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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  by  B.Sc,  D a n i e l C. O f f i n U n i v e r s i t y o f 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 o f Mathematics  We accept t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d  THE UNIVERSITY OF BRITISH COLUMBIA April,  ©  1979  D a n i e l C. O f f i n , 1979  DE-6  In p r e s e n t i n g t h i s  thesis in partial  an a d v a n c e d d e g r e e a t the L i b r a r y I further for  shall  the U n i v e r s i t y  make i t  agree that  this  thesis for  It  extensive  f i n a n c i a l gain shall  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  RT/C ^  Columbia  the requirements I agree  r e f e r e n c e and copying of  this  that  not  copying or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  i s understood  Mft-TH E f t i  of  B r i t i s h Columbia,  available for  permission.  Department of  75-51 ! E  of  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  written  BP  freely  permission for  by h i s r e p r e s e n t a t i v e s . of  fulfilment  or  publication  be a l l o w e d w i t h o u t  my  (ii)  ABSTRACT  In the c l a s s i c a l l e a d s , under g e n e r a l minimum.  c a l c u l u s o f v a r i a t i o n s , the H a m i l t o n - J a c o b i  hypotheses, t o s u f f i c i e n t c o n d i t i o n s f o r a l o c a l  The o p t i m a l  c o n t r o l problem as w e l l has i t s own Hamilton -  J a c o b i approach t o s u f f i c i e n t  conditions f o r optimality.  we extend t h i s approach t o the d i f f e r e n t i a l  Hamilton - J a c o b i e q u a t i o n (chapter  In p a r t i c u l a r , the f a m i l i a r  i s g e n e r a l i z e d and a c o r r e s p o n d i n g  2) i s o b t a i n e d .  where i t i s shown how :this  may l e a d t o c o n s i d e r a b l e  simplification.  how the c l a s s i c a l t h e o r y  o f canonical transformations  i n c l u s i o n problem. a set valued  necessary  The s u f f i c i e n c y c o n d i t i o n (chapter  i s d e r i v e d and an example i s p r e s e n t e d  bear on c e r t a i n H a m i l t o n i a n  In t h i s t h e s i s  i n c l u s i o n problem; a g e n e r a l ,  nonconvex, n o n d i f f e r e n t i a b l e c o n t r o l problem.  condition  theory  result  F i n a l l y , we show (chapter 4)  i n c l u s i o n s associated with  may be brought t o the d i f f e r e n t i a l  Our main t o o l w i l l be the g e n e r a l i z e d  gradient,  d e r i v a t i v e f o r L i p s c h i t z f u n c t i o n s which reduces t o the  s u b d i f f e r e n t i a l o f convex a n a l y s i s 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 t h e C"*" case.  3)  (iii) TABLE OF CONTENTS  Chapter 0 - I n t r o d u c t i o n  page 1  Chapter 1 - P r e l i m i n a r i e s (A)  Generalized Gradients  page 6  (B)  The D i f f e r e n t i a l  page 9  I n c l u s i o n Problem  Chapter 2 - Necessary C o n d i t i o n s  page 13  Chapter 3 - S u f f i c i e n t  page 35  Conditions  Chapter 4 - C a n o n i c a l T r a n s f o r m a t i o n s  page 47  Bibliography  page 62  (iv)  Acknowledgements  I would l i k e to express  my a p p r e c i a t i o n t o the U n i v e r s i t y o f  B r i t i s h Columbia f o r t h e i r generous f i n a n c i a l a s s i s t a n c e and the use of t h e i r r e s o u r c e s  d u r i n g my s t a y a t UBC.  S p e c i a l thanks a r e due t o  Frank C l a r k e f o r p r o p o s i n g  the t o p i c and p r o v i d i n g my i n i t i a l  and  H i s c r i t i c i s m s and comments, made throughout  a sense o f d i r e c t i o n .  impetus  the d u r a t i o n o f the work, p r o v i d e d a constant source o f s t i m u l a t i o n . Many thanks to C o l i n C l a r k and Frank C l a r k e f o r r e a d i n g the f i n a l manuscript. typed  F i n a l l y , I would l i k e t o thank P a t r i c i a B o t t who  the manuscript,  depleted  reserves.  and who g e n e r o u s l y  carefully  added h e r enthusiasm t o my  (1)  0.  Introduction The  b a s i c problem i n t h e c a l c u l u s o f v a r i a t i o n s c o n s i s t s o f m i n i -  mizing the f u n t i o n a l  /J L(x(t),x(t))dt  over the f u n c t i o n s  x(') i n a s p e c i f i e d c l a s s which s a t i s f y the endpoint  constraints  x(0)  = x  Under c e r t a i n c o n d i t i o n s  and  Q  x ( l ) = x.^ .  i t i s p o s s i b l e t o a s s o c i a t e with the f u n c t i o n  L(x,v) the H a m i l t o n i a n H ( " , Q  d e f i n e d by  H(x,p) = <p,v> - L ( x , v ) ,  where v = v(x,p) i s o b t a i n e d by s o l v i n g f o r v i n t h e e q u a t i o n  P = L U,v) . v  A w e l l known n e c e s s a r y c o n d i t i o n i n o r d e r t h a t the a r c x(') f u r n i s h a minimum f o r the problem i s t h a t t h e a r c x ( « ) ,  t o g e t h e r with the a r c p(«)  d e f i n e d by  p(t) = L ( x ( t ) , x ( t ) ) , v  s a t i s f y Hamilton's c a n o n i c a l  (0.1)  x(t) = H (x(t),p(t))  equations:  ,  -P(t) = H (x(t),p(t)) .  In the n i n e t e e n t h c e n t u r y the c e n t r a l importance t o t h i s problem o f a r e l 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 e q u a t i o n , the Hamilton J a c o b i e q u a t i o n d e f i n e d 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 o f the c a l c u l u s o f v a r i a t i o n s depends upon the e x i s t e n c e o f a s o l u t i o n W(t,x), i n an a p p r o p r i a t e domain, o f 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 o f (0.2), depending on n independent parameters W(t,x,X^,...,X ) , may  be used to generate a t r a n s f o r m a t i o n o f  coordinates  (x,p) +  (X,P)  f o r which the t r a n s f o r m e d v e r s i o n o f e q u a t i o n s (0.1) i s  X(t) = 0  ,  P(t) = 0 .  T h i s t r a n s f o r m a t i o n i s a s p e c i a l case o f a " c a n o n i c a l t r a n s f o r m a t i o n " ; see Courant and H i l b e r t [ 1] f o r d e t a i l s . The s t a n d a r d problem i n o p t i m a l c o n t r o l t h e o r y c o n s i s t s o f m i n i m i z i n g the  functional  /jL(x(t),u(t))dt  s u b j e c t to the t r a j e c t o r y x(«)  (0.3)  (0.4)  satisfying  x(t) = f(x(t),u(t))  u ( t ) ^ U almost everywhere where U i s a compact s e t i n R  x(0) = x  Q  ,  x(l) =  x1  m  .  N a t u r a l l y a s s o c i a t e d w i t h t h i s problem i s the v a l u e f u n c t i o n ,  and  (3)  defined  f o r (t,y) e [0,1]  S(t,y)  X R  n  = infimum  by  /^L(x(s),u(s))ds  where the infimum i s taken over the c l a s s o f t r a j e c t o r i e s x : [ t , l ] -> R which s a t i s f y  (0.3),  (0.4) and the endpoint  x(t) = y  ,  n  constraints  x(l) =  x1  .  T h i s f u n c t i o n , when d i f f e r e n t i a b l e , can be shown t o be a s o l u t i o n o f the f o l l o w i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n (see W.H.  Fleming and R.W.  Rishel  [1]),  (0.5)  max { S ( s , x ( s ) ) + <S ( s , x ( s ) ) Z X u e u  , f(x(s),u(s))> - L(x(s),u(s))  } = 0  which i s , as we s h a l l see, the n a t u r a l analogue o f the Hamilton - J a c o b i e q u a t i o n f o r t h i s problem. Related Suppose t h e r e  t o e q u a t i o n (0.5) i s a s u f f i c i e n t c o n d i t i o n f o r o p t i m a l i t y . exists a  s o l u t i o n o f (0.5) S ( t , x ) , such t h a t f o r some  c o n t r o l u*(t) with corresponding t r a j e c t o r y x * ( t ) ,  (0.6)  S ( t , x * ( t ) ) + < S ( t , x * ( t ) ) , f ( x * ( t ) , u * ( t ) ) > _L ( x t  x  *  (t)jU  *  (t))=  o .  then u*(«) i s an o p t i m a l c o n t r o l w i t h x*(«) the o p t i m a l t r a j e c t o r y . To demonstrate t h i s we r e w r i t e  e q u a t i o n (0.5) as f o l l o w s :  /jL(x(t),u(t))dt > /J{S (t,x(t)) t  = S(l,  X l  +  <S (t,x(t)),f(x(t),u(t))>}dt x  ) - S(0,x ) . 0  We need o n l y n o t i c e t h a t e q u a t i o n (0.6) i m p l i e s t h a t e q u a l i t y h o l d s when x = x* and u = u*. An  equivalent  reformulation  o f the o p t i m a l c o n t r o l problem i s as  (4)  follows:  minimize (0.7)  <j>(x(l))  x £ E(x)  (0.8)  subject to  and  x(0) = x, 0  where the m u l t i f u n c t i o n E(') i s g i v e n by  E(x) = f(x,U)  When we c o n s i d e r a l a r g e r c l a s s o f m u l t i f u n c t i o n s , the d i f f e r e n t i a l i n c l u s i o n problem, as t h e l a t t e r f o r m u l a t i o n i s c a l l e d , i s more g e n e r a l than the o p t i m a l c o n t r o l problem.  T h i s can be seen f o r example from t h e  case when the c o n s t r a i n t s e t U i s i t s e l f a m u l t i f u n c t i o n dependent upon the s t a t e : U = U(x).  In t h i s case we s i m p l y d e f i n e  E(x) = f ( x , U ( x ) )  F o l l o w i n g F.H. C l a r k e [ 1 ] , we i n t r o d u c e the H a m i l t o n i a n H(•, •) for a multifunction E(-):  (0.9)  H(x,p) = max { <p,v>: v £ E(x) }  i n the case i n which we a r e m i n i m i z i n g a f u n c t i o n <(>(x(lj) o f the endpoint. I f we a r e m i n i m i z i n g an i n t e g r a l f u n c t i o n a l f ^h(x,x)dt,  H(x,p) = max { <p,v> - L ( x , v ) : v S E(x)  } .  For the o p t i m a l c o n t r o l 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 }  then we d e f i n e  (5)  so t h a t e q u a t i o n (0.5) may  be w r i t t e n  S.(t,x) + max { <S u G U Z  ( t , x ) , f ( x , u ) > - L(x,u)  } = 0  or  X  S (t,x) + H(x,S (t,x)) = 0 . t  x  T h i s f o r m a l i s m suggests t h a t a Hamilton  - J a c o b i t h e o r y might  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. o f a p p l y i n g Hamilton  o f the v a l u e f u n c t i o n .  o f c h a r a c t e r i z i n g the d i f f e r e n t i a l  W.H.  Fleming and R.W.  s u i t a b l e smoothness assumptions,  problem  difficulties  - J a c o b i t e c h n i q u e s i n o p t i m a l c o n t r o l t h e o r y are  compounded by the problem  In f a c t , as we  However the  R i s h e l [ l]  properties  show t h a t under  the v a l u e f u n c t i o n i s l o c a l l y  s h a l l see, f o r the more g e n e r a l d i f f e r e n t i a l  Lipschitz.  inclusion  t h e r e are examples t o show t h a t the v a l u e f u n c t i o n i s not  differentiable.  be  To circumvent  necessarily  t h i s d i f f i c u l t y we w i l l develop a Hamilton -  J a c o b i t h e o r y 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 u s i n g a c a l c u l u s  f o r nonsmooth f u n c t i o n s i n t r o d u c e d and developed by F.H.Clarke [ 3 ] . In so doing, the Hamilton  - J a c o b i e q u a t i o n f o r the d i f f e r e n t i a l  inclusion  problem w i l l be s t a t e d i n terms o f " g e n e r a l i z e d g r a d i e n t s " , s e t v a l u e d d e r i v a t i v e s which reduce t o 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 o f convex a n a l y s i s i n the convex case. In Chapter 2, we w i l l  d e r i v e the Hamilton  - J a c o b i e q u a t i o n from the  p r o p e r t i e s o f the v a l u e f u n c t i o n and s u b s e q u e n t l y o b t a i n a n e c e s s a r y condition f o r optimality.  In Chapter 3 we  s u f f i c i e n c y c o n d i t i o n o f Hamilton we  develop the  characteristic  - J a c o b i t h e o r y and f i n a l l y  i n Chapter  show t h a t the c l a s s i c a l t h e o r y o f c a n o n i c a l t r a n s f o r m a t i o n s can be  brought t o bear upon the " H a m i l t o n i a n i n c l u s i o n s " t h a t correspond'to differential inclusion  problems.  4,  (6)  1.  Preliminaries We  b e g i n by r e c a l l i n g some elements o f the c a l c u l u s o f  gradients  and  concerning  (A)  subsequently s t a t i n g p e r t i n e n t information  the d i f f e r e n t i a l  Generalized  n  -> R.  The  Gradients. w i l l be d e a l i n g w i t h r e a l v a l u e d  f u n c t i o n f i s s a i d to be  an a r b i t r a r y compact s e t K o f R , n  depending on K,  l o c a l l y L i p s c h i t z on R  there e x i s t s a constant  functions n  i f given  k < °°, perhaps  such t h a t  |f(x) - f ( y ) | < k|y  whenever x,y  and d e f i n i t i o n s  i n c l u s i o n problem.  Throughout t h i s paper we f:R  generalized  belong to K.  -  x|  A theorem o f Rademacher, (see H.  F e d e r e r [ 1])  a s s e r t s t h a t a l o c a l l y L i p s c h i t z f u n c t i o n 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 s e t o f n-dimensional Lebesque measure 0. h u l l of a set E contained  in R  n  i s denoted by  The  convex  coE.  Definition: I f f i s l o c a l l y L i p s c h i t z on R ,  the g e n e r a l i z e d g r a d i e n t  n  denoted 9 f ( x ) , i s g i v e n  generalized  mapping from R  n  gradient  e x i s t s , lim Vf(y.)  e x i s t s and y. -> x }  i s an example o f a s e t v a l u e d  to the subsets o f R  n  E:R  -> 2  R  mapping.  A  n  , i s called a multifunction.  Definition: The  m u l t i f u n c t i o n E(.)  the f o l l o w i n g h o l d s : and  x,  by  co { l i m V f ( y . ) : V f ( y . ) i-x»  The  o f f at  given  i s s a i d to be upper semicontinuous any  sequence o f p o i n t s  a sequence o f elements v^ o f E ( x ) n  provided  x^ c o n v e r g i n g to x  c o n v e r g i n g to a p o i n t v i n  R, n  (7)  then v belongs to E ( x ) . Some p r o p e r t i e s o f the g e n e r a l i z e d  gradient  as a s e t v a l u e d  mapping  include:  (1.1)  3f(-) ^ $  (1.2)  3 f ( " ) i s convex and  (1.3)  3f(«)  We  i n F.H.  s t a t e a p r o p o s i t i o n g i v i n g the r e l a t i o n between  gradients  and  the u s u a l  gradient.  The  proof  i s supplied  C l a r k e [ 2].  Proposition The  i s an upper semicontinuous m u l t i f u n t i o n .  s h a l l now  generalized  compact.  1.  f o l l o w i n g are  (1.4)  Sf(x) = iO  (1.5)  Vf(x)  equivalent:  a singleton;  e x i s t s , Vf(x)  = ? and  Vf i s continuous a t x r e l a t i v e  to  the s e t upon which i t e x i s t s . Example:  C o n s i d e r the  l o c a l l y L i p s c h i t z f u n c t i o n f:R -> R d e f i n e d  f(x) = x  2  by  if x < 1  = 1  if x > 1  x We  attempt to e v a l u a t e  f o r which f ' ( x ^ ) has  at most two  9f(l).  We  consider  an a r b i t r a r y sequence x  e x i s t s f o r a l l n and n o t i c e t h a t the sequence convergent subsequences w i t h l i m i t s  T a k i n g the convex h u l l , we  2 and  -*- 1  f'(x^)  -1 r e s p e c t i v e l y .  f i n d t h a t 3 f ( l ) = [-1,2],  Definition: I f f i s l o c a l l y L i p s c h i t z on R , n  the g e n e r a l i z e d  o f f a t x i n the d i r e c t i o n v, denoted f°(x;v) i s g i v e n  directional derivative by  (8)  f°(x;v) = l i m sup f ( y + Av) - f ( y ) y+x;UO A  The u s u a l o n e - s i d e d d i r e c t i o n a l d e r i v a t i v e o f a f u n c t i o n f : R  -> R, denoted  n  f ' ( x ; v ) , i s g i v e n by  f ' ( x ; v ) = l i m f ( x + Av) - f ( x ) A+0 A  whenever t h i s  limit exists.  A l o c a l l y Lipschitz function f:R  -> R i s  n  c a l l e d r e g u l a r a t x i n case f°(x;v) = f ( x ; v ) f o r every v i n R . 1  n  Example:  The f u n c t i o n f ( x ) c o n s i d e r e d i n the p r e v i o u s example  regular.  However the f u n c t i o n g(x) = - f ( x ) does have the p r o p e r t y o f  regularity.  i s not  To show t h i s we observe t h a t f o r y < 1,  g(7 + A) - g(y) A  < 0 <  g ( l + A) - g ( l ) A  .  Therefore, g°(l;l) = l i m sup g(y+A) - g(y) = l i m 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 t h a t g°(l;-l) = g'.(l;.-l)>  t h a t g i s r e g u l a r a t 1.  a n  d we may  . .  deduce  F o r o t h e r v a l u e s o f x, g i s c o n t i n u o u s l y  differ-  e n t i a b l e and hence r e g u l a r . Proposition  2.  f°(x;v) = maximum { <£,v>  The p r o o f may be found  i n F.H.  From p r o p o s i t i o n 2, we may  (1.6)  3f(x) =.{  : ? £ 3f(x) }  Clarke [ 2 ] .  deduce t h a t  ?: <?,v> <^ f°(x;v) f o r every v i n R° }  The c h a i n r u l e f o r composition  o f mappings  as w e l l as the mean v a l u e  (9)  p r o p e r t y have been adapted f o r g e n e r a l i z e d g r a d i e n t s . P r o p o s i t i o n 3. Let g be a c o n t i n u o u s l y d i f f e r e n t i a b l e mapping g:R h:R  m  -> R be L i p s c h i t z .  R.  n  m  Let  Then i f f : R -* R i s g i v e n by f = h o g we have n  3 f ( x ) C 3h(g(x)) o Dg(x) .  The p r o o f may be found  i n F.H. C l a r k e [ 3 ] .  P r o p o s i t i o n 4. I f x and y a r e d i s t i n c t p o i n t s o f R  n  then t h e r e i s a p o i n t z i n the  open l i n e segment between x and y such t h a t  f ( y ) - f ( x ) € <y-x,3f(z)> .  The p r o o f may be found (B)  i n G. Lebourg [ 1 ] .  The D i f f e r e n t i a l I n c l u s i o n Problem. When c o n s i d e r i n g t h e problem of. e x i s t e n c e o f s o l u t i o n s f o r t h e  optimal  c o n t r o l problem, A. F i l i p p o v found  i t convenient  t o study the  s e t v a l u e d mapping  x•+ f(x,U)  = {f(x,u)  : u e U}  and was a b l e t o show t h a t i f t h e t r a j e c t o r y x : [ 0 , l ] -> R relation  n  s a t i s f i e d the  (0.7), x ? E ( x ) , w i t h E ( x ) d e f i n e d as  E(x)  = f(x,U)  then x ( . ) a l s o 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 a p p r o p r i a t e u,  x ( t ) = f ( x ( t ) , u ( t ) ) and u ( t ) G U r e s p e c t i v e l y .  As mentioned i n the  i n t r o d u c t i o n , we w i l l undertake a study o f t h e more g e n e r a l i n c l u s i o n problem.  differential  (10)  In g e n e r a l , possessing  (1.7)  we  will  (t,x) £ [0,1]  X  the f o l l o w i n g p r o p e r t i e s :  f o r each t i n [0,1] compact  (1.8)  a multifunction E(t,x),  consider  and  E(t,x) w i l l  x in R , n  be  a nonempty  set. the m u l t i f u n c t i o n t •+ E ( t , x ) i s measureable;  f o r each x i n R , n  that i s  ' { t e [o,i]  : E(t,x)  n s / o l  i s Lebesque measureable f o r every c l o s e d s e t E ( t , x ) i s L i p s c h i t z i n x; t h a t i s , t h e r e  (1.9)  w i t h the f o l l o w i n g p r o p e r t y : and  v^  i n E ( t , x ^ ) , there  l l " v  In the k ( t ) w i l l be (1.10)  V 2  I  any  t  lxi  "  exists a k £  two  X 2  I  E(t,x)  a number  2> 2  Lipschitz  x  ^  ^  n  constant  k.  i s i n t e g r a b l y bounded; t h a t i s , t h e r e  such t h a t f o r a l l t i n [0,1]  |v| 1  x  '  event t h a t E i s independent o f t , the r e p l a c e d by  points  L*(0,1)  i n E ( t , X 2 ) such t h a t  i s some  1 C ) k  given  S.  b(t)  x in R  n  and  exists b £  L^(0,1)  v in E(t,x),  .  Definition: An  a b s o l u t e l y continuous mapping x : [ 0 , l ]  R  t r a j e c t o r y f o r the m u l t i f u n c t i o n E ( t , x ) p r o v i d e d  n  i s s a i d to be that  x ( t ) G E ( t , x ( t ) ) f o r almost every t i n [0,1]  The  differential  i n c l u s i o n problem can  then be  a  . „  formulated: given  a  locally  R  n  (11)  L i p s c h i t z f u n c t i o n <t>:R -»• R and a m u l t i f u n c t i o n s a t i s f y i n g p r o p e r t i e s n  (1.7) through (1.10) we attempt t o minimize <}>(x(l)) s u b j e c t t o  (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  .  Definition: A r e l a x e d t r a j e c t o r y f o r the m u l t i f u n c t i o n E ( t , x ) continuous mapping x:[0,l]  -»- R  n  which  i s an a b s o l u t e l y  satisfies  x ( t ) S c o E ( t , x ( t ) ) a. e.  The  r e l a x e d d i f f e r e n t i a l i n c l u s i o n problem c o n s i s t s o f m i n i m i z i n g the  f u n c t i o n a l 4>(x(l))  s u b j e c t t o x . £ c o E ( t , x ) and x(0) = X . Q  I f a m u l t i f u n c t i o n i s compact, convex, L i p s c h i t z and i n t e g r a b l y bounded then i t s t r a j e c t o r i e s a r e s e q u e n t i a l l y compact i n the t o p o l o g y o f u n i f o r m convergence. allows  The f o l l o w i n g p r o p o s i t i o n from F.H. C l a r k e [4]  us t o conclude t h a t  (1.13)  i n f { o>(x(l)): x e E ( t , x )  , x(0) = X  Q  =--inf { <J>(x(l)): x G c o E ( t , x ) , x(0) = X  } Q  } .  P r o p o s i t i o n 5. Let E ( t , x )  be measureable i n t L i p s c h i t z i n x and i n t e g r a b l y bounded.  Then f o r every r e l a x e d t r a j e c t o r y y ( * ) j g i v e n any p o s i t i v e 6 t h e r e i s a t r a j e c t o r y x(«) f o r E w i t h x(0) = y(0) and  IIx - yll J  The  proof  < 6 . CO  -  i s s u p p l i e d i n F.H. C l a r k e [4] .  (12)  Definition: Given t ^ [0,1] and y  R  n  the value function for the differential  inclusion problem denoted V(t,y), i s given by V(t,y) = inf K x ( l ) ) where the infimum i s 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) t e l l s 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 C o n d i t i o n s . In Chapter 1 we i n t r o d u c e d the v a l u e f u n c t i o n ;  V(t,y) = i n f { ( K x ( l ) ) : x e E ( x ) , x ( t ) = y } .  Our  immediate g o a l w i l l be t o demonstrate t h a t under s u i t a b l e r e s t r i c t i o n s  on the c h o i c e o f m u l t i f u n c t i o n E , the v a l u e f u n c t i o n 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 R .  Subsequently we w i l l d e r i v e the g e n e r a l i z e d  n  Hamilton - J a c o b i e q u a t i o n and p r e s e n t a n e c e s s a r y with t h i s  condition associated  equation.  We s h a l l assume t h a t t h e m u l t i f u n c t i o n E ( t , x ) s a t i s f i e s c o n d i t i o n s (1.7)  through  (2.1)  (1.10), and  E ( t , x ) i s L i p s c h i t z i n t ; t h a t i s , t h e r e e x i s t s a number k' < »  w i t h the f o l l o w i n g p r o p e r t y :  g i v e n any two p o i n t s t ^ and t ^ i n [0,1]  and v^ i n E ( t ^ , x ) , t h e r e i s some  \\  i n E ( t , x ) such t h a t 2  ~ v | lk'|t 2  1  We w i l l need the f o l l o w i n g d e f i n i t i o n s  , t | . 2  (see F.H. C l a r k e [ 4 ] ) :  Definition:  The  function p:[0,l] X R  X R  n  n  [0,«>) i s d e f i n e d by,  p(t,x,v) = d[v, E(t,x)] .  Definition: I f x(«) i s an a b s o l u t e l y continuous [t,l]  a r c d e f i n e d on t h e i n t e r v a l  w i t h t G [ 0 , 1 ) , we d e f i n e d (x) t  = f  t  1  p(s,x(s), x(s))ds  .  (14)  The  f o l l o w i n g p r o p o s i t i o n i s taken from the same  Proposition  source.  1:  There e x i s t s a p o s i t i v e number K w i t h the f o l l o w i n g given any  a r c x(*)  d e f i n e d on [0,1]  E d e f i n e d on [0,1]  such t h a t x(0)  IIx - yll J  Corollary  co  property:  t h e r e e x i s t s a t r a j e c t o r y y(«)  = y(0)  for  and  < Kd_(x) . 0  —  1:  P r o p o s i t i o n 1 remains v a l i d i f we  are c o n s i d e r i n g a r c s  on a s m a l l e r i n t e r v a l [ t , l ] where t belongs to  defined  [0,1).  frroof: Define  the  change o f parameter 6 :[ t , l ]  vs) = f ^ r Given an a r c x(«)  we  u(p)  = x ( ( l - t ) p + t)  = x(6^(p))  to E.  Let v(«)  Proposition  be  v  n  X R.  properties  satisfies  " " [ 0 1] u  the a r c y ( - ) over [ t , l ]  K /  0 by  by  E(p,u)]  I t f o l l o w s t h a t E has  n  on [0,1]  .  the t r a j e c t o r y f o r E, whose e x i s t e n c e  1, which  " Define  X R  d e f i n e the a r c u(«)  and  P(P,u,v) = d[v,  (p,u,v) £ [0,1]  by  ;  d e f i n e d on [ t , l ]  Set E(p,u) = ( l - t ) E ( ( l - t ) p + t , u)  for  -+[0,1]  P CP.u-CpD» u(p))dp  .  identical  i s asserted i n  (15)  y(s) = v(e (s)) t  Notice  that  y ( s ) = J _ v(8 1-t  ( s ) ) E _ 1 _ E(8. ( s ) , v ( 6 1-t  1  and  p(p,u(p),  u(p))  = d[u(p),  (s))) = E(s,y(s))  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 t h a t  1 1 K/ p(p,u(p),u(p))dp = K/ p(s,x(s),x(s))ds Q  h  -  x l l  [  and  t  t j l  ]  =  l l v  "  u l l  [0,l] - / t P K  (  s  '  x  (  s  )  ^  C  s  )  )  d  s  •  F i n a l l y we need o n l y n o t i c e t h a t  y(t)  = v(0)  = u(0)  = x(t) .  Q.E.D.  Lemma 1. Suppose t h e r e e x i s t s a t r a j e c t o r y x(«) t G [ 0 , 1 ) , with i n i t i a l  c o n d i t i o n x ( t ) = y.  y t h e r e e x i s t s a constant following property: d e f i n e d on [ t , l ]  f o r E d e f i n e d on [ t , l ] , Given a compact s e t C c o n t a i n i n g  n < °°, perhaps depending on C, w i t h t h e  i f y ^ S C t h e r e e x i s t s a t r a j e c t o r y x^(*)  with  initial  c o n d i t i o n x ^ ( t ) = y^ which s a t i s f i e s  | K x ( i ) ) - K x ^ l ) ) | < n|y - y \ ±  R e c a l l <f):R -+ R i s assumed t o be l o c a l l y L i p s c h i t z . n  Proof: Define  for E  t h e a r c x(-) over [ t , l ]  by  •  (16)  x(s)  Notice  = x(s)  + y  - y  1  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, t h e r e  i s some v £ E ( s , x ( s ) )  which  satisfies  |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), Corollary x^(«)  .  X'(s))ds  <  /Jk(s) |y  1 a p p l i e s and allows  f o r E which  - y j d s  <  IlkHJy  -  us t o a s s e r t the e x i s t e n c e  .  of a trajectory  satisfies  IIx  n  x^t)  = X(t)  - xll  < K l I k I L |y  1  = y  «> —  and  1  -  yJ  1  .  1'  1  Therefore,  »x  - xll^<  1  (Kllkl^  +  l)|y - y j  .  *  There e x i s t s a compact s e t C t r a j e c t o r y x^f*)  w i t h the f o l l o w i n g p r o p e r t y :  o f E with i n i t i a l  p o i n t y^ contained  f o r any  i n the compact  * set  C, x ^ s ) i s c o n t a i n e d  in C  fora l l s in [t,l] .  To see t h i s  we n o t i c e t h a t f o r any such t r a j e c t o r y ; x (s) 1  so  = x^t)+  / ^ ( ^ d u  that |x Cs) I 1  <  Ix  1  Ct) I  +  /^k(u)du  <  |x CtD I 1  +  Bkllj  .  (17)  Hence t h e class o f such t r a j e c t o r i e s i s u n i f o r m l y bounded i n the supremum norm. To complete t h e p r o o f o f Lemma 1 we l e t for  be a L i p s c h i t z c o n s t a n t  *  the f u n c t i o n $(•) v a l i d on t h e compact s e t C .  - x ( l ) | < K CKllklt + 1)|y - y \  UCxjCl)) - 4»(x(l)) | •< K j x ^ l )  We s e t n = KjCKllkll  Then  1  x  1  +1).  Q.E.D.  P r o p o s i t i o n 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 s e t C we w i l l show t h a t  |v(t, ) - v(t,y )| < nl/j - y | 7l  whenever y  and y  2  2  2  belong t o t h e s e t C.  The c o n s t a n t n i s t h e same as  t h a t i n Lemma 1. I f x^(') i s a t r a j e c t o r y f o r E w i t h i n i t i a l  data ( t , y ^ ) , Lemma 1  asserts the existence o f a t r a j e c t o r y x ( * ) with i n i t i a l 2  data  (t,y ) 2  which s a t i s f i e s  (2.2)  V(t,y ) - V ( t , 2  7 l  ) < <Kx (l)) - V ( t , y ) 2  x  < <))Cx Ci)) - V ( t , y p 1  + n|y - y | :  2  .  Let x™ denote a m i n i m i z i n g sequence o f t r a j e c t o r i e s with i n i t i a l ( t , y ^ ) and x^ t h e c o r r e s p o n d i n g t r a j e c t o r i e s w i t h i n i t i a l which s a t i s f y  |Kx"(l))  .  -  Kx (D)| 2  i  nlxj - y  2  |•  data  data (t,y ) 2  (18)  We deduce from t h e i n e q u a l i t i e s  V ( t , y ) - V{t, ) 2  (2.2) t h a t  < lim  Yl  - V(t,  R e v e r s i n g t h i s argument, t h a t i s deducing with i n i t i a l  7 l  ) + n|y - y | = n|y 1  2  x  - y |  the e x i s t e n c e o f a t r a j e c t o r y  d a t a ( t , y ^ ) from a g i v e n t r a j e c t o r y w i t h the i n i t i a l  data  ( t , y ) , a l l o w s us t o deduce t h a t 2  v(t, ) - v(t,y ) < n | y - y | yi  which concludes  2  1  2  the p r o o f .  Q.E.D.  Lemma 2: Given e > 0 t h e r e e x i s t s a constant g(e) < °° w i t h the f o l l o w i n g p r o p e r t y : i f t ^ and t for  2  belong t o [0,1 - e] and x ^ - ) i s a t r a j e c t o r y  E d e f i n e d on [ t ^ , l ]  exists a trajectory  x 2  with i n i t i a l  conditions x ^ ( t p  ( - ) d e f i n e d on [ t , l ] 2  Y> t h e r e  =  which s a t i s f i e s  x (t ) = y 2  and  2  | K x ( l ) ) - K x ^ l ) ) | £ S(e) | t 2  - t |  2  .  1  Proof: C o n s i d e r the mapping x : [ t , 1]  x(r)  [ t ,1] d e f i n e d by  = 1 - t ( r- t ) + t x  1  2  .  T-rr-  2  It follows that T  _ 1  ( S ) = Us  - t ^ + t  2  where m = 1 - t  .  1  l _ "  m  t  2  S i n c e x ^ f s ) belongs t h e r e i s some v  t o E ( s , x ( s ) ) i t f o l l o w s from (2.1) t h a t 1  b e l o n g i n g t o E(x ( s ) , x ( s ) ) such t h a t 1  s  |v  1  s  - x ^ s ) ] < k'|s - x  _ 1  ( s ) | = 2sk'|t 1  1  - t |  - h  2  .  2  (19)  Therefore,  dlxjfs)  , lE(x" (s),x (s))] < | ly m m ~ < I lv — '— s m  -  1  1  <  X l  (s)|  - V  s'  |.v ||l s  -  I +  1|  +  Iv ' s |v  - x (s) 1^ ' 1  - x (s)|  m  Since v  belongs to E ( x ( s ) , x ( s ) ) we may 1  g  1  dlxjCs) ,  1E(T" (S),X (S))] < b(x 1  _ 1  1  deduce from (1.10) t h a t  (s))\t  m  I f E\s,y)  1 - ^  follows  < b ( T ( s ) ) + 2sk'|t e 1  1  From C o r o l l a r y 1 we may  deduce the e x i s t e n c e  d e f i n e d on the i n t e r v a l [ t ^ , l ] ,  v ( t p  Iv  -  =  x, II  x  ( t  y  < K(llblL  +  2k )  o  v  e  r  1  t  n  2  e  1  '1  2  2  j  2'  by  = v(x(r))  that = mv (x(r» 2  2  o f a t r a j e c t o r y v(») f o r E,  It, - t  interval [ t , l ]  x (r)  - t | .  and  =  1  1  which s a t i s f i e s  )  1  1°°—  D e f i n e the a r c * ( ' )  2  , E(s,y)]  that  1  x (r)  1 - t,  1  p(s,x (s),x (s))  It follows  - t |  = l E ( x " ( s ) , y ) and m p(s,y,v) = dlv  it  - t | + 2sk'|t  ±  £ mE(x(r) ,v(x ( r ) ) ) = E ( r , x ( r ) ) 2  (20)  so t h a t x ( « ) i s a t r a j e c t o r y f o r E which 9  x (t ) 2  2  satisfies  and  = v(t ) = y x  x (D = 2  v(l)  Therefore  | x ( l ) - x ^ l ) | < KCllbllj. + 2k') | t 2  -t j  2  £  and |4,(x (l)) - ( K X j U ) ) ! 2  for  an a p p r o p r i a t e  Proposition  i M K C l l b l ^ +.2k')|t  L i p s c h i t z constant  2  - t  x  M.  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  The p r o o f  i s i d e n t i c a l to that given f o r P r o p o s i t i o n  [0,1-E].  1 and w i l l  be  omitted. Here i s an example property:  the v a l u e  f o r which the o p t i m a l  a r c z(«) has the f o l l o w i n g  f u n c t i o n V ( t , y ) i s not d i f f e r e n t i a b l e anywhere on  the p o i n t s e t  {(s,z(s))  Our example by ( > y ) • x  takes p l a c e Define  : s belongs t o [0,1]} .  in R  2  and we w i l l  denote a g e n e r i c p o i n t  the m u l t i f u n c t i o n E(x,y) by  E(x,y) = {(0,0)} U  {(-1,-1)}  and the o b j e c t i v e f u n c t i o n a l <j> by  <j>(x,y) =  | (x,y) | .  2 i n R'  (21)  We consider the problem of minimizing <Kx(l),y(l)) over a l l trajectories (x(•),y(•)) of E, defined on [t,l] and satisfying the i n i t i a l condition (x(t),y(t)) = (x ,y ) 0  0  .  For this particular problem the optimal trajectory for the i n i t i a l data (t,XQ,  XQ)  can be shown to satisfy  (x(s),y(s)) =  '(0,0) (-1,-1)  i f t < s < (l-t)-x +t Q  i f 1-x  <s<l  provided (1-t) > x^ and (x(s),y(s)) = (-1,-1) i f (1-t) < x  f  In order to consider arbitrary i n i t i a l data (t,x,y) with 0 _< t <_ 1 and 2 (x,y) in the f i r s t quadrant of the plane, let D be the diagonal in R : D = {(x,y) : y = x, 0 <_ x <_ 1} . Given (x ,y ) let P = (p ,p ) be the point of D closest to (x ,y ) Q  0  Figure 1  Q  Q  0  Q  0  (22)  Case 1:  1-t <_  .  The o p t i m a l t r a j e c t o r y  satisfies  i f t < s < 1-p  (0,0) (x(s),y(s))=  L(-l,-l)  In t h i s  i fl-p < s < 1 Q  case  V(t,x , y ) Q  Case 2:  Q  1-t < p  Q  =  |(x ,y ) - (P ,P )| = d [ ( x , y ) , 0  0  0  0  V(t,x ,y ) = Q  D] .  0  (-1,-1)  and  |(x ,y ) - (1-t,1-t)| . 0  0  t o summarize t h i s i n f o r m a t i o n i s as f o l l o w s :  be the v e c t o r (-1,-1).  let h  Then  V ( t , x , y ) = d[ (x,y),D] =  0  satisfies  (x(s),y(s))-=  A c o n v e n i e n t way  0  .  The o p t i m a l t r a j e c t o r y  If  0  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 s the h y p e r p l a n e i n d i c a t e d i n F i g u r e 2 then  V ( t , x , y ) = d[(x,y),D] p r o v i d e d (x,y) belongs to the h a l f s p a c e below =  L  t  |(x,y) - ( 1 - t , 1 - t ) | p r o v i d e d (x,y) belongs t o the h a l f s p a c e above  (23)  Figure  2  This r e p r e s e n t a t i o n properties of V . e >  i s convenient when d e r i v i n g  F o r example, i t can be seen t h a t  differential  (see F i g u r e  3) for  0  V(S,1-S,1-S+E)  =  V(s,1-S,1-S-E)  =  £  and £  J2 Figure  3  I t may be concluded t h e r e f o r e t h a t V i s nowhere d i f f e r e n t i a b l e on the s e t {(s,l-s,l-s):  0 £ s <_ 1 }  .  (24)  For the i n i t i a l  d a t a (0,1,1) the o p t i m a l a r c i s g i v e n by  Z(s) = ( l - s , l - s ) .  In s  n  two  o r d e r t h a t we may  -»- s; x  -»• 1-s;  n  compute  V ( s , l - s , l - s ) we  y -> 1-s; and VV(s ,x ,y ) -> v 'n ' n n'-'n'  s e p a r a t e cases t o c o n s i d e r when computing  Case 1:  c o n s i d e r sequences  v  as n -»• °°.  V(  s n  >  x n  There are  >v ). n  (x ,y )*h < h * ( l - s ,1-s ) i n which case n n — n' n^ J  v  J  v  VV(s ,x ,y ) = (x +y - 2 ( l - s ),x -(1-s ),y -(1-s )) ^ n' n " n ^ n n *• n ' n ^ n^'^n n" (x ,y ) - ( l - s ,1-s ) n n n n J  J  v  (x ,y )*h > h - ( l - s  VV(s  ,x ,y ) = n n n  ,1-s  ( 0  f, 1  K  J  v  Case 2:  v  )  i n which case  ' n-Pn' n-Pn x  y  r-r  •>  ^  ( n ,Y ) n ) - n(p n,p ' x  J  r  r  where ( P > P ) i - the c l o s e s t p o i n t i n D t o ( s  n  n  t h a t i n Case 2, l i m VV(s  n  )  1  x n  >y )' n  I t i - easy to see s  ,x ,y ) = ± 1 (0,-1,1). n n —pr  In  Case 1 we n o t i c e t h a t i t i s s u f f i c i e n t t o c o n s i d e r l i m VV(s ,x ,y ) ^ n n n x - (1-s 1 when the r a t i o n n^ remains c o n s t a n t as n •> ». The q u a n t i t i e s y -(1-s ) •'n n^ v s x r l i m ^C n ' n '> n ) y then be p a r a m e t e r i z e d w i t h t h i s r a t i o . In p a r t i c u l a r : J  m a  for  k b e l o n g i n g t o [-1,0],  (l+k,k,l) belongs to 8V(s,1-s,1-s); 717k 2  for  p b e l o n g i n g t o [0,°°),  (l+p,l,p) belongs to 3 V ( s , l - s , l - s ) ; TITp^  J  (25)  for  m b e l o n g i n g t o [-1,0]  (1+m,l,m) belongs  TT+m  to 3 V ( s , l - s , l - s ) .  2  In a l l cases the (x,y) components o f 3V l i e on the u n i t c i r c l e o f the (x,y) plane w h i l e the t component v a r i e s from 0 when (x,y) = _1_ (-1,1), J2 to a maximum v a l u e o f -Jl when (x,y) = _1_ (1,1) and back t o 0 when 72 (x,y) = 1 (1,-1). Taking the convex h u l l o f these p o i n t s g i v e s us  71  the two-dimensional  s e t shown i n F i g u r e 4.  Figure 4  To b e g i n the d i s c u s s i o n o f n e c e s s a r y : c 6 h a i t i o n s w e : : p r o v e a'-result which has been adapted from W.H.  Fleming  and R.W.  Rishel [ l ] .  P r o p o s i t i o n 4. If  x(») i s a t r a j e c t o r y f o r E d e f i n e d on [ t , l ]  a nondecreasing for  initial  f u n c t i o n o f s on [ t , l ] .  then V ( s , x ( s ) ) i s  I f z(«) i s an o p t i m a l  trajectory  c o n d i t i o n s z ( t ) = y then V ( s , z ( s ) ) i s c o n s t a n t on [ t , l ] .  Proof: For any - ^ 2 which s a t i s f y S  S  t <^ s^ <_ s^ ^_ 1 ;  V(s ,x(s )) < V(s ,x(s )) . 1  If  1  2  z(-) i s o p t i m a l , V ( t , z ( t ) ) = cj>(z(l)).  2  T h e r e f o r e 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 t h a t e q u a l i t y must h o l d and V ( s , z ( s ) ) i s c o n s t a n t on [ t , l ] P r o p o s i t i o n 5: For every  0 <  ( t , y ) b e l o n g i n g t o [0,1] X R ; n  inf v E E(t,y)  { l i m sup V(t+ ,y+Av) - V ( t , y ) } A+0 A  Proof: We c o n s i d e r the d i f f e r e n c e q u o t i e n t  V(t+  for  A > 0 and v £ E ( t , y ) .  ,y+Av) - V ( t , y ) A D e f i n e the a r c 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 t o t h a t p r e s e n t e d  i n C o r o l l a r y 1 may  be used t o show the f o l l o w i n g : t h e r e e x i s t s a t r a j e c t o r y x (•) f o r E A d e f i n e d f o r s E [t,t+A]  which  satisfies  x^(t) (2.3)  It  Hx -  x  x  l l  [  t  ^  t  +  x  <  ]  0, we may conclude  1 A It  and  p(t+s,x(t+s),x' (t+s))ds  .  f o l l o w s e a s i l y from our assumptions on the m u l t i f u n c t i o n E t h a t  p (*,•,•) i s continuous. to  K/J  = y  f o l l o w s from  T h e r e f o r e i f we c o n s i d e r a sequence o f A t e n d i n g  that  P(t+s,x(t+s),x'(t+s))ds  (2.3) t h e r e f o r e , t h a t  ->  p ( t , x ( t ) , x ' (t))  = 0 .  (27)  Now f o r A > 0, i f x (•) denotes t h e t r a j e c t o r y o f e q u a t i o n A  (2.3)  and n  i s an a p p r o p r i a t e L i p s c h i t z c o n s t a n t 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) __  S i n c e x,(•)  + (t.+A,x (t+A)) - V ( t , y ) __ V  x  <  V(t+A,y+Av) - V ( t , y ) _  i s a t r a j e c t o r y f o r E, P r o p o s i t i o n 4 i m p l i e s  V(t+A,x (t+A)) - V(t,y) > 0 A  .  It follows that  -o(A) A  < V(t+A,y+Av) - V ( t , y ) A  and  0 <_ l i m 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 ) . The  Q.E.D.  f o l l o w i n g Lemmas w i l l be needed i n t h e d e r i v a t i o n o f t h e g e n e r a l i z e d  Hamilton - J a c o b i e q u a t i o n . Lemma 3: I f V(*,») i s d i f f e r e n t i a b l e a t ( t , y ) and z(«) t r a j e c t o r y with i n i t i a l  i s an o p t i m a l r e l a x e d  c o n d i t i o n z ( t ) = y, then t h e r e e x i s t s a v e c t o r  £ b e l o n g i n g t o 3 z ( t ) n c o E ( t , y ) such t h a t  V (t,y) + V (t,y)t  ?  = 0  (28)  Proof:  We w i l l  show t h a t i n f a c t  8z(t) C coE(t,y)  and  that there  i s an element c. o f 9 z ( t ) which V (t,y) t  Given a sequence A^ t e n d i n g existence  + V (t,y)-c y  satisfies  = 0  .  t o 0, t h e mean v a l u e  property  implies the  o f A' G (0.A ") such t h a t  z(t+A ) = z ( t ) + A c n n n v  v  f o r some t,^ G 9z(t+A^). we may assume t h a t 8z(»)  that  £  By c o n s i d e r i n g  ->  ? G 3z(t).  I t follows  We may conclude  an a p p r o p r i a t e  subsequence A.. ,  from u p p e r s e m i - c o n t i n u i t y  of  therefore  z(t+A.) = z ( t ) + A.£ + o(A.) J J •J V(t+A. z(t+A.)> - V ( t , z ( t ) ) = V ( t + A . , z ( t ) + A . c : ) 3  V (t,z(t)) t  and - V ( t , z ( t ) ) + o(\.)  + V ( t , z ( t ) ) - C + o(X.) y  A.  However i t f o l l o w s  from P r o p o s i t i o n 4, s i n c e  z(») i s o p t i m a l ,  V(t+X , z ( t + A ) ) - V ( t , z ( t ) ) = 0  that  .  Therefore; V (t,z(t)) + V (t,zCt)D-c t  To  demonstrate t h a t  = 0  .  8 z ( t ) C c o E ( t , y ) we l e t B denote t h e s e t  (29)  {s:t<s<l  and z ( s ) does n o t e x i s t } U {s:t<_s<l  and z ( s ) £ c o E ( s , z ( s ) ) } .  Then B has measure 0 and as i s shown i n F.H. C l a r k e [ 2 ] ,  9z(t) = 3 z ( t ) B  where 9_z(t) i s t h e convex h u l l o f a l l l i m i t s B that s  n  belongs  to [ t , l ] \ B .  z ( s ) w i t h s -> t such ^ n n  F o r any such sequence s , ( z  s n  )  belongs  • to c o E ( s  n J )  z(s )).  Therefore l i m ( z  n  of the m u l t i f u n c t i o n E(«,«).  s n  )  e  c o E ( t , z ( t ) ) by u p p e r s e m i c o n t i n u i t y  Taking t h e convex h u l l s o f a l l such  limits  we f i n d t h a t 9 z(t) C coE(t,z(t))  .  D  D  Q.E.D.  Lemma 4: For every  ( t , y ) b e l o n g i n g t o [0,1]X  R, n  min min { a + b«v }^_0 . 9V(t,y) E ( t , y ) Proof: We c o n s i d e r (a,b) £ 9V(t,y) as r e p r e s e n t e d by a convex  combination  o f l i m i t s o f g r a d i e n t s o f V; t h a t i s , m (a,b) = l i m y a.[ a. , b. ] . , li i n-x» 1=1 where  a ,b i i n  n  = VV(t ,y ) l i n  v  n  y  : t l  n  -»- t ; y\ •+ y f o r i = l,...,m. l  Given  v G E ( t , y ) , s i n c e E(«,«) i s L i p s c h i t z , we may deduce t h e e x i s t e n c e o f u^ €E E ( t , y " ) and v ^ E E ( t " , y ) which s a t i s f y , n  l i u  ~ l 1 ( t ) | y ? - y| v  k  i n ni .. j n . v. - u. < k' t . - t . 1  I  I — 1  1  I  1  respectively,  and  (30)  I t f o l l o w s t h a t v } -+ v  since  1  \\  ~ v|  < k'|t?  - t|  k(t)|yj  +  -  y|  Therefore, .  a + b«v  m v r , n n-, £ a..{a. + b.*v.} . l l i i  i •  n  = lim  1  m lim Z a.V(t ,y n-x» i = l n  We  may  n  n  n  ; l,v ) > 0 n  consequently  a + b-v  for  ; l,v )  conclude from P r o p o s i t i o n 5 t h a t  V'(t ,y  and  n  arbitrary  (a,b)  £  9V(t,y).  min 3V(t,y)  Theorem  >_ 0  It follows  min {a E(t,y)  that  + b-v}>_0.  Q.E.D.  1: For every ( t , y ) b e l o n g i n g  to [0,1]  X  R, n  min { a - H(t,y,-b) } = 0 9V(t,y)  Proof: We optimal exist  suppose t e m p o r a r i l y  t h a t V i s d i f f e r e n t i a b l e at  relaxed t r a j e c t o r y with i n i t i a l  (t,y).  The  d a t a ( t , y ) , which i s known to  (see d i s c u s s i o n i n Chapter 1 ( B ) ) , w i l l be  denoted z ( * ) .  From  (31)  Lemma 3 we  deduce the e x i s t e n c e o f a v e c t o r t, b e l o n g i n g t o c o E ( t , y ) take c = z ( t ) ) such  ( i f z ( t ) e x i s t s we may  that  V (t,y) + V (t,y)-? = 0 . t  S i n c e z, £ c o E ( t , y ) t h e r e i s some convex combination such  {a^} and v^  E(t,y)  that m Z a [V i=l  (t,y) + V (t,y)-v ] = 0 y  However,  V (t,y) + V (t,y)-v. t  = V ( t , y ; l,v.) > 0  y  and c o n s e q u e n t l y f o r some i  V (t,y) + V (t,y)-v. t  We  may  conclude from t h i s and P r o p o s i t i o n 5 t h a t whenever V i s d i f f e r e n t i a b l e  (2.4)  Given  = 0 .  y  inf V ' ( t , y ; l , v ) = 0. E(t,y)  (a,b) G 3V(t,y) such t h a t  (a,b) = l i m V V ( t , y ) n-*=° n  f o r sequences  t -*"t and Y^*y> l  V ( t ,y 1  e  t  v  e n  n  E ( t ,y ) s a t i s f y  (see e q u a t i o n  ; l,v ) = 0 .  By c o n s i d e r i n g a s u i t a b l y chosen subsequence, v^, we may v^->v.  (2.4))  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 v G E ( t , y ) .  assume t h a t Consequently  (32)  (2.5)  a + b«v = l i m V f t . , y . ) + V ( t . , y . ) - v . = 0 .  From (2.5) and Lemma 4 we may conclude t h a t  (2.6)  min min 3V(t,y) E ( t , y )  R e c a l l t h a t H(t,y,-b) =  max v€E(t,y)  { a + b - v } = 0 .  {(-b)«v}  so t h a t e q u a t i o n  (2.6) may be  rephrased as  min 3V(t,y)  { a - H(t,y,-b) } = 0 .  Q.E.D.  We w i l l say a L i p s c h i t z f u n c t i o n W(t,y) i s a s o l u t i o n o f t h e g e n e r a l i z e d Hamilton  - J a c o b i e q u a t i o n i f f o r every  (2.7)  ( t , y ) b e l o n g i n g t o [0,1] X R  max { a + H(t,y,b) 3W(t,y)  n  = 0 .  Theorem 1 c o u l d be r e p h r a s e d then by s t a t i n g t h a t - V ( t , y ) i s a s o l u t i o n o f t h e g e n e r a l i z e d Hamilton  - J a c o b i equation.  The f o l l o w i n g r e s u l t i s a n e c e s s a r y c o n d i t i o n i n o r d e r t h a t z(») be an o p t i m a l  trajectory.  Theorem 2: I f z(«) i s an o p t i m a l 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 ] , t h e f o l l o w i n g h o l d s : t h e r e e x i s t s £(s) £ 3 z ( s ) i ~ E ( s , z ( s ) ) and  ( a ( s ) , b ( s ) ) G 3 V ( s , z ( s ) ) such t h a t  (2.8) Furthermore,  a(s) + b(s)-s(s) = 0 .  (33)  (2.9)  -b(s)-C(s)  and we may  =  H(z(s),-b(s))  take £(s) = z ( s ) whenever  z i s d i f f e r e n t i a b l e a t s.  Proof: Suppose z ( * ) i s d i f f e r e n t i a b l e a t an i n t e r i o r p o i n t s o f the interval [t,l] .  C o n s i d e r f o r A £ [ 0 , 1 - s ] , the a r c  z(s+A) = z ( s ) + Az(s) .  We  d e f i n e the L i p s c h i t z f u n c t i o n f ( A ) by  f ( A ) = V(s+A,z(s+A))  and note t h a t s i n c e z i s c o n t i n u o u s l y for  generalized  gradients  ( P r o p o s i t i o n 1.3) s t a t e s  9f(A) C 3V(s+A,z(s+A))  We w i l l now  demonstrate t h a t 0  (2.10)  • [l,z(s)]  • [l,z(s)]  imply  .  that  V(s+A,z(s+A)) - V(s+A,z(s+A)) = o(A) as  and  that  z ( s ) e x i s t s , z(s+A) = z ( s ) + A z ( s ) + o(A) as A4-0 .  thereby f o l l o w s  A+0  consequently  f ( A ) - f ( 0 ) = V(s+A,z(s+A)) - V ( s , z ( s ) )  From P r o p o s i t i o n 4 we n o t i c e  that  V(s+A,z(s+A)) - V ( s , z ( s ) )  rule  .  3f(0) which w i l l  0 G 3V(s,z(s))  Since  d i f f e r e n t i a b l e , the c h a i n  = 0  + o(A) .  It  (34)  and  conclude  that  f(A)  An  identical  and  f (0) 1  a r g u m e n t shows t h a t f ( - A )  = 0.  which implies from t h i s  Therefore,  0 = -f (0) 1  Z(S^)-H;(S).  From e q u a t i o n  (a^,b^) b e l o n g i n g  (2.10)  follows  considering  1  that  ( 2 . 1 0 ) we d e d u c e t h e e x i s t e n c e  t o 3V(s^,z(s^))  v  appropriate  (a^,b^)->(a(s) , b ( s ) )  s ^ £ ( t , l ) a n d s^-»-s  which s a t i s f y ,  f o r every i ,  + b.•z(s.) = 0 .  1  x  J  s u b s e q u e n c e s , we may assume  and again  (a(s),b(s)) e 3V(s,z(s))  .  a(s)  To  a sequence o f p o i n t s  i tfollows by uppersemicontinuity  riE(s,z(s)).  a.  By  < f°(0;l)  < f°(0;-l) a n d 0 = f ' ( 0 ) Equation  exists  fact.  £(s) G 3 z ( s ) of points  - £(0) = o ( A ) s o t h a t f ( 0 )  ( P r o p o s i t i o n 1.2) t h a t 0 G 3 f ( 0 ) .  We w i l l now c o n s i d e r such that  - f(0) = o(A) .  by uppersemicontinuity  that  i t follows  that  Therefore,  + b(s)•S(s)  complete t h e statement o f t h i s  = 0 .  t h e o r e m , we n o t i c e  f r o m Theorem 1  that  a(s)  - (-b(s))'C(s)  > a(s) - H(s,z(s),-b(s))  > 0 .  E q u a l i t y thereby holds 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  for  classical  calculus  an o p t i m a l s o l u t i o n were t a c k l e d w i t h the  o f extremals".[ R.  Courant [ l ] ] .  d i f f i c u l t i e s were awesome and t h i s manner.  For  few  the  o p t i m a l c o n t r o l problem, the  difficulties  increased  In t h i s chapter, we  the  technical  completely analysed i n not  conditions.  s i t u a t i o n i s somewhat  o f the v a l u e f u n c t i o n .  i n direct proportion  wanted to s p e c i f y the  "fields  purposes, the method was  same sense as the n e c e s s a r y  owing to a more e x p l i c i t use  of  e l e g a n t but  problems c o u l d be and  conditions  construction  T h i s method was  Hence, to a l l i n t e n t s  p r a c t i c a l i n the  one  of v a r i a t i o n s , s u f f i c i e n t  However,  better  technical  to the p r e c i s i o n w i t h which  d i f f e r e n t i a l properties  of t h i s  function.  p r e s e n t 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  s e r v e to e l i m i n a t e  computational problems encountered i n the  analysis  some o f  of optimal  the  control  problems. We will  b e g i n by  l a y the  functional  s t a t i n g a converse to P r o p o s i t i o n  groundwork f o r the main theorem.  cf>:R ->- R i s by n  Proposition  X R  Recall  the  assumption a l o c a l l y L i p s c h i t z  objective function.  1.  Let W(s,y) be on [0,1]  4 o f Chapter 2 which  n  such  a r e a l valued, l o c a l l y L i p s c h i t z function  defined  that  W(l,y) =  for  every y b e l o n g i n g to R .  and  suppose f o r any  x(t  ) = y , W(s,x(s)) i s n o n d e c r e a s i n g on [ t , 1 ] .  n  Let  Ky)  t r a j e c t o r y x(«)  (t ,y ) be  given i n i t i a l  o f E, d e f i n e d  conditions  on [ t , l ]  and  satisfying  I f z(-)  is a  trajectory  Q  (36)  for is  E d e f i n e d on [ t , l ] ,  satisfying  Q  constant on [ t ,1];  conditions  then  z(«)  z  ( t ) = YQ> Q  a n  d  such  t h a t W(s,z(s))  i s o p t i m a l f o r the g i v e n  initial  and  W(t v  o  ,y ) = V ( t ,y ) o o o  where V(-,«) i s the v a l u e f u n c t i o n . Proof: For any  t r a j e c t o r y x(-) such t h a t x ( t ) = y , Q  with e q u a l i t y f o r x = z.  That i s ,  4>(z(l)) 1  and  z(>)  cf»(x(l))  i s thereby o p t i m a l f o r the i n i t i a l  c o n d i t i o n s ( t -,y ) .  W ( t , y ) = 4>(z(l)) = ( o ^ o v  Q  Theorem Let Hamilton  t  )  Q > E  o  Therefore  - D  1: -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 o f the g e n e r a l i z e d - Jacobi  equation;  max { a + H(t,y,b)} 3W(t,y) Suppose W s a t i s f i e s  = 0 .  the boundary c o n d i t i o n s  W(l,y) =  for  every y b e l o n g i n g t o R .  and  suppose the t r a j e c t o r y  n  Let z(«)  <Ky)  ( t ,^ ) be g i v e n i n i t i a l  satisfies  z  ( t ) = YQ Q  and  conditions  (37)  3W(s,z(s)) • [ l , z ( s ) ]  then z(-) i s an o p t i m a l  = 0  a.e.;  t r a j e c t o r y f o r ( t »y )  ,Y  W(t ^ o  ) = V(t o  and  ,y ) . o  J  o  Proof: We  b e g i n by n o t i n g t h a t i f -W(s,y) i s a s o l u t i o n o f (2.7)  W(s,y) i s a s o l u t i o n o f the f o l l o w i n g t h r e e  (3.1)  equations:  max { a + H(t,y,b) } = 0 ; -3W(t,y) min {a 3W(t,y)  - H(t,y -b) } = 0  T h i s serves  We  } = 0  to show more c l e a r l y the c o n n e c t i o n  c o n d i t i o n and  the Hamilton - J a c o b i e q u a t i o n  will first  as a f u n c t i o n o f s.  o f E.  We  I t f o l l o w s t h a t the s e t G,  Lebesgue measure 1-t to G we  by  exists}  i f * ( • ) i s d e f i n e d over [ t , l ] . Q  (T)  =  For a p o i n t  W(S+T,X(S)+TX(S))  f o r some e > 0.  are met  defined  define  f  property  (2.7).  observe t h a t W(s,x(s)) i s L i p s c h i t z  G = {s:d_ W(s,x(s)) e x i s t s } n {s:x(s) ds  f o r T G [0,e]  between the s u f f i c i e n c y  demonstrate t h a t W(s,x(s)) i s a n o n d e c r e a s i n g f u n c t i o n  o f s f o r every t r a j e c t o r y x(*)  s belonging  ;  3  min min { a + vb 3W(t,y) E ( t , y )  has  then  Then the c o n d i t i o n s f o r the c h a i n r u l e  (see P r o p o s i t i o n 1.3)  and we  may  deduce t h a t  (38)  3f (0)  C 9W(s,x(s))  -[l,x(s)]  .  Since s G G i t follows that  x ( s + i ) = x ( s ) + x x ( 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)) T+0 ^  • [ l,x(s)] .  s  T  However 9W(s,x(s)*  [ l , x ( s ) ] >^ 0 from e q u a t i o n (3.1)  so t h a t 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 t h e t r a j e c t o r y z(«)  d_W(s,z(s))  i t follows that  G 9W(s,z(s))  • [l,z(s)l  a.e.;  ds however by h y p o t h e s i s 9W(s,z(s))  • [l,z(s)]  = 0 a.e.  so t h a t W ( s , z ( s ) ) i s c o n s t a n t 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(»,«) t h e r e b y 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  c o n c l u d e t h a t z(»)  i s optimal.  Q.E.D.  The d e r i v e d n e c e s s a r y c o n d i t i o n s f o r t h e d i f f e r e n t i a l i n c l u s i o n problem  c o u l d 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  aw  that  • [ 1,'z]  = 0 .  Could the s u f f i c i e n c y theorem be sharpened so as t o decrease the above discrepancy?  To show t h a t i n f a c t t h i s i s not p o s s i b l e , we w i l l  s t r a t e t h a t when the v a l u e  f u n c t i o n i s r e g u l a r the n e c e s s a r y  demon-  condition  may be s t r e n g t h e n e d t o c o i n c i d e w i t h the s u f f i c i e n t c o n d i t i o n .  Recall  that r e g u l a r i t y requires  V°(t,y;i,v) = V ' ( t , y ; i , v )  for  i = ±1 and v €E R . n  T h i s c o n d i t i o n c o u l d be shown t o always be the  case when t h e problem i s "convex" i n a c e r t a i n sense. Theorem 2: I f z(») i s an o p t i m a l d a t a ( t , y ) and i f the v a l u e  8V(s.,z(s))  In o r d e r  t r a j e c t o r y f o r the problem with function i s regular  •• [ l , z ( s ) ]  initial  then  = 0 a.e.  t o prove t h i s theorem i t w i l l be n e c e s s a r y t o i n t r o d u c e the  notions  o f g e n e r a l i z e d normal and tangent cones t o an a r b i t r a r y c l o s e d  set  F o r a more d e t a i l e d d i s c u s s i o n see F.H. C l a r k e  Q.  We d e f i n e the f u n c t i o n d:R  d(x)  n  [2].  -+ R by  = min {|x-e| : e £ Q} .  I t i s e a s i l y v e r i f i e d t h a t d(x) i s u n i f o r m l y  L i p s c h i t z w i t h constant  1.  Definition: The  cone o f normal .vectors  o f the s e t  t o Q a t e, denoted N ^ f e ) ,  i s the closure  (40)  { p £ R  n  : 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 i s : (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 R . n  set Q = {x : F(x)r 0}  We  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  V  Ad - e  G Q satisfy A  = d(Ad) . A  It follows that 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) = l i m sup AIO  for  every d G Q(0)•  F(Ad) d  That i s  T  <£,d> £  for  a l l ? G 3F(0)  and  d G T (0).  3F(0)  P r o o f o f Theorem We  recall  be o p t i m a l [t,l]  Therefore,  Q  Q  .  Q.E.D.  2: t h a t a n e c e s s a r y c o n d i t i o n f o r the t r a j e c t o r y z(«) d a t a ( t , y ) i s t h a t V ( s , z ( s ) ) be  (see P r o p o s i t i o n 2.4). to the  We  c o u l d r e s t a t e t h i s by  interval [ t , l ] ,  {  I f we  0  C [T (0)]° = N (0)  f o r the i n i t i a l  s belonging  <_ 0  (s,x)  we  may  for a l l  set  conclude t h a t the a r c z(*) d e f i n e d  (s,z(s)) =  by  z(s)  belongs t o the c l o s e d s e t Q f o r every s b e l o n g i n g recall  on  : V(s,x) = V ( t , y ) } .  denote t h i s s e t by Q,  p o i n t we  constant  saying  ( s , z ( s ) ) belongs t o the  to  the f o l l o w i n g theorem from F.H.  to [ t , l ] .  At  this  Clarke [ 2].  Theorem: The  t r a j e c t o r y z(s) belongs t o the c l o s e d s e t $ f o r s > t i f and  o n l y i f z'(s)  i s tangent to $ a t z ( s ) almost everywhere; t h a t i s , z'(s)  belongs to T ( z ( s ) ) $  It  a.e.  .  f o l l o w s from t h i s theorem t h a t  z'(s)  = [l,z(s)]  belongs to T ( s , z ( s ) ) a.e. n  .  (42)  Therefore,  f o r every  (a,b)  belonging  [l,z(s)]  to N Q ( S , Z ( S ) ) ,  • (a,b)  < 0 .  In p a r t i c u l a r , Lemma 1 a s s e r t s t h a t  (3.3)  Now  [l,z(s)]  z(«)  • 9V(s,z(s))  <0  a.e.  .  and V(«,») s a t i s f y the g e n e r a l i z e d Hamilton - J a c o b i  equation,  Theorem (2.1), min 9V(s,z(s))  { a + <b,z(s)> } = 0  a.e.  .  It follows that  (3.4)  [l,z(s)]  Inequalities  (3.3)  and  [l,z(s)J  • 3V(s,z(s)) ^ 0  (3.4)  imply  a.e.  .  a.e.  .  that  • 8V(s,z(s)) = 0  Q.E.D.  •Example:  2 We  c o n s i d e r the problem i n R  over a l l t r a j e c t o r i e s  of minimizing  ( x ( s ) , X 2 ( s ) ) which s a t i s f y  {%x(l)  1  2  T h i s problem c o u l d be r e f o r m u l a t e d  minimize {%x(l)  + y  : -1  < v < 1} .  as  2  - /^|x|ds} I *  over a l l a r c s x(s) such t h a t x ( t ) = y, and  X2(l)}  ( x ^ ( t ) , X 2 ( t ) ) = (y^,y2)  and  ( x ( s ) , x ( s ) ) .£ { ( v . - l x ^ s ) |)  +  I  |x| <_ 1.  (43)  A reasonable will  guess as to the nature  o f the o p t i m a l  arc x(s) i s that i t  look, f o r some s w i t c h i n g time T, as f o l l o w s :  Figure 5  X*  To v e r i f y t h i s s u p p o s i t i o n we g e n e r a l i z e d Hamilton - J a c o b i  min {a 3W(t,y y ) r  attempt to c o n s t r u c t a s o l u t i o n W o f the equation;  - H(y  } = 0  ,y ,-b)  2  which s a t i s f i e s the boundary c o n d i t i o n s  W ( l , y , y ) = <$>(y ,y ) 1  and  1  2  f o r which the a r c i n q u e s t i o n  SWfSjX^s) , x ( s ) ) 2  To b e g i n w i t h , we trajectory.  We  = & h  1  + y  2  .  (x^(s),x (s)) satisfies 2  • [ l j X j f s ) , x ( s ) ] = 0 a.e. 2  .  must d e c i d e which s w i t c h i n g time w i l l g i v e an  consider  arcs  x(s) = y  + ( s - t ) on [ t , T ]  = y  1  and  2  + (T-t) - (s-T) on [ T , l ]  an a s s o c i a t e d performance f u n c t i o n  optimal  (44)  f(T) = %(y 2T-t-l) - /J|x(s)|ds + y . 1+  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 -3T +[ 3-4(y t)] T+fy^t) (3/2+2t-y )+%t -l+y 2  2  r  1  2  i f T < hi 1 - ( y t ) ] r  f(T) =< T  2  _  + ( _y ( t )  T  t  y r  ht  2  +  + 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 ) : 0 < t < 1 , y  >0}  ±  x  as in the following diagram; Figure 6  c B A  > then the minimum for f(T) i s 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^,y ); 2  W(t,y y ) = -J t -y t+3/2y -y -l+y 2  r  2  2  2  1  1  2  in A,  (45)  = ht^ - ht + y ^ t - y  + y  1  i n B,  2  2~  = -ht  2  + t/2  y  + y t -  - h + y  1  i n C.  2  2~" It i s e a s i l y v e r i f i e d  that  W ( l , y , y ) = hy 1  2  +  1  y  r  To show t h a t W ( l , y ^ , y ) s a t i s f i e s the H a m i l t o n - J a c o b i e q u a t i o n ,  we  2  note t h a t the H a m i l t o n i a n f o r the  E(y y ) r  i s given  =  2  UvHyJ)  1  2  1 >  p ) = 2  =  d e r i v e the f u n c t i o n  p  1  - P 1I  1  -P  v<  1}  if P  2  - P  X  l  2  y  i l  i  £  X  Pi  > o  <  0  •  VW;  v W ( t , y , y ) = [-t  - y  2  = [t -  = [-t +  The  : - l<  by  H(y ,y ,p  We  multifunction  x  + 3/2,-t + 3/2  h + y , t - h,l] 1  y  1  + h,t - % , l ]  - 2y l] r  in A  in B  i n C.  s e t 9W(t,y^,y ) i s e a s i l y computed on the boundary o f these 2  regions.  To demonstrate t h a t W(t,y^,y ) i s a s o l u t i o n o f the g e n e r a l i z e d Hamilton 2  Jacobi equation, A,  B,  one  C o f which we  - t + 3/2  - 2y  1  l a s t computation i s n e c e s s a r y i n each o f the g i v e one,  > 0 so  that  the o t h e r  two  being  similar.  regions  In r e g i o n  A,  (46)  = (-t - y±  + 3/2)  - [ - t + 3/2  F i n a l l y , t o prove o p t i m a l i t y  - 2y±  +  =  0 .  o f the t r a j e c t o r y w i t h the chosen s w i t c h i n g  time, suppose f o r example the i n i t i a l  d a t a i s {t,y^,y^){  =  (%,%,y ). 2  Then the o p t i m a l a r c would be  + (s -• 1/4)  for  1/4  < s  =. 3/4  - (s - 1/2)  for  1/2  < s < 1  1  x (s) 2  For a l l s such t h a t  = y  1/4  2  - / J  / 4  < s < 1/2  3W(s,x (s),x (s)) 1  = (-S  f o r 1/2  <  x ( s ) = 1/2  +  X l  2  ( s ) + 1/2)  |  X l  | d s  1/2  .  ,  • [l,x x ] r  2  + (s - 1/2)  - |x (s)| = 0 1  < s < 1 , 8W(s,x (s),x (s)) 1  2  •  [l,^,^]  = (s - 1/2 + x ( s ) ) - (s - 1/2) - Ix^s)! = 0 x  which proves o p t i m a l i t y by Theorem 1.  .  (47)  4. Canonical Transformations. Let U be a given open subset of R . n  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 l i e in Q and satisfy the boundary conditions x(0) e C  and x(l) e C .  Q  1  When E(',«) satisfies conditions (1.7) through (1.10), <)>(•) is locally Lipschitz and C^, C^ are closed sets in R the following theorem from n  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 N  c  o  (z(0)),  -p(l) £ N „ (z(l)) + X9<J.(z(l)) l  and  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, w i t h the concept  i n the a n a l y s i s o f H a m i l t o n i a n  systems.  We  of canonical  first  develop  transformations, some  terminology  and n o t a t i o n . We  c o n s i d e r the problem which c o n s i s t s o f  minimizing  /jF(t,y(t),y(t))dt  s u b j e c t to the a r c y(*) s a t i s f y i n g v a r i o u s endpoint s h a l l be suppressed  i n our d i s c u s s i o n ) .  i s the c l a s s o f a b s o l u t e l y continuous the open subset ft o f R .  The  n  l o c a l l y L i p s c h i t z and y(«)  The  c o n s t r a i n t s (which  c l a s s o f admissable  f u n c t i o n s y:[0,l]->R  n  which l i e i n  i n t e g r a n d F(t,«,-) w i l l be assumed to be  Lebesgue measureable i n the t argument.  i s c a l l e d an extremal  f o r the i n t e g r a n d F p r o v i d e d  The  a r c £(•) which  C l a r k e [6]  We f(t,Y)  =  a.e.  for details).  s h a l l consider a C y > with  2  t r a n s f o r m a t i o n o f c o o r d i n a t e s f:R  (t,Y) r e p r e s e n t i n g the new  coordinates.  The  " t r a n s f o r m a t i o n o f c o o r d i n a t e s " i s i n t e n d e d t o imply t h a t the f y ( t , Y ) i s n o n s i n g u l a r everywhere so t h a t , l o c a l l y , we inverse  exists  satisfies  (|(t),C(t)) E 8F(t,y(t),y(t))  (see F.H.  arc  i t satisfies  the E u l e r - Lagrange d i f f e r e n t i a l i n c l u s i o n , t h a t i s , i f t h e r e an a b s o l u t e l y continuous  arcs  n n X R -+R , term  matrix  can o b t a i n  the  transformation  h(t,y) = Y .  We  w i l l a l s o assume t h a t the range o f the mapping f i n c l u d e s the open  subset ft  (49)  Q C range £ .  Then g i v e n an a r c y ( ' ) which l i e s i n fi, we d e f i n e the i n v e r s e image Y'(«) by the i m p l i c i t  equation  f(t,Y(t)) = y(t) .  W i t h i n the s e t tt, an e q u i v a l e n t r e p r e s e n t a t i o n c o u l d be g i v e n by t h e inverse  transformation  Y(t) = h ( t , y ( t ) ) .  We s h a l l have o c c a s i o n to c o n s i d e r an a p p r o p r i a t e mapping from the tangent  space  t o the tangent  space R .  We n o t i c e t h a t whenever  y  the d e r i v a t i v e Y ( t ) e x i s t s  f (t,Y(t)) + f (t,Y(t))oY(t) = y(t) t  y  and we d e f i n e the t r a n s f o r m a t i o n j : R X R  (4.2)  n  X R  n  -»• R  n  by  j(t,Y,V) = f ( t , Y ) + f ( t , Y ) o V . t  In equation o f l e n g t h n.  (4.2) f  y  Y  i s an nXn m a t r i x and f  and V a r e column v e c t o r s  F i n a l l y , we c o n s i d e r a l o c a l l y L i p s c h i t z  F ( t , y , v ) and the a s s o c i a t e d mapping  F*:R X R  n  X R  n  -> R  integrand d e f i n e d 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 i n t e g r a n d F ( t , y , v ) a r e i n one t o one  correspondence, v i a the change o f c o o r d i n a t e s f ( t , Y ) = y, w i t h the  (50)  extremals o f the i n t e g r a n d F*(t,Y,V) . Proof: I f we s e t  g(t,Y,V) = [ f ( t , y ) , j ( t , Y , V ) ]  = (y,v)  then g i s a t l e a s t c o n t i n u o u s l y d i f f e r e n t i a b l e ; whenever VF e x i s t s , VF* e x i s t s and  VF*(t,Y,V) = VF(t,y,v)oDg(t,Y,V)  where Dg i s the J a c o b i a n o f the t r a n s f o r m a t i o n g.  I t f o l l o w s 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 s e t  ft, t h e r e i s an a r c £(•) which s a t i s f i e s  (|(t),?(t)) G 9F(t,y(t),y(t))  From the i n c l u s i o n  (4.4)  Now  a.e. .  (4.3) we may deduce t h a t  (5(t),C(t))oDg(t,Y(t),Y(t))  G 9F*(t,Y(t),Y(t))  Dg(t,Y,V) i s a l i n e a r t r a n s f o r m a t i o n from R^  n  matrix representation  f (t,Y)  0  y  Dg(t,Y,V) j (t,Y,V) y  We may,  therefore, rewrite inclusion  f (t,Y) Y  (4.4) as  a.e..  i n t o R^  n  which has the  (51)  U(t)of (t,Y(t)) + 5(t)oJ (t,Y(t),Y(t))  (4.5)  Y  , ?(t)of (t,Y(t))]  Y  y  belongs t o 9 F * ( t , Y ( t ) , Y ( t ) )  From equation  .  (4.2), we may d e r i v e  j (t,Y,V) = f ( t , Y ) y  t Y  It follows that.the  + £ (t,Y)oV yY  .  i n c l u s i o n (4.5) may be w r i t t e n as  [ | f ( t , Y ) + 5°f (t,Y) + 5 o f ( t , Y ) Y 0  a.e.  y  tY  Y Y  , 5o.£ (t,Y)]  u  belongs t o 9F*(t,Y,Y)  To demonstrate t h a t Y(*) i s an extremal,  Y  a.e.  i t s u f f i c e s t o show t h a t  d_£(t) f ( t , Y ( t ) ) = 5°f (t,Y) + C o f ( t , Y ) + 5 o f ( t , Y ) o Y dt ' • y  Y  ty  y Y  .  For n o t a t i o n a l convenience, we s e t  G(t,Y,£) = ? o f ( t , Y ) Y  Then  dJ(t)of dt  (t,Y(t))  = G (t,Y, 5 ) + G (t,Y,£>Y + G (t,Y,£)u f ^ 1  .  1  We make the following observations:  G ( t , Y , 0 = Co£ (t,Y) = 5 o f C t , Y ) t  Yt  G (t,Y,g)= C o f ( t , Y ) y  yY  G (t,Y,£)(•) = G(t,Y,«)  since f i s C ; 2  t y  so that  G (t,Y,g)«Y = &»f (t,Y)oY ; Y  yY  which i s just the dual l i n e a r mapping  so that G ( t , Y , 5 ) o | = G(t,Y,|) = 5 o f ( t , Y ) c  y  .  [f (t,Y)] y  (52)  I t f o l l o w s from t h i s and our p r e v i o u s remark t h a t Y(«) i s an extremal trajectory. The extremal  c o n c l u s i o n o f the theorem f o l l o w s p r o v i d e d t h a t g i v e n an Y(*) f o r the i n t e g r a n d 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 i n t e g r a n d F ( t , Y , V ) .  T h i s demonstration  i s identical  to the one g i v e n when we use the i n v e r s e t r a n s f o r m a t i o n g ( t , y ) = Y i n s t e a d o f 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 H a m i l t o n i a n  (-P(t) , y ( t ) ) e 3 H ( t , y ( t ) , p ( t ) )  We are i n t e r e s t e d i n those  transformations  inclusion  .  o f 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 t u r n s a t i s f y a Hamiltonian  inclusion  (-P(t) , Y ( t ) ) G 9 H * ( t , Y ( t ) , P ( t ) )  f o r some H a m i l t o n i a n call  f u n c t i o n H*(t,Y,P).  F o r o u r p u r p o s e s , we s h a l l  the t r a n s f o r m a t i o n c a n o n i c a l i f t h i s c o n d i t i o n i s met.  general d i s c u s s i o n o f canonical transformations  F o r a more  in a classical  setting,  see Caratheodory [ l ] . An  important  c l a s s o f c a n o n i c a l t r a n s f o r m a t i o n s , used w i d e l y i n  c l a s s i c a l mechanics, a r e those  d e r i v a b l e from a g e n e r a t i n g f u n c t i o n .  2 We a r e g i v e n a C , r e a l v a l u e d f u n c t i o n i s t o be induced  (4.6)  <f>(t,y,Y).  through the f o l l o w i n g r e l a t i o n s :  p = cfy(t,y,Y)  The t r a n s f o r m a t i o n  (53)  (4.7) .  -P = <|> (t,y,Y) Y  (4.8)  H*(t,Y,P) = H(t,y,p) + <j> (t,y,Y) t  I t w i l l be assumed t h a t t h e m a t r i x (4.6)  may be s o l v e d  i s nonsingular  everywhere so t h a t  (locally) f o r Y  Y = f(t,y,p)  E q u a t i o n (4.7) then  .  .  implies  P = g(t,y,p)  and  the d e s i r e d transformation  Notice  i s [Y,P] = [ f ( t , y , p ) , g ( t , y , p ) ]  t h a t n o n s i n g u l a r i t y i m p l i e s t h a t e q u a t i o n (4.7) may be s o l v e d  l o c a l l y f o r y so t h a t t h e i n v e r s e t r a n s f o r m a t i o n s Equation  are r e a d i l y a v a i l a b l e .  (4.8) g i v e s us a r u l e f o r forming t h e new H a m i l t o n i a n .  H* can be o b t a i n e d  by a s t r a i g h t f o r w a r d  c a l c u l a t i o n , canonical  ions generated i n t h i s f a s h i o n a r e e s p e c i a l l y u s e f u l . c a l l e d a generating  f u n c t i o n f o r the canonical  transformat-.  The f u n c t i o n < j > is  transformation.  f o l l o w i n g Lemma w i l l be used t o show t h a t i n f a c t , such are  Since  The  transformations  canonical.  Lemma 1: 2 Suppose t h a t f o r some C functions  F(t,y,y)  f u n c t i o n cj>, the two l o c a l l y L i p s c h i t z  and G(t,y,v) a r e r e l a t e d as f o l l o w s :  F(t,y,v)  = G(t,y,v) + (j> (t,y) + <j> (t,y) v t  Then F and G generate t h e same e x t r e m a l s . Proof:  y  0  .  (54)  Clearly, i f VF exists then VG exists and conversely. VF = VG + (<|>+<|> o v , <j> ), t y  Furthermore,  and from the definition of generalized  gradient 9F(t,y,v) = 3G(t,y,v) + (*  (t,y) + *  ( t . y ) o v , <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) + (<|> + <fr o y , <jy) t y  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)) dt y  a.e. .  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). the following observations: VH(t,y,p)  VF(t,y,p,v,u)  We make  exists i f and only i f  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 p  y  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 , y , Y ) + <f> (t,y,Y)-v + 4> (t,y,Y).V t  y  = H*(t,Y,P)  Y  - H(t,y,p) + p-v - P-V .  F o r n o t a t i o n a l convenience, we s e t  * ( t , y , p ) = <t>(t,y,f(t,y,p))  and d e r i v e t h e f o l l o w i n g :  *t It follows  *  t  + *  =  *t  +  V t f  J *y = *y +  V y f  > ^p  =  V p' f  •  that  * v + ^ - u = <ft  + *  p  Y  '  f  t  +  v  = 4> + <l>'V + <f> -[ t  Y  +  v  Y  f t  (  +  v  f v  V  °v  v  V"u  +  + f ou] p  d>. + < j > -v + cb.,-V t y Y Y  T  = H*(t,Y,P)  r  - H(t,y,p) + p-v - P-V  .  T h e r e f o r e t h e c o n d i t i o n s o f Lemma 1 a r e met and we may conclude t h a t  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 t h e same e x t r e m a l s .  Q.E.D.  Example: 2 We denote a g e n e r i c element o f R set  {(x,y) : x<0} .  by ( x , y ) .  L e tftbe t h e open  D e f i n e t h e m u l t i f u n c t i o n E over [0,1] X ft by  (58)  E(t,x,y) = cb{(2x , x e  _ t  - |ln(-x)-t|)} U { ( o , - x e  _t  - | ln(-x)-t|) }  Figure 7 (0,-xe  _t  - |ln(-x)-t|)  (2x,xe  It is easily verified that the Hamiltonian for this multifunction is given by H(t x,y,p,q) = 2xp + xqe  - q[ln(-x)-t|  t  3  -xqe  - q|ln(-x)-t|  i f q+e^ < 0  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 ^ p j q ) by  (p>q)  and  = (  f l  » y) n  x  =  O >- ) u  v  ( ? , ? ) = (-n ,-fi ) = (x,y) u  v  (59)  The Hamiltonian, defined for a l 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 = cve  i f v+e < 0 .  + v|ln(-£)-t|  -t  i f v+e u > 0  + v|ln(-?)-t|  1  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 )v - e X  The new coordinates  (X,Y,P,Q)  t + X  u .  are related to (u,v,?,C) by  (C,E) = C« ,n ) = (-e ,-Y-e ) t+X  u  X  v  and (-P,-Q) = (ft ,ft ) = (-ve -ue ,-v) . X  x  t+X  Y  These relations imply that X = ln(-e) - t ; Y =  +e  _ t  ;Q= ;P= V  -ve ? - u? . _t  The new Hamiltonian H**, is calculated from equation ( 4 . 8 ) ; H**(t,x,Y,p,Q) = n + H(t,u,v,n ,n ) t  uc  u  - 2u£ - ?ve  1  uz + Cve + v|X 1  It follows, after simplifying, that  v  + v X|  i f v+e^i > 0  i f v+e u < 0 t  (60)  H**(t,X,Y,P,Q) = P + Q|X[ = -P + We notice that since  £ = -e  t+X  i f Pe  i f Pe"  Q|X|  > 0  X  X  < 0 .  , H** is defined on [0,1]  X R  2  2 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. conditions  The actual value of c depends on the boundary  (X(0),Y(0),Q(1),P(1)) which in turn are specified by  transforming the boundary conditions original system. constant.  (x(0),y(0),p(l),q(l)) for the  Theorem 1 from F.H. Clarke [ 1] implies that H** is  The trajectories are, therefore, easily identifiable.  system is analysed for the case (X,P) phase plane is given below.  Figure 8  c = -1  This  in F.H. Clarke [ 1] and the  (61)  As an example.to i l l u s t r a t e the i n v e r s e v e r i f i e d that the t r a j e c t o r i e s  t r a n s f o r m a t i o n s , i t may be  (X(t),Y(t),P(t),Q(t))  X ( t ) = - | t - h\ ; Y ( t ) = / Q | X ( S ) | d s  g i v e n by  ;  = h - t ; Q ( t ) = -1  P(t)  with boundary d a t a  ( X ( 0 ) , Y ( 0 ) ) = (-%,0) ; (P(1),Q(1)) =  {-h,-l)  transformed i n t o the t r a j e c t o r i e s  x(t) = - e  = -e  p ( t ) = (t-k)e ~ h  2  = -e~^ + ^ x e ~ 2  S  ~  t  2  -t - e "  h < t < 1  - |ln(-x(s))  S  -  + f£[-xe"  h < t < 1  2  2t  = (t-%)e~  y ( t ) = -e~^ + / J [ x ( s ) e "  0 < t < h  2 t _ J s  - s|]ds  i f0 < t < h  |ln(-x)-s|]ds  - |ln(-x)-s|]ds  S  q(t)  =1  i f% < t < 1 .  0 £ t <_ 1  w i t h boundary d a t a  ( x ( 0 ) , y ( 0 ) ) = (-e" ,-e" ) ; ( P ( l ) , q ( l ) ) = %  %  fte^-e ,1) -1  .  (62)  BIBLIOGRAPHY  C.Caratheodory: [ l] C a l c u l u s o f V a r i a t i o n s and P a r t i a l E q u a t i o n s o f t h e F i r s t Order, P a r t San F r a n c i s c o  1, Holden  Differential - Day,  (1965).  F.H.Clarke: [ l] Optimal C o n t r o l and the True H a m i l t o n i a n , t o appear. [ 2] G e n a r a l i z e d G r a d i e n t s and A p p l i c a t i o n s , T r a n s a c t i o n s Amer. Math. Soc. 205 (1975), 247 - 262. [ 3] G e n e r a l i z e d G r a d i e n t s o f L i p s c h i t z F u n c t i o n a l s , Advances i n Math., t o appear Center, Madison, [4]  Admissable  (Tech. Rep. #1687, Mathematics  Research  Wisconsin)  R e l a x a t i o n i n V a r i a t i o n a l and C o n t r o l  Problems,  J . Math. A n a l . A p p l . 51 (1975), 557 - 576. [ 5 ] N e c e s s a r y C o n d i t i o n s f o r a General C o n t r o l Problem, i n C a l c u l u s o f V a r i a t i o n s and C o n t r o l Theory, Mathematics  ( e d i t e d by D . L . R u s s e l ) ,  Research C e n t e r ( U n i v e r s i t y o f W i s c o n s i n - Madison)  Pub. No. 36, Academic P r e s s , N.Y. [6] The E u l e r - Lagrange  (1976).  Differential  Inclusion, J . Differential  E q u a t i o n s , 19 (1975), 80 - 90. R.Courant:  [ l ] C a l c u l u s o f V a r i a t i o n s , Courant I n s t i t u t e New U n i v e r s i t y , N.Y.  R.Courant  York  (1962).  and D . H i l b e r t : [ 1] Methods o f M a t h e m a t i c a l P h y s i c s , Volume 2, I n t e r s c i e n c e , N.Y.  (1962).  H.Federer: [ l] Geometric Measure Theory, S p r i n g e r - V e r l a g W.H.Flemming and R.W.Rishel:  (1969).  [ l] D e t e r m i n i s t i c and S t o c h a s t i c Optimal  Control, Springer - Verlag  (1975).  (63)  BIBLIOGRAPHY  I.M.Gelfand and S.V.Fomin: [ l ] C a l c u l u s R.Silverman), P r e h t i c e  of Variations,  ( t r a n s l a t e d by  - H a l l , Englewood C l i f f , N.J. (1963).  G. Lebourg: [ 1] Comptes Rendus de l'Academic des S c i e n c e s de P a r i s , November  10, (1975).  H. Rund: [ l] The Hamilton - J a c o b i  Theory i n the C a l c u l u s  Van Nostrand, London (1966).  of Variations,  

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