(t,y,p,v,u) = P-V - H*(t,Y,P) are, by Theorem 1, the image trajectories under the transformation Y=f(t,y,p) , P=g(t,y,p) of the extremals for the integrand G. An argument identical to that given for equation (4.10) implies that the extremals for G*(t,Y,P,V,U) are the trajectories for the Hamiltonian inclusion (-P(t),Y(t)) G 8H*(t,Y(t),P(t)) . To complete the proof, i t suffices to show that F and G generate the same extremals. From equations (4.6), (4.7) and (4.8) we may deduce that (57) * t(t,y,Y) + <f>y(t,y,Y)-v + 4>Y(t,y,Y).V = H*(t,Y,P) - H(t,y,p) + p-v - P-V . For notational convenience, we set *(t,y,p) = <t>(t,y,f(t,y,p)) and derive the following: * t = * t + V f t J *y = *y + V f y > ^ p = V fp' • It follows that * t + * * v + ^ p - u = <ft + * Y ' f t + v v + ( v V v + V " u = 4>t + <l>Y'V + <f>Y-[ft + f v ° v + f pou] d>. + <j> -v + cb.,-V Y t T y rY = H*(t,Y,P) - H(t,y,p) + p-v - P-V . Therefore the conditions of Lemma 1 are met and we may conclude that F(t,y,p,v,u) = p-v - H(t,y,p) and G(t,y,p,v,u) = P-V - H*(t,Y,P) generate the same extremals. Q.E.D. Example: 2 We denote a generic element of R by (x,y). Let ft be the open set {(x,y) : x<0} . Define the multifunction E over [0,1] X ft by ( 5 8 ) E(t,x,y) = cb{(2x , xe _ t - |ln(-x)-t|)} U { ( o , -xe _ t - | ln(-x)-t|) } Figure 7 (2x,xe (0,-xe_t - |ln(-x)-t|) It is easily verified that the Hamiltonian for this multifunction is given by H(t3x,y,p,q) = 2xp + xqe t - q[ln(-x)-t| i f q+e^ < 0 -xqe - q|ln(-x)-t| i f q+e^ > 0 . The Hamiltonian system is then defined provided that we specify the boundary conditions for (x,y,p,q) satisfy x < 0. We will attempt to obtain information about the solutions to this system with the help of canonical transformations. We consider the generating function ft(t,x,y,u,v) = -xu - yv . Then the new coordinates (u,v,£,r;) are related to (x^pjq) by and (p>q) = ( f l x» ny) = O u > - v ) ( ? , ? ) = (-n u,-fi v) = (x,y) (59) The Hamiltonian, defined for al l (t,u,v,£,£) such that t G [0 ,1] and c; < 0, satisfies H*(t,u,v,?,£) = H(t,x,y,p,q) = -2<;u - ?ve + v|ln(-?)-t| i f v+e u > 0 = cve - t + v|ln(-£)-t| i f v+e1 < 0 . This Hamiltonian in turn may be simplified with the aid of the canonical transformation generated by the function ft(t,u,v,X,Y) = -(Y + e X)v - e t + Xu . The new coordinates (X,Y,P,Q) are related to (u,v,?,C) by (C,E) = C«u,nv) = (-et+X,-Y-eX) and (-P,-Q) = (ft x,ft Y) = (-veX-uet+X,-v) . These relations imply that X = ln ( - e ) - t ; Y = + e _ t ; Q = V ; P = -ve _ t? - u? . The new Hamiltonian H**, is calculated from equation ( 4 . 8 ) ; H**(t,x,Y,p,Q) = n t + H(t,u,v,nu,nv) uc - 2u£ - ?ve 1 + v X| i f v+e^i > 0 uz + Cve 1 + v|X i f v+etu < 0 It follows, after simplifying, that (60) H**(t,X,Y,P,Q) = P + Q|X[ i f Pe X > 0 = -P + Q | X | i f Pe"X < 0 . t+X 2 2 We notice that since £ = -e , H** is defined on [0,1] X R X R : To begin the analysis of extremal trajectories for this Hamiltonian system, we notice that H** is independent of Y which implies that Q is constant, Q = c. The actual value of c depends on the boundary conditions (X(0),Y(0),Q(1),P(1)) which in turn are specified by transforming the boundary conditions (x(0),y(0),p(l),q(l)) for the original system. Theorem 1 from F.H. Clarke [ 1] implies that H** is constant. The trajectories are, therefore, easily identifiable. This system is analysed for the case c = -1 in F.H. Clarke [ 1] and the (X,P) phase plane is given below. Figure 8 (61) As an example.to i l l u s t r a t e the inverse transformations, i t may be v e r i f i e d that the t r a j e c t o r i e s ( X ( t ) , Y ( t ) , P ( t ) , Q ( t ) ) given by X(t) = - | t - h\ ; Y(t) = /Q|X(S)|ds ; P(t) = h - t ; Q(t) = -1 with boundary data (X(0),Y(0)) = (-%,0) ; (P(1),Q(1)) = {-h,-l) transformed into the t r a j e c t o r i e s x(t) = - e 2 t _ J s 0 < t < h = -e 2 h < t < 1 p(t) = (t-k)eh~2t ~t -t = ( t - % ) e ~ 2 - e " h < t < 1 y(t) = -e~^2 + / J [ x ( s ) e " S - |ln(-x(s)) - s|]ds i f 0 < t < h = -e~^2 + ^ x e ~ S - |ln(-x)-s|]ds + f£[-xe" S - |ln(-x)-s|]ds i f % < t < 1 . q(t) = 1 0 £ t <_ 1 with boundary data (x(0),y(0)) = (-e" %,-e" %) ; ( P ( l ) , q ( l ) ) = fte^-e-1,1) . (62) BIBLIOGRAPHY C.Caratheodory: [ l] Calculus of Variations and P a r t i a l D i f f e r e n t i a l Equations of the F i r s t Order, Part 1, Holden - Day, San Francisco (1965). F.H.Clarke: [ l] Optimal Control and the True Hamiltonian, to appear. [ 2] Genaralized Gradients and Applications, Transactions Amer. Math. Soc. 205 (1975), 247 - 262. [ 3] Generalized Gradients of L i p s c h i t z Functionals, Advances i n Math., to appear (Tech. Rep. #1687, Mathematics Research Center, Madison, Wisconsin) [ 4 ] Admissable Relaxation i n V a r i a t i o n a l and Control Problems, J. Math. Anal. Appl. 51 (1975), 557 - 576. [ 5 ]Necessary Conditions f o r a General Control Problem, i n Calculus of Variations and Control Theory, (edited by D.L.Russel), Mathematics Research Center (University of Wisconsin - Madison) Pub. No. 36, Academic Press, N.Y. (1976). [6] The Euler - Lagrange D i f f e r e n t i a l Inclusion, J . D i f f e r e n t i a l Equations, 19 (1975), 80 - 90. R.Courant: [ l] Calculus of Variations, Courant I n s t i t u t e New York University, N.Y. (1962). R.Courant and D.Hilbert: [ 1] Methods of Mathematical Physics, Volume 2, Interscience, N.Y. (1962). H.Federer: [ l] Geometric Measure Theory, Springer - Verlag (1969). W.H.Flemming and R.W.Rishel: [ l] Deterministic and Stochastic Optimal Control, Springer - Verlag (1975). (63) BIBLIOGRAPHY I.M.Gelfand and S.V.Fomin: [ l] Calculus of Var i a t i o n s , (translated by R.Silverman), Prehtice - H a l l , Englewood C l i f f , N.J. (1963). G. Lebourg: [ 1] Comptes Rendus de l'Academic des Sciences de P a r i s , November 10, (1975). H. Rund: [ l] The Hamilton - Jacobi Theory i n the Calculus of Va r i a t i o n s , Van Nostrand, London (1966). 

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