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Journal of Mathema tical Sciences, Vo l. 78, No. 5, 1996

G E N E R A L I Z E D S O L U T I O N S O F 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 S

O F T H E F I R S T O R D E R . T H E I N V A R I A N C E O F G R A P H S R E L A T I V E T OD I F F E R E N T I A L I N C L U S I O N S

A . I . S u b b o t i n U D C 5 1 7. 97 4 , 5 17 .9 78

1 . I n t r o d u c t i o n

T h e p r e s e n t p a p e r c o n t a i n s r e s u l ts o n t h e t h e o r y o f p a r ti a l d i f f er e n ti a l e q u a t i o n s ( P D E ) o f fi rs t o r d e r.

T h e g e n e r a l iz e d s o l u t i o n s o f t h e s e e q u a t i o n s , w h i c h a re d e f i n e d o n t h e b a s is o f c o n s t r u c t i o n s f r o m d i f fe r e n ti a l

g a m e th e o r y , a r e c o n s i d e r e d .

A s is k n o w n , i n m a t h e m a t i c a l c o n t r o l t h e o r y a n i m p o r t a n t p l a c e i s o c c u p i e d b y t h e i n v e s t i g a t i o n o f

p r o b l e m s , w h i c h r e q u i r e t h e c o n s t r u c t i o n o f f e e d b a c k c o n t r o l ac c o r d i n g t o w h i c h t h e w a r r a n t e d r e s u l t is

g u a r a n t e e d u n d e r m o s t u n f a v o r a b l e d i s t u r b a n c e s o r co u n t e r a c t i o n s . S u c h p r o b l e m s a r e i n c l u d e d i n t h e fi el d

o f d i ff e re n t ia l g a m e t h e o r y . D i f f er e n ti a l g a m e t h e o r y h a s b e e n w i d e l y s tu d i e d . A n i m p o r t a n t i n f l u e n c e o n t h ed e v e l o p m e n t o f t h i s t h e o r y w a s b r o u g h t a b o u t b y [ 1 , 3 , 8 , 9 , 1 1 - 1 5 , 3 4] ..

I n t h e i n v e s t i g a t i o n o f g a m e - c o n t r o l p r o b l e m s , r es u l ts a n d m e t h o d s f r o m v a r i o u s f i el d s o f m a t h e m a t i c s

a r e u s e d . M o r e o v e r , t h e n e w k n o w l e d g e o b t a i n e d i n d i f fe r e n ti a l g a m e t h e o r y m a y p r o v e t o b e u s e f u l w i t h i n t h e

f r a m e w o r k o f t h i s t h e o r y a n d i t s d i re c t a p p l i c at i o n s . F o r e x a m p l e , t h e c o n s t r u c t i o n s o f u - s t a b l e a n d v - s t a b le

f u n c t i o n s , d e v e l o p e d i n p o s i t i o n a l d i f f e re n t i a l g a m e t h e o r y [ 5 ], s e r v e a s a b a s i s f o r d e t e r m i n i n g t h e g e n e r a l i z e d

( m i n i m a x ) s o l u t i o n s o f p a r t i a l d i f f e r e n t ia l e q u a t i o n s o f f i rs t o r d e r . T h i s a p p r o a c h i s p r e s e n t e d i n t h e g i v e n

p a p e r .

I n S e c . 2 , t h e d e f i n i t i o n o f a m i n i m a x s o l u t i o n o f a 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 o f f i r st o r d e r

F ( x , u , D u ) = O , x 9 G C R "

i s g iven . A m i n i m a x s o l u t i o n o f th i s e q u a t i o n i s d e f in e d a s a c o n t i n u o u s f u n c t i o n u ( x ) ( x 9 C 1 G ) , t h e g r a p ho f w h i c h is w e a k l y i n v a r i a n t r e l a ti v e t o s o m e s y s t e m o f d i ff e r en t ia l e q u a t io n s ( t h i s s y s t e m m a y b e c o n s i d e r e d

a s a r e d u c t i o n o r r e l a x a t i o n o f t h e c h a r a c t e ri s t ic s y s t e m o f o r d i n a r y d i f fe r e n ti a l e q u a t i o n s ) . I n o t h e r w o r d s ,

o v e r t h e s e t

g r u = : 9 9 C l G }

t h r o u g h a n y p o i n t ( z 0 , Z o) 9 g r u p a s s t r a j e c t o r i e s o f t h e ~ c h a r a c t e r i s t ic " d i f f e r e n t ia l i n c l u s i o n . I n i n f i n i t e s i m a l

f o r m , t h i s d e f i n i t i o n i s e x p r e s s e d a s f o l l o w s : t h e i n t e r s e c t i o n

{ ( f , g ) G R " x R : g = ( f , p ) - F ( x , z , p ) } n T ( u ) ( z ) #

for a l l ( z , u ) 9 g r u a n d p 9 R " . H e r e T ( u ) ( z ) is a con t ingen t cone ( a Boul igand cone) o f t h e s e t g r u ,

c o n s t r u c t e d a t t h e p o i n t ( z , u ) E g r u . I n S e c . 2 , t h e c o m p a t i b i l i ty o f t h e n o t i o n s o f m i n i m a x a n d c la s s ic a ls o l u t io n s o f 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 s o f f i rs t o r d e r i s a ls o s h o w n .

I n S e c. 3, t h e n o t i o n s o f u p p e r a n d l o w e r s o l u t io n s a r e c o n s i d e re d . T h e y p l a y a n i m p o r t a n t r o l e i n th e

t h e o r y o f 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 s o f f i rs t o r d e r . A n upper ( lower) so lu t ion i s d e f i n e d a s a f u n c t i o n w h o s e

e p i g r a p h ( s u b g r a p h ) i s w e a k l y i n v a r i a n t r e l a t i v e to t h e c h a r a c t e r i s t i c d i f f e r e n t ia l i n c l u s i o n . A m i n i m a x s o l u t i o n

c a n b e d e f in e d a s a f u n c t i o n w h i c h p o s s e s se s s i m u l t a n e o u s l y t h e p r o p e r t i e s o f u p p e r a n d l o w e r s o lu t i o n s .

T h e o r i g in o f t h e d e f i n i t i o n s p r o p o s e d h e r e is r e la t e d w i t h t h e c o n s t r u c t i o n s o f p o s i t i o n a l d i f fe r e n ti a l

g a m e th e o r y . I t i s k n o w n ( s e e , e .g . , [ 5] ) t h a t t h e v a l u e f u n c t i o n o f a d i f f e r e n t ia l g a m e p o s s e s s e s t h e p r o p e r t i e s

T r a n s l a t e d f r o m I t o g i N a u k i i T e k h n i k i , S e r iy a S o v r e m e n n a y a M a t e m a t i k a i E e P r i l o z h e n i y a . T e m a t i c h e s k i e O b z o ry .

Vol . 13 , Dinamichesk ie S i s t emy-1 , 1994 .

594 1072-3374 /96/7805-059451 5.00 9 Plenu m Publ ishing Corpora tion



o f u - s t a b il i ty a n d v - s t a b i l it y w h i c h r e p r e s e n t t h e o p t i m a l i t y p r in c i p le . T h e s e p r o p e r t i e s c a n b e e x p r e s s e d i n

v a r io u s f o r m s , i n c l u d i n g t h e i r e x p r e s s i o n s a s a p a i r o f i n e q u a l it i e s f o r d i r e c t io n a l d e r i v a t i v e s , w h i c h s a t i sf ie s

o n e a n d o n l y o n e g e n e r a l iz e d ( m i n i m a x ) s o l u t i o n o f t h e C a u c h y p r o b l e m f o r t h e I s a a c s - B e l l m a n e q u a t i o n .

T h i s d e f in i ti o n w a s i n t r o d u c e d i n [1 6, 1 9 ] . T h e p r o p o s e d a p p r o a c h c a n b e u s e d f o r s t u d y i n g a w i d e r a n g e o f

b o u n d a r y - v a l u e p r o b l e m s a n d C a u c h y p r o b l e m s f or v a ri o u s t y p e s o f p a r t i a l d i f f e re n t ia l e q u a t i o n s o f fi rs t o r d e r

[ 17 ]. I n S e c . 3 , t h e c o i n c i d e n c e o f t h e n o t i o n o f a u - s t a b l e ( v - s t a b le ) f u n c t i o n w i t h t h a t o f a n u p p e r ( lo w e r )

s o l u ti o n o f t h e H a m i l t o n - J a c o b i - B e l l m a n - I s a a c s e q u a t i o n is p r o v e d .S t a r t i n g f r o m t h e e a r l y 1 9 8 0 s , t h e i n v e s t i g a t i o n s o f g e n e r a l i z e d s o l u t io n s o f 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 s

o f f ir st o r d e r , w h i c h h a v e b e e n i n t e n s iv e l y ca r r ie d o u t b y f o r e ig n m a t h e m a t i c i a n s , w e r e b a s e d o n t h e n o t i o n

o f v i s c o s i ty s o l u t i o n , i n t r o d u c e d b y M . G . C r a n d a l l a n d P . L . L i o n s - [3 0 , 3 1] . I t is i m p o r t a n t t o n o t e t h a t ,

d i f f e r e n t b y t h e i r f o r m a n d o r i g i n , t h e n o t i o n s o f m i n i m a x a n d v i s c o u s s o l u t i o n s i n t h e f i n a l a n a l y s i s p r o v e t o

b e e q u i v a l e n t . T h i s r e s u l t i s f o r m u l a t e d i n S e c . 3 ,

T h e m a t e r i a l i n S e c s . 2 a n d 3 is o f a r e v i e w c h a r a c t e r . I n th e s e s e c t i o n s , t h e f u n d a m e n t a l n o t i o n s a r e

g i v e n , t h e r e s u l t s o b t a i n e d a r e f o r m u l a t e d , a n d c e r t a i n c l a r if i c a ti o n s a r e p r e s e n t e d . A l l p r o o f s , a s a r u l e , a r e

o m i t t e d .

I n S e c. 4 , t h e b o u n d a r y - v a l u e p r o b l e m o f D i r i c h l e t t y p e f o r t h e 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 o f f i rs t o r d e r

H ( z , D u ) - u = O,

w i t h th e b o u n d a r y c o n d i t io n

u ( z ) = ~ ( z ) ,

w h e r e O G is a b o u n d a r y o f t h e d o m a i n G , is co n s i d e re d .

x E G c R ~'

x E O G ,

T h e p e c u l i a r i t y o f t h i s p r o b l e m c o n s i s t s i n t h e

f a c t t h a t i t i s n e c e s s a r y t o d e f i n e t h e n o t i o n o f a d i s c o n t i n u o u s s o l u t i o n f o r i t. T h i s d e f i n i t i o n i s g ! v e n i n

S e c. 4 . H e r e t h e e x i s te n c e a n d u n i q u e n e s s t h e o r e m o f a m i n i m a x s o l u t io n o f t h e b o u n d a r y - v a l u e p r o b l e m i s

a ls o f o r m u l a t e d . T h e m e a n i n g f u l n e s s o f t h e p r o p o s e d d e f i n i ti o n i s s h o w n b y t h e e x a m p l e o f a t i m e - o p t i m a l

c o n t ro l p r o b l e m . I t is k n o w n t h a t t h e o p t i m a l t i m e a s a f lm c t i o n o f t h e i n i ti a l s t a te o f t h e c o n t r o l s y s t e m m a y

b e d i s c o n ti n u o u s . I t i s s h o w n t h a t t h i s fu n c t i o n ( a f te r s o m e s im p l e t ra n s f o r m a t i o n ) c o i n c id e s w i t h a m i n i m a x

s o l u ti o n o f t h e c o r r e s p o n d i n g b o u n d a r y - v a l u e p r o b l e m .

T o c o n c l u d e t h e p a p e r , i n S e c. 5 t h e p r o o f o f t h e u n i q u e n e s s o f a m i n i m a x s o l u t i o n f o r a D i r i c h l e t- t y p e

b o u n d a r y - v a l u e p r o b l e m i s g iv e n . L e t u s n o t e t h a t i n th i s p r o o f t h e m e t h o d o f L y a p u n o v f u n c t i o n s i s u s e d .

2 . C l a s s i c a l a n d G e n e r a l i z e d C h a r a c t e r i s t i c s

2 . 1 . C o n s i d e r t h e 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 o f f i rs t o r d e r

F ( x , u , D u ) = 0, z e G C 1r (2.1)

H e r e D u : = au aua;7~'"" 9 a -~ ,) is t h e g r a d i e n t o f t h e f u n c t i o n u a n d G i s a n o p e n s e t i n R ~ .

O n e o f t h e f u n d a m e n t a l r e s u l t s o f t h e c l a s si c al th e o r y o f p a r t i a l d i f f e re n t i a l e q u a t i o n s o f f ir s t o r d e r c o n s i s ts

i n t h e f a c t t h a t , u n d e r c e r t a i n c o n d i t i o n s , t h e s o l u t i o n o f E q . ( 2 .1 ) ( t o b e m o r e p r e c i s e , t h e s o l u t i o n o f t h e

C a u c h y p r o b l e m f o r t h i s e q u a t i o n ) i s r e d u c e d t o t h e i n t e g r a t i o n o f t h e c harac t e r i s t i c s y s t e m o f o r d i n a r yd i f f e re n t i a l e q u a t i o n s

{ ~ = D p F ( x , z , p ) ,

[a = - D ~ F ( x , z , p ) - p D z F ( x , z , p ) , (2 .2 )

= (p , D v F ( x , z , p ) ) .

H e r eO F O F ~, O_FF

a n in n e r p r o d u c t o f tw o v e c t o r s p a n d f is d e n o t e d b y t h e s y m b o l ( p , f ) . N o t e t h a t t h e f u n c t i o n F i s a n

i n t e g r a l o f t h e s y s t e m ( 2 . 2) , t h a t i s, i t a s s u m e s a c o n s t a n t v a l u e a l o n g a n y o f t h e s o l u t i o n s o f t h e s y s t e m ( 2 .2 ) .

595



M o r e o v er , f o r th e s o l u t io n s o f th i s s y s t e m , u s e d i n th e m e t h o d o f c h a r a c te r i st i c s, t h e f u n c t i o n F a s s u m e s a

z e ro v a l u e. T h e r e f o r e , t h e t h i r d e q u a t i o n o f t h e s y s t e m ( 2 .2 ) ( i t i s c a l l ed t h e characterist ic band equation) is

e q u i v a l e n t t o t h e e q u a t i o n

= ( ~ , p ) - F ( z , z , p ) . (2 .3 )

T h i s t y p e o f n o t a t i o n i s n ec e s s a r y f o r th e d e f i n it io n o f a g e n e r a li z e d ( m i n i m a x ) s o l u t i o n , g i v e n i n S e c .

2 .4 . T h e c o n s t r u c t i o n u s e d i n t h i s d e f i n i t io n m a y b e c o n s i d e r e d a s a r e d u c t i o n o r r e l a x a t i o n o f t h e c la s si ca l

m e t h o d o f c h a r a c t e r i s t i c s .

2.2.

n o n s m o o t h a n a l y s i s a r e u s e d . L e t u s re c a l l t h e s e n o t i o n s .

L e t W b e a c l o s e d s e t i n t h e s p a c e l~ ~ . L e t

d i s t ( y ; W ) : = m i n ~ e w [ [ y - w l] .

T h e s e t

I n t h e d e f i n i t io n o f a m i n i m a x s o l u t i o n , s o m e n o t i o n s fr o m t h e t h e o r y o f d i f fe r e n t ia l i n c l u s io n s a n d

T ( w ; W ) : = { h E R " : l i m i n f d i s t ( w + g h ; w ) = 0 } ( 2 . 4 )sW g

i s c a l l e d t h e contingent cone ( t h e Boul igand cone, t h e tangent cone; s o m e o t h e r n a m e s f o r t h i s s e t a r e a ls o

u s e d ) . I f • > 0 a n d h q T ( w ; W ) , t h e n i t i s n o t d i f f i c u lt t o s e e t h a t ,~h E T (w; W ) , t h a t i s , t h e s e t T ( w ; W )i s r e a ll y a c o n e . O n e c a n s h o w t h a t i t is c lo s e d .

T h e s e t W C R " i s c a l le d weakly inv ariant r e l a t i v e t o t h e d i f f e re n t i a l i n c l u s i o n

~](t) e E(y(t)) C R " ( 2 . 5 )

i f, f o r a n y p o i n t Y0 E W , t h e r e e x i s t s a s o l u t i o n o f t h e d i f f er e n t ia l i n c l u s i o n ( 2 . 5 ) w i t h y ( 0 ) = y o s u c h t h a t

V(t) E W f o r a l l t > 0 . I f t h i s c o n d i t i o n h o l d s , o n e a l s o s a y s t h a t t h e s e t W p o s s e s s e s t h e viability property.

I n v a ri a n t s e t s a n d t h e c o n t r o l p r o b l e m s a s s o c i a t e d w i t h t h e m h a v e b e e n i n v e s t i g a t e d b y m a n y a u t h o r s ( se e,

e . g . , [ 7 , 2 1 , 2 7 , 2 8 , 3 6 , 4 1 ] , a n d t h e r e f e r e n c e s t h e r e i n ) .

W e s h a l l a s s u m e t h a t t h e s e t s E ( y ) a r e c o n v e x a n d c lo s e d f o r a ll y E R = , t h e m u l t i v a l u e d m a p p i n g

y ~ E(y ) i s u p p e r s e m i c o n t i n u o u s w i t h r e s p e c t t o a n i n c l u si o n a n d s a t is f ie s t h e c o n d i t i o n

m ax I lh l l < (1 + I lYl l)# .heS(~) --

I t i s k n o w n ( s e e [ 36 ]) t h a t u n d e r t h e s e a s s u m p t i o n s t h e c l o s e d s e t W i s w e a k l y i n v a r i a n t r e l a t i v e t o th e

d i f fe r e n ti a l i n c l u s i o n ( 2 .5 ) i ff t h e c o n d i t i o n

T ( w ; W ) f3 E ( w ) # z , V w e W (2 .6 )

h o l d s .

2 . 3 . L e t u s p o i n t o u t t h e b a s ic a s s u m p t i o n s r e g a r d i n g t h e f u n c t i o n F , u n d e r w h i c h a m i n i m a x s o l u t io n

o f E q . ( 2 . 1 ) w i l l b e d e f i n e d .

W e s h a l l a s s u m e t h a t t h e f u n c t i o n ( z , z , p ) ~ F ( z , z , p ) i s c o n t i n u o u s , n o n i n c r e a s i n g i n t h e v a r i a b l e z ,t h a t i s ;

F( z , Z l , p ) >_ F ( z , z 2 , p ) , V ( z , Z l , z 2 , p ) e C x R x g x x ~ , Z l < z 2 ,

a n d s a t i s f i e s t h e c o n d i t i o n s

I F ( z , z , 0 )1 < ( 1 + I lz ll + I z l ) ~ , I F ( z , z , p ) - F ( z , z, q ) l < l i p - q l l p ( z ) , ( 2 .7 )

V (z , z , p ,q ) E G I t R ~' R ~,

w h e r e p ( z ) : = ( 1 + I lz l l) # , ~ i s a p o s i t i v e n u m b e r .

T h e a b o v e c o n d i t i o n s m a k e i t p o s s i b le t o s i m p l if y t h e d e f i ni ti o n ; w i t h o u t t h e m t h e c o n s t r u c t i o n o f a

m i n i m a x s o l u t i o n m u s t c o n t a i n s o m e a d d i t i o n a l a p p r o x i m a t e e l em e n t s .

59 6



2 . 4 . L e t u s p a s s t o t h e d i r ec t d e f i n it i o n o f a mi n i m ax s o lu t i o n . Co n s i d e r t h e s y s t e m w h i ch co n s is t s o f

t h e d i f f e r en t i a l i n eq u a l i t y an d t h e d i f f e r en t ia l eq u a t i o n

I1~11 p(x ), ~ = (~:,p) - F ( z , z , p ) . (2.8)

H e r e p is s o m e v e c t o r f r o m R " . N o t e t h a t t h e d i f f e re n t ia l e q u a t i o n f o r t h e v a r i a b l e z c o i n c id e s w i t h t h e

ch a rac t e r i s t i c b a n d eq u a t i o n . Sy s t e m (2. 8) c an b e r ew r i t t en i n t h e " fo rm o f t h e d i f f e r en t i a l ( ch a r ac t e r i s t i c )

inc lus ion( & ( t ), /, .( t) ) e E ( x ( t ) , z ( t ) , p ) , (2.9)

w h e r e

E ( x , z , p ) : = { ( f , g ) e R " x R : I lfll -< p ( x ) , g = ( f , p ) - F ( x , z , p ) } , " ( x , z , p ) E G x R x R n. (2 .10)

Let us def ine th e so lu t io n o f Eq . (2 .9 ). A ssum e R + := [0, c r Let (x ( . ) , z ( . ) ) : 1r + ~ I r x l~ be a con t in uou s

fu n c t i o n , ( X T ( . ) , Z T ( ' ) ) be a r es t r i c t ion to the c losed in terva l T : = [ t l , 12] C R + o f th i s funct ion . A c on t in uou s

fu n c t i o n (x ( - ) , z ( - ) ) , s a t i s fy i n g t h e co n d i t i o n : i f

{ ( x ( t ) , y ( t ) ) : t e T } C G ,

t h e n t h e f u n c t i o n ( X T ( ' ) , Z T ( ' )) i s ab s o l u t e l y co n t i n u o u s an d , f o r a l mo s t a l l t e T , t h e i n c l u s i o n (2 . 9 ) h o l d s ,

wi l l be ca l l ed a s o l u t i o n o f t h e s y s t em (2 . 1 ) .

By t h e s y m b o l In v (2 .9 )(P) w e s h a ll d en o t e t h e f am i l y o f c lo s ed s e ts W C c l G x R w eak l y i n v a r i an t r e l a t i v e

to th e d i f f eren t i a l inc lus ion (2 .9 ) fo r f ixed p E R ~.

D e f i n i t i o n . A m i n i m a x s o l u t i o n of Eq . (2 .1 ) i s a con t inuo us func t ion u : c l G - -~ R such t ha t

gru E A Inv(2.s)(p), (2.11)

peR"

w h e r e g r u : = { ( x , u ( x ) ) : x e c l G } is t h e g r a p h o f t h e f u n c t i o n u .

T h u s , t h e c o n t i n u o u s f u n c t i o n u i s c a l l e d t h e m i n i m a x S o l u t i o n of Eq . (2 .1 ) i f, fo r an y p E R ~ and

x 0 E c l G , t h e r e ex i s t s a s o l u t i o n ( a g e n e r a l i z e d c h a r a c t e r i s t i c ) o f t h e d i f f e r en t i a l i n c l u s i o n (2 . 9) s u ch t h a t

z (O ) = x o , z ( t ) = u ( x ( t ) ) fo r a ll t > 0 . O n e can a ls o s ay t h a t t h e m i n i m a x s o l u t i o n i s t h e f u n c t i o n u , t h e

g rap h o f w h i ch co n t a i n s v i ab l e t r a j ec t o r i e s o f t h e d i f f e r en t i a l i n c l u s i o n (2. 9 ) u n d e r a n a r b i t r a ry ch o i ce o f t h e

p a r a m e t e r p.

2 . 5 . L e t u s v e r i fy t h e co m p a t i b i l i t y o f t h e d e f i n i ti o n o f a mi n i m ax s o l u t i o n w i t h t h e n o t i o n o f a c l a s si ca l

s o l u t i o n . N o t e f i r s t t h a t

E ( x , z , p ) A E ( x , z , q ) # z , V ( x , z , p , q ) 6 G x R x R '~ x • " .

In f ac t , f o r p # q , t h i s i n t e r s ec t i o n co n t a i n s an e l eme n t ( f . , g . ) o f t h e fo r m

( 2 . 1 2 )

f . : = [ F ( x , z , p ) - F ( x , z , q ) ] ( p - ' q )

l i p - q l l 2

g . : = ( f . , p ) - F ( x , z , p ) = ( f . , q ) - F ( x , z , q ) .

L e t u b e a c l a s s i ca l so l u t i o n o f E q . ( 2 .1 ) , t h a t is , t h e fu n c t i o n u i s co n t i n u o u s l y d i f f e r en t i ab l e an d s a t is f ie s E q .

(2.1 ). L e t u s s h o w t h a t t h i s f u n c t i o n is a mi n i ma x s o l u t io n o f t h i s eq u a t i o n . L e t u s ch o o s e an y (x 0 , z0) E g ru ,

p E Ir " . Us in g (2 .12) , i t i s no t d i f f i cu l t to ver i fy tha t the re ex i s t s a so lu t io n ( x ( t ) , z ( t ) ) o f t h e d i f f e r en t i a l

inc lus ion

(~ ,1 ;) E E ( z , u ( x ) , D u ( x ) ) A E ( x , u ( z ) , p ) , (2.13)

597



D e f i n i t i o n .

u : c l G ---* R , s a t i s f y i n g t h e f o l lo w i n g c o n d i t i o n ( 3 . 1 ) ( c o n d i t i o n ( 3 .2 ) ):

e p i u E N I n v ( 2 .9 ) ( p ) ,p E R n

h y p o u E ["1 Inv(2 .D)(p),

pER ~

W e s h a l l d e f i n e a n u p p e r ( l o w e r ) s o l u t i o n o f E q . ( 2 .1 ) a s a lo w e r (u p p e r ) s e m i c o n t i n u o u s f u n c t i o n

(3 .1 )

(3 .2 )

w h e r e e p i u : = { ( x , z ) E c l G x • : z > u ( x ) } a n d h y p o u : = { ( z , z ) E c l G x R : z < u ( x ) } a r e t h e e p i g r a p h

a n d s u b g r a p h o f t h e f u n c t i o n u r e s p e c t i v e l y .

T h u s , t h e l o w e r s e m i c o n t i n u o u s f u n c t i o n u is c a ll e d t h e u p p e r s o l u t i o n i f , f o r a n y p G R ~ , x 0 E c l G , a n d

zo >_ u (x o ) , t h e r e e x i s t s a s o l u t i o n ( z ( t ) , z ( t ) ) o f t h e d i f f e r e n t ia l i n c l u s i o n (2 . 9) s u c h t h a t ( x ( 0 ) , z ( 0 ) ) = (Zo, Zo) ,

z ( t ) > u ( x ( t ) ) f o r a ll t > 0 . T h e e x p l a n a t i o n o f t h e d e f i n i t i o n o f a lo w e r s o l u t i o n is c o m p l e t e l y a n a l o g o u s .

N o t e t h a t t h e m i n i m a x s o l u t i o n d e f i n e d i n S e c . 2 . 4 s a ti s fi e s c o n d i t i o n s ( 3 .1 ) a n d ( 3 .2 ) . U s i n g t h e

c o n n e c t e d n e s s p r o p e r t y o f t h e s e t o f t r a j e c t o r i e s o f t h e d i f f e re n t i a l i n c l u s i o n , i t is n o t d i f f ic u l t t o g e t t h e

c o n v e r s e s t a t e m e n t : i f t h e c o n t i n u o u s f u n c t i o n u s a t is f i es c o n d i t i o n s (3 . 1) a n d ( 3 . 2 ), t h e n t h i s f u n c t i o n i s

a m i n i m a x s o l u t i o n o f E q . ( 2. 1) . T h u s , a m i n i m a x s o l u t i o n ca n b e d e f i n ed as a f u n c t i o n w h i c h p o ss e s s es

s i m u l t a n e o u s l y t h e p r o p e r t i e s o f lo w e r a n d u p p e r s o l u t io n s .

3 . 2 . S o m e m o d i f i c a t i o n s o f t h e d e f i n it i o n s o f u p p e r a n d l o w e r s o l u t i o n s a re p o s s i b l e . F o r e x a m p l e , i n t h e

d e f in i ti o n s o f u p p e r a n d l o w e r m i n i m a x s o l u ti o n s , i n s t e a d o f t h e m u l t iv a l u e d m a p p i n g E o f t h e f o r m ( 2 .1 0 ),

o n e c a n u s e t h e m a p p i n g s ,

( x , z , q ) ~ U ( x , z , q ) , ( x , z , p ) ~ V ( x , z , p ) , ( x , z , q , p ) e I ~ ~ x R x Q x P ,

u p p e r s e m i c o n t i n u o u s w i t h r e s p e c t t o a n i n c lu s i o n , s u c h t h a t U ( x , z , q ) a n d V ( x , z , p ) b o t h a r e c o n v e x s u b s e t s

i n W ' x R , s a t i s f y i n g t h e f o l l o w i n g c o n d i ti o n s :

( ( f , g ) e g ( x , z l , q ) , g l ~_~ z 2 ) ~ { ( f , g + r ) : r ~ > 0 } [ ..J V ( ~ , z2 , q ) # 0 ,

(3 .3 )( ( I , g ) e v ( ~ , z 2 , p ) , ~1 < ~ ] ~ { ( f , g - r ) : r > O } n V ( ~ , ~ l , p ) # ~ ,

s u p m i n [ ( s , / ) - g ] = F ( ~ , z , s ) = i n f m a x [ ( f , s l - g ] , V ( ~ , z , s ) e C • 2 1 5 ~,qeQ f , g ) E U ( . ~ , z , q ) pEP(f,g)eV(x,z,p)

m a x { l l f l l + I g l : ( f , g ) e g ( x , z , q ) u y ( x , z , p ) } < (1 -I -I I~ l l + I~ l ) r (q ,p ) ,

w h e r e r ( q , p ) i s s o m e p o s i t i v e n u m b e r . N o t e t h a t t h e m u l t i v a l u e d m a p p i n g E ( 2. 10 ) s a t is f ie s a ll t h e s e c o n d i -

t i o n s .

I n s t e a d o f t h e d i f f e r e n ti a l -i n c l u s i o n ( 2 . 9 ) , l e t u s c o n s i d e r t w o d i f f e r e n ti a l i n c lu s i o n s :

( ~ , ~ ) e u ( ~ , ~ , q ) , ( 3 . 4 )

( ~ , ~ ) e y ( ~ , z , p ) . ( 3 .5 )O n e c a n s h o w t h a t c o n d i t i o n s ( 3 . 1 ) a n d ( 3 . 2 ) a r e e q u i v a l e n t t o t h e c o n d i t i o n s

e p i u G A I n v ( a . 4 ) (q ) , ( 3 . 6 )qe Q

h y p o u e N I n v ( a .s ) ( p ) ( 3 .7 )p E P

r e s p e c t i v e l y .

C o n d i t i o n ( 3 .6 ) m e a n s t h a t , f o r a n y p E P , x 0 E cl G , a n d Zo >_ U(X o), t h e r e e x i s t s a s o l u t i o n ( x ( t ) , z ( t ) )

o f t h e d i f fe r e n ti a l i n c l u s io n ( 3 . 4 ) s u c h t h a t ( x ( 0 ) , z ( 0 ) ) = ( X o , Z o ) , z ( t ) > u ( z ( t ) ) f o r a l l t _> 0 . N ot e t ha t ,

599



i n s t e a d o f t h e i n i t i a l v a l u e s z 0 _> u ( z 0 ) , i t is s u f f ic i e n t to c o n s i d e r t h e v a l u e s Zo = U(Xo) . S u c h a m o d i f i c a t i o n

t a k e s p l a c e a l s o f o r c o n d i t i o n ( 3 . 7 ) .

3 . 3 . T h e a p p r o a c h p r o p o s e d h e r e i s t h e d e v e l o p m e n t o f t h e c o n s t r u c t i o n e la b o r a t e d i n t h e t h e o r y o f

p o s i t i o n a l d if f e r e n t i a l g a m e s [ 5 ] . L e t u s sh o w t h a t f o r t h e B e l l m a n - I s s a c s e q u a t i o n , c o n d i t i o n s ( 3 . 6 ) a n d ( 3 .7 )

c o i n c i d e w i t h t h e u - s t a b i l i t y a n d v - s t a b i l i ty o f t h e v a l u e f u n c t i o n o f t h e d i f f e r e n ti a l g a m e .

C o n s i d e r t h e d i ff e r en t ia l g a m e i n w h i c h t h e m o t i o n o f t h e c o n t r o l s y s t e m i s d e s c r i b e d b y t h e o r d i n a r y

d i f fe r e n t ia l e q u a t i o n

7 ) (t ) = h ( t , y ( t ) , p ( t ) , q ( t ) ) , t o <_ t <_ 0 , y ( t o ) = Y o e R m . ( 3 . 8 )

H e r e p ( t ) E P a n d q ( t ) E Q a r e c o n t r o l s o f t h e f i rs t a n d o f t h e s e c o n d p l a y e r s r e s p e c t i v e ly , P a n d Q a r e

t w o c o m p a c t s e t s . T h e s e c o n t r o l s ar e c h o s e n a c c o r d i n g t o t h e f e e d b a c k l a w , i. e. , w i t h r e s p e c t t o t h e c u r r e n t

p o s i t i o n ( t , y ( t ) ) . T h e f i rs t p l a y e r se e k s t o g u a r a n t e e a m i n i m i z a t i o n o f t h e v a l u e o f t h e p e r f o r m a n c e i n d e x

0

7 ( Y ( ' ) , P ( ' ) , q ( ' ) ) : = a ( y ( O ) ) - / g ( t , y ( t ) , p ( t ) , q ( t ) ) d t .

to

( 3 . 9 )

T h e s e c o n d p l a y e r , o n t h e c o n t r a ry , s e e k s t o g u a r a n t e e a m a x i m i z a t i o n o f t h e v a l u e o f t h is f u n c t i o n a l .

W e s h a l l a s s u m e t h a t t h e f u n c t i o n s h a n d g b o t h a r e c o n t i n u o u s o n [0 , 0 ] x 1 ~" x P x Q a n d s a t i s f y t h e

L i p s c h i t z c o n d i t i o n w i t h r e s p e c t t o t h e v a r i a b l e y , a n d t h e f u n c t i o n ~ r is. c o n t i n u o u s . I t i s a l s o a s s u m e d t h a t

m i n m a x [ ( s , h ( t , y ,p , q ) ) - g ( t , y , p , q ) ] = m a x m i n [ ( s , h ( t , y , p , q ) ) - g ( t , y , p , q ) l := H ( t , y , s ) , (3 .10)pEP qEQ qEQ pE P

( t , y , s ) c [ 0 , 0 ] x R ~ x R " .

I n t h is c a s e , t h e o p t i m a l g u a r a n t e e d r e s u lt s o f t h e f i rs t a n d s e c o n d p l a y e r s c o in c i d e; t h e i r c o m m o n v a l u e i s

c a l l e d t h e v a l u e o f t h e p o s i t io n a l d i f f er e n ti a l g a m e . T h e v a l u e o f t h e g a m e d e p e n d s o n t h e i n i t ia l p o s i t i o n .

T h e r e f o r e , o n e c a n d e f i n e t h e v a l u e f u n c t i o n ( t o , y o ) ~ u ( t o , y0 ) : [0 , 0 ] x R ~ -* R .

I t is k n o w n t h a t t h e v a l u e f u n c t i o n sa ti sf ie s t h e H a m i l t o n - J a c o b i - B e l l m a n - I s a a c s e q u a t i o n

O u0--[ + H ( t , z , D = u ) = 0 ( 3 . t l )

a t t h o s e p o i n t s w h e r e i t i s d i f f e r e n t i a b l e .

I t is a ls o k n o w n t h a t t h e r e e x i st s o n e a n d o n l y o n e c o n t in u o u s " f u n c t i o n w h i c h s a ti sf ie s t h e b o u n d a r y

c o n d i t i o n u ( 8 , y ) = a ( y ) a n d p o s s e ss e s t h e u - s ta b i l it y a n d v - s t a b i l it y p r o p e r t i es . T h i s f u n c t i o n i s th e v a l u e

f u n c t i o n o f t h e d i f f e r e n t ia l g a m e o f t h e f o r m ( 3 .8 ) , ( 3 . 9 ).

L e t u s r e c a ll t h e d e f i n i t i o n o f u - s t a b i l i t y f o r t h e d i f f e r e n t i a l g a m e u n d e r c o n s i d e r a t i o n . A f u n c t i o n u ( t , y )

i s c a l l ed u - s t a b l e i f , f o r a n y ( t 0 , y 0 ) E [ 0 , 0 ] x R ~ a n d q E Q , t h e r e e x i s t s a s o l u t i o n o f t h e d i f f e r e n t i a l i n c l u s i o n

( i / ( t ) , ~ ( t ) ) e c o { ( h ( t , y ( t ) , p , q ) , g ( t , y ( t ) , p , q ) ) : p e P } , t o < t < O , y ( t o ) = y o , z ( t o ) = 0 ,

s u c h t h a t t h e i n e q u a l i t y

~ ( t , y ( t ) ) < ~ ( t o , y o ) + z ( t )

h o l d s f o r all t E [to, 0].

T h e v - s t a b i l i t y p r o p e r t y i s d e f i n e d i n a s i m i l a r w a y . L e t u s s h o w t h a t d e f i n i t i o n s o f u - s t a b i l i t y a n d o f t h e

u p p e r s o l u t i o n o f E q . ( 3 .1 1 ) c o i n c i d e . I t is c l e a r t h a t E q . ( 3 .1 1 ) m a y b e c o n s i d e r e d a s a p a r t i c u l a r c a s e o f

E q . ( 2 . 1 ) , i n w h i c h

= m + 1 , 9 = ( t , y ) , p = ( s o , s ) e R • R ~ ,

F C ~ , ~ , p ) = s o + H C t , ~ , ~ )

600



( n o t e t h a t h e r e o n e m a y n o t r e q u i r e th a t t h e f u n c t i o n F s a t is f y t h e L i p sc h i tz c o n d i t i o n w i t h r e s p e c t t o t h e

v a r ia b l e t) . L e t u s d e f i ne t h e m u l t i v a l u e d m a p p i n g

U ( t , y , q ) : = c o { ( 1 , h ( t , y , p , q ) , g ( t , y , p , q ) ) : p E P } c R • 2 1 5 ( t, y , q ) e [ o , a ] • 2 1 5 (3 .12)

N o t e t h a t i t s a ti s f ie s t h e c o n d i t i o n ( 3 . 3 ) , w h i c h , in t h i s c a s e , h a s t h e f o r m

m a x r a i n { S o + ( s , h ) - g : ( 1 , h , g ) 9 U ( x , z , q ) } = m a x m i n [ s o + ( s , h ( t , y , p , q ) ) - g ( t , y , p , q ) ] = s o + H ( t , y , s ) .qEQ qEQ pEP

T a k i n g i n to a c c o u n t t h e r e m a r k g i v e n a t t h e e n d o f S ec . 3 . 2 , w e g e t t h a t t h e u - s t a b i l i ty c o n d i t i o n m e a n s , i n

e s s e n c e , t h e w e a k i n v a r i a n c e o f t h e e p i g r a p h o f t h e f u n c t i o n u ( t , x ) r e la t i v e to t h e d i f f e r e n t i a l i n c l u s i o n

( i , y , k ) 9 U(3.x2)(t , y, q)

f o r e a c h q 9 Q . T h u s , t h e n o t i o n s o f a u - s ta b l e f u n c t i o n a n d a n u p p e r s o l u t i o n a re e q u i v a l e n t. A n a l o g o u s l y ,

o n e c a n s h o w t h e e q u i v a l e n c e o f t h e n o t i o n s o f a v - s t a b l e f u n c t i o n a n d a l o w e r s o l u t i o n . T h e r e f o r e , t h e v a l u e

f u n c t i o n o f t h e d i f fe r e n ti a l g a m e ( 3 . 8 ), ( 3 .9 ) c o i n c id e s w i t h t h e m i n i m a x s o l u t i o n o f t h e C a u c h y p r o b l e m f o r

E q . ( 3 .1 1 ) ( t h e b o u n d a r y c o n d i t i o n i n t h i s p r o b l e m : u ( 0 , y ) = o ' (y ) ) .

3 . 4 . L e t u s c o n t i n u e w i t h t h e c o n s i d e r a t io n o f t h e u p p e r a n d l o w e r s o l u t io n s o f E q . ( 2 .1 ) . T h e w e a ki n v a r i a n c y p r o p e r t y c a n b e e x p r e s s e d i n d i f f e r e n t w a y s , i n c l u d i n g t h a t i n t h e f o r m o f i n e q u a l i t i e s f o r t h e

d i r e c t i o n a l d e r i v a t i v e s . W e s h a l l u s e t h e f o l lo w i n g n o t a t i o n s f o r t h e u p p e r a n d l o w e r D i n i s e m i - d e r i v a t i v e s o f

t h e f u n c t i o n u a t t h e p o i n t x 0 i n t h e d i r e c t i o n f :

d - u ( x o ; f ) = s u p i n f [ u( xo + 6h) - U(Xo)]lg,~>o (6,h)~a

w h e r e A : = { ( 6 , h ) :

d+u(xo; f ) = i n f s u p [ U ( X o + g h ) - u ( x o ) ] / 6 ,~>o (&h)EA

6 9 ( 0 , c ) , l l f - h l l < ~ , S o + g h 9 G } .

P r o p o s i t i o n . Let the fun ct io n F sa t i s fy the condi t ions poin ted out in Sec . 2 . 4 . For the lower (upper)s emi co n t i n u o u s f u n c t i o n u : c l G - - * R to be an upper ( lower) min ima x so lu t ion o f Eq . ( 2 . 1 ) , i t i s necessary

and su f f i c ien t tha t , fo r any Xo E G and p E R" , the inequal it y (3 .13) ( inequal i t y ( 3 . 1 4 ) ) be valid:

i n f { d - u ( x o ; f ) - ( p , f ) + F ( x o , p , u ( x o ) ) : I I f [ I -< p(xo)} _ o, (3 .13)

s u p { d + u ( X o ; ) - ( p, ) - 4 - F ( x o , p , u ( X o ) ) : Ef[l-< ( x o ) } _ > o. ( 3 . 1 4 )

T h e p r o o f i s, e s s e n ti a l ly , s i m i l a r t o t h e p r o o f s o f t h e a n a l o g o u s s t a t e m e n t s i n [ 1 7, 42 ].

T h u s , w e g e t t h a t t h e m i n i m a x s o l u t i o n c an b e d e f in e d as fo ll ow s : t h e m i n i m a x s o l u t i o n o f E q . ( 2 . 1 ) i s

t h e c o n t i n u o u s f u n c t i o n u : cl G ---* R w h i c h , f o r a n y Z o E G a n d p E R " , s a t i s f ie s t h e p a i r o f i n e q u a l i t i e s ( 3 . 1 3 )

a n d ( 3 . 1 4 ) .

A l o n g w i t h c o n d i t i o n s ( 3 . 1 3) a n d ( 3 .1 4 ) , o n e c a n c o n s i d e r th e ( e q u i v a l e n t to t h e m ) i n e q u a l i t i e s

s u p i n f { d - u ( x o ; f ) - g : ( f , g ) E U ( x o, u ( x o ) ,q ) } < O , V x o E G , ( 3 . 1 5 )qEQ

i n f s u p { d + u ( x o ; f ) - g : ( f , g ) E V ( x o , U (X o),p )} > 0 , V x o E G , ( 3 .1 6 )pE P

w h e r e U a n d V b o t h a r e m u l t i v a l u e d m a p p i n g s w h i c h s a ti s fy t h e c o n d i t i o n s p o i n t e d o u t i n S e c . 3 . 2 .

R e m a r k . C o n d i t i o n s ( 3. 13 ) a n d ( 3. 15 ) a r e e q u iv a l e n t i f e a c h of t h e s e i n e q u a li t ie s h o l d f o r a ll x o f r o m s o m e

o p e n d o m a i n . A n a n a l o g o u s r e m a r k i s v a l id f o r c o n d i t i o n s ( 3 .1 4 ) a n d ( 3 . 1 6 ).

601



3 . 5 . L e t u s p r e s e n t t h e d e f i n i t i o n o f a v i s c o s i t y s o l u t i o n , g i v e n b y M . G . C r a n d a l l a n d P . L . L i o n s [ 3 0,

31].

D e f i n i t i o n . A superso lu t ion (subso lu t ion) o f E q . ( 2 .1 ) i s a l ow e r ( u p p e r ) s e m i c o n t i n u o u s f u n c t i o n u : c l G

R , s a ti s f y i n g t h e f o l l o w i n g c o n d i t i o n : i f t h e d i f f e r e n c e u ( z ) - ~ ( z ) a t ta i n s a l oc a l m i n i m u m ( m a x i m u m ) a t t h e

p o i n t x 0 E G a n d i f a t t h i s p o i n t t h e f u n c t i o n ~ i s d i f f e re n t i a b le , t h e n t h e f o l l o w i n g i n e q u a l i t i e s h o l d :

F ( Z o , u ( x 0 ) , D ~ ( z o ) ) < 0 , ( 3 . 17 )

F (z o , u ( z o ) , D ~ (xo ) ) >_ O . (3 .18)

A vi scos i t y so lu t ion i s d e f i n e d a s a f u n c t i o n w h i c h i s s i m u l t a n e o u s l y a s u b s o l u t i o n a n d a s u p e r s o l u t i o n .

N o t e t h a t t h e s i g n s o f t h e i n e q u a l i t i e s in ( 3 .1 7 ) a n d ( 3 .1 8 ) a r e o p p o s i t e t o t h o s e o f t h e c o r r e s p o n d i n g

i n e q u a li t ie s in t h e d e f i n i t i o n g iv e n i n [ 30 , 3 1] . T h e p o i n t i s t h a t , i n t h e p a p e r s m e n t i o n e d , t h e n o n d e c r e a s i n g

f u n c t i o n s z ~ F ( z , z , p ) w e r e c o n s i d e r e d . F o r t h e c o n s t r u c t i o n s p r o p o s e d i n t h i s w o r k , i t i s m o r e c o n v e n i e n t

t o c o n s i d e r t h e n o n i n c r e a s i n g f u n c t i o n s z ~ F ( z , z , p ) .

I t is k n o w n t h a t c o n d i t i o n ( 3 . 1 7) i s e q u i v a l e n t to t h e c o n d i t i o n

F ( z , u ( x ) , p ) <_ O , V x e G , p 9 D - u ( x ) ; (3 .19)

h e r e D - u ( x ) i s t h e subdi f f eren t ia l o f t h e f u n c t i o n u a t t h e p o i n t x , w h i c h i s d e f in e d a s t h e s e t

D - u ( z ) : = { p 9 I V ' : O ; u ( z ) < 0}, (3 .20 )

w h e r e

O ; u ( x ) : = s u p { ( p , f ) - O T u ( z ) : f 9 R " } . ( 3 . 2 1 )

A n a l o g o u s l y , c o n d i t i o n ( 3 . 1 8 ) i s e q u i v a l e n t t o t h e c o n d i t i o n

w h e r e

F ( z , u ( z ) , p ) >_ O , V z 9 G, p 9 D + u ( x ) ,

D + u ( x ) : = { p 9 R " : O.pu(x) > 0 } ,

O.pu(x) : = in f { ( p , f ) - O ] u ( z ) : f 9 R " }

( t h e s e t D + u ( x ) i s c a l l e d t h e superd i f f eren t ia l o f t h e f u n c t i o n u a t t h e p o i n t x ) .

( 3 . 2 2 )

3 . 6 . T h e f o ll o w i n g s t a t e m e n t o n th e e q u i v a l e n c e o f t h e v i sc o s i ty a n d m i n i m a x s o l u t i o n s is v a l i d .

P r o p o s i t i o n . Le t t h e f u n c t i o n F s a t i s f y t h e co n d it io n s p o i n ted o u t i n S ec . 2 .4 . Th en , f o r t h e l o wer s em i -

cont inuous funct ion u : c lG - -* R , condi t ions (3 .13 ) a nd (3 .19) are e4uiva len t . Analogous ly , fo r the upper

s em i co n t i n u o u s f u n c t i o n u , co n d i t io n s (3 .14 ) a n d (3 .22 ) are also equivalent .

T h e i m p l i c a t i o n ( 3 . 1 3 ) =~ ( 3 .1 9 ) i s e a s i l y v e r i f i e d . L e t x E G a n d p 9 D - u ( z ) . A c c o r d i n g to ( 3 . 1 3 ) , t h e r e

ex i s t s f 9 R " s u c h t h a t

0 i u C z ) - ( p , f ) + F C x , u ( z ) , p ) < 0.

F r o m ( 3 :2 0) w e h a v e t h e i n e q u a l i t y

< p , f > - a T e ( x ) < 0 .

J o i n i n g t h e s e i n e q u a l i t i e s w e g e t ( 3 .1 9 ) .

T h e p r o o f o f t h e i m p l i c a t i o n ( 3 . 1 9) ~ ( 3 .1 3 ) i s o m i t t e d h e r e ( se e t h e p r o o f o f t h e a n a l o g o u s s t a t e m e n t s

in [17 , 181).

3 . 7 . A s h a s a l r e a d y b e e n s h o w n in S e c . 3 .3 , t h e c o n s t r u c t i o n o f t h e m i n i m a x s o l u t i o n h a s i t s o r i g i n i n

t h e t h e o r y o f p o s i t i o n a l d i f f e r e n t i a l g a m e s .

60 2



Note a l so tha t inves t iga tions on the junc t ions o f v is cos ity so lu t ion theo ry and o f op t im al con t ro l theo ry

and d i f fe ren t ia l gam es a re now very subs tan t i a l in num ber .

Am ong th em , in a n um be r of works (see, e .g . , [24, 33, 39, 43]) i t i s shown th at , for d if ferent types of

op t im al con t ro l p rob lem s and d i f fe ren ti a l gam es , the va lue func t ion co inc ides w i th th e v i s cos i ty so lu t ion o f the

c o r re s p o n di n g B e l l m a n - I s a a c s e q ua t i o n. O n t h e o t h e r h a n d , i t is k n o w n t h a t t h e H a m i l t o n - J a c o b i e q u a t i o n o f

a ( su ff i cien tly ) genera l fo rm m a y be cons ide red as the Be l lm an- I s aacs eq ua t ion fo r a d i f f e ren t i a l gam e, chosen

in a ce r ta in way . Such con s t ruc t ions were desc r ibed and used fo r the p roof o f the ex i s tence o f the v i s cosi tyso lut ions o f bounda ry -va lue p rob lem s and o f Cauchy p rob lem s fo r pa r t i a l d i f f e ren t ia l equa t ions o f the f ir st

order (see [32, 33, 35]) . N ote th at the chara cter is t ic d if ferent ia l inclus ion (2 .9) co ntains th e elem ents of such

cons t ruc t ions . A g rea t num ber o f pub l ica t ions is devo ted to th e app l ica t ions o f v i s cos i ty so lu t ion theo ry to

the s tu dy o f va rious p rob lem s o f op t im al con t ro l theo ry and d i f fe ren t ia l gam es .

Th e p roof o f the equ iva lence o f m in im ax and v i s cos i ty so lu t ions g iven in [17 , 18 ] is based on the f ac t o f

s eparab i l i ty o f a convex com pac t and an ep ig raph o f the f i r s t va r ia t ion o f a lower s em icon t inuous func t ion

which is , genera l ly speak ing , nonconvex . Th e d ua l i ty be tween the de f in i tion o f a v i s cos i ty so lu tion , in the

fo rm o f inequa l i t i e s in wh ich the subd i f f e ren t i a l s a r e used , and the de f in i t ion o f a m in im ax so lu t ion , where

the inequal i t ies for the d irect io nal der ivat ives are employed, appe.ars here . This is only one aspec t of the

dua l i ty in t r in s ic to the genera l i zed so lut ions o f pa r t i a l d i f fe ren t ia l equa tions o f th e f i r s t o rder , wh ich conf irm s

the L . Young r em a rk th a t Ham i l ton ians a re in t r in s ica l ly in te r l inked w i th the dua l i ty id ea ( s ee [20 , p . 74 ]) . I ti s a lso appropr ia te to n o te th a t the r ep lacem en t o f the th i rd eq ua t ion o f the sys te m (2 .2 ) ( the cha rac te r i s t i c

band equa t ion ) by equa t ion o f the fo rm (2 . 3 ) i s ve ry s im i la r to the p rob lem o f choos ing o f the "p roper"

Hamil tonian for the opt imal control problem, cons idered in [2 , 20, 40] .

4 . D i s c o n t i n u o u s S o l u t i o n s o f B o u n d a r y - V a l u e P r o b l e m s f o 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 s o f

t h e F ir s t O r d e r

4 . 1 . Le t u s cons ide r the D i r ich le t - type boundary -va lue p rob lem fo r Eq . (2 .1 ) . W e sha l l a s sum e tha t ,

fo r any x e G and p E R ~ , the func t ion z ~ F ( x , z , p ) is s t r ic t ly decreas ing. Th erefo re , Eq. (2 .1) may be

r e p la c e d b y t h e e q u a t i o n

H ( x , D u ) - u = O , x 9 G C R '~. (4.1)

Thus , we sha l l cons ide r the bound ary -va lue p rob lem fo r Eq . (4.1 ) w i th the b oun dar y co nd i t ion

u ( z ) = a ( x ) , x 9 o a , (4.2)

where O G i s a b o u n d a r y o f t h e d o m a i n G .

Note tha t , in the theo ry o f -genera l ized so lut ions o f pa r t i a l d i f fe ren t ia l equa t ions o f th e f i r s t o rde r , the m ain

resu lt s a r e ob ta ined fo r con t inhous so lu tions . A t the s am e t im e , the re i s a need to in t rod uce d i s con t inuous

so lu t ions . Fo r exam ple , in op t im a l con t ro l theo ry and d i f fe ren ti a l gam es , th e B e l lm a n- I s aac s equa t ion i s

cons idered wh ich satis f ies th e t im e-op t ima l funct ion at a l l points w here i t is d if ferent iable . As is know n, th is

func t ion can be d i s con t inuous . Thus , in th is p rob lem , one and on ly one d i s con t inuous so lu t ion fo r a pa r t i a ld i f fe ren ti a l equ a t ion o f fi r st o rd er i s subs tan t i a l ly de f ined . A t p resen t , a num ber o f works r e la te d to th i s a spec t

have been pub l i shed ( s ee , e .g . , [ 22 , 23 , 37 ] ) . Bu t they d o no t exhaus t the p rob lem . In pa r t i cu la r , ou t o f the

f r am e o f p ropos i t ions p rov id ing the ex i s tence and un iquenes s o f the d i s con t inuous v i s cos i ty so lu t ion i s the

boundary -va lue p rob lem fo r the Be l lm an- I s aacs equa t ion , wh ich appear s in the d i f f e ren t i a l gam e o f pu r su i t

( th is is a typical , pro ble m w ith a d iscon t inuous solut ion) . Below, in Sec. 4 .4 , the def in i t ion of a general ize d

m in im ax (pos s ib le d i s con t inuous ) so lu t ion o f the p rob lem (4 . 1 ) - (4 . 2 ) i s in t roduced and the ex i s tence and

un iqueness cond i t ions fo r it a r e fo rm ula ted . I t i s shown tha t the p roposed no t ion i s adeq ua te fo r th e above-

m en t ioned op t im al con t ro l p rob lem s and d i f fe ren ti a l gam es .

4 . 2 . I t is know n ( see , e . g ., [ 22] ) tha t the boun dary -va lue p rob lem for the Be l lm an equa t ion , w h ich

503



a p p e a r s i n t h e t i m e - o p t i m a l c o n t r o l p r o b l e m , is r e d u c e d to ( 4 . 1) , ( 4. 2 ). L e t u s re c a l l t h e c o r r e s p o n d i n g

t r a n s f o r m a t i o n .

L e t t h e c o n t r o l s y s t e m b e d e s c r i b e d b y t h e o r d i n a r y d i ff e re n t ia l ~ q u a t i o n

i ( t ) = f C x ( t ) , p ( t ) ) , t 9 R + , x ( 0 ) = x 0 9 R " , ( 4 . 3 )

w h e r e p ( . ) : g + ---* P i s a m e a s u r a b l e f u n c t i o n ( a n a d m i s s i b l e c o n t r o O , P C I t ~ i s a c o m p a c t s e t . L e t a c l o s e d

t e r m i n a l s e t M b e g i v e n i n t h e p h a s e s p a c e g " . L e t xCt , xo , p ( . ) ) b e t h e . t r aj e c to r y o f t h e s y s t e m ( 4 .3 ) a n d l e t

X ( z o ) = { z ( . , x o , p (. ) ) : p ( .) is a n a d m i s s i b l e c o n t r o l } .

L e t u s a s s u m e

t 0 ( x ( .) ) : = m i n { t _ > 0 : x ( t ) 9 M } , ( 4 .4 )

r 0( X o ) : = m i n { t 0 ( x ( . ) ) : x ( . ) 9 X ( x 0 ) } . ( 4 .5 )

( H e r e t o ( x ( . ) ) = o o i f x ( t ) • M f o r a l l t > 0 . ) A s s u m e t h a t t h e f u n c t i o n f i s c o n t i n u o u s , s a ti s f ie s t h e L i p s c h i t z

c o n d i t io n w i t h r e s p e c t t o t h e v a r i ab l e x , a n d t h e s e t { f ( x , p ) : p 9 P } i s c o n v e x . I n t h i s c a s e , t h e r e e x i s t s a

m i n i m u m in ( 4 . 4 ).

I t is w e l l k n o w n t h a t t h e f u n c t i o n x ~ r 0 ( x ) : R " ~ [ 0, o o] s a t is f ie s t h e B e l l m a n e q u a t i o n

m i n ( D T o ( x ) , f ( z , p ) ) + 1 = 0 (4 .6 )pEP

a t a l l p o i n t s w h e r e i t i s d i f f e r e n t i a b l e .

C o n s i d e r t h e f u n c t i o n ( t h e S . N . K r u z h k o v ' s t r a n s f o r m a t i o n [6])

= 1 - e x p ( -T 0 ( ) ) : R [ 0 , 1 ] . ( 4 . 7 )

F r o m ( 4 .6 ) , ( 4 .7 ) i t f o ll o w s t h a t t h e f u n c t i o n u s a t is f ie s t h e e q u a t i o n

r a in ( D u ( x ) , f ( x , p ) ) - u ( z ) + 1 = 0 (4.8)pEP

a t t h o s e p o i n t s w h e r e i t i s d i f f e re n t i a b l e . I t i s c l e a r t h a t u ( x ) = 0 fo r z 9 M . T h u s , w e c o m e t o t h e p r o b l e m

( 4 . 1 ) , ( 4 . 2 ) , i n w h i c h

G = W ' \ M ' H ( x , s ) = m i n { s , f ( x , p ) } + l , c r ( x ) = O f o r a l l x E O G . ( 4 ' 9 )pEP

T h e t i m e - o p t i m a l f u n c t i o n To i s lo w e r s e m i c o n t i n u o u s ; t h e s e t o f p o i n t s o f d i s c o n t i n u i t y o f t h i s f u n c t i o n i s, a s

u s u a l, n o n e m p t y . I t i s c l e a r t h a t t h e f u n c t i o n u i s al so l o w e r s e m i c o n t i n u o u s a n d h a s t h e s a m e s e t o f p o i n t s

o f d i s c o n ti n u it y . I t i s i m p o r t a n t t o n o t e t h a t t h e f u n c t i o n u ( 4 .7 ) c o i n ci d e s w i t h t h e m i n i m a x s o l u t i o n o f t h e

p r o b l e m ( 4 . 1 ) , ( 4 . 2 ) , ( 4 . 9 ) .

4 . 3 . L e t u s p o i n t o u t t h e a s s u m p t i o n s u n d e r w h i c h w e s h a ll c o n s i d e r t h e p r o b l e m ( 4 . 1 ) - ( 4 .2 ) .

( H 1 ). T h e f u n c t i o n H ( - , 0 ) i s b o u n d e d , i .e .,

] H ( x , 0 ) [ < a f o r a l l x E G . ( 4 . 1 0 )

( H 2 ). T h e r e e x i st s a n u m b e r i t s u c h th a t

[ g ( z , p O - H ( z , p 2 ) [ < _ ( l + [ l x l l ) l [ p l - p 2 ] [ # f o r a l l z e G , p , 9 ( i = 1 , 2 ) . ( 4 . 1 1 )

( H 3 ). L e t A b e a n o p e n , b o u n d e d s u b s e t in G . A s s u m e

r = e • - y l l < 1 } . ( 4 . 1 2 )

I t i s a s s u m e d t h a t , f o r a n y ~ > 0 a n d f o r a n y o p e n b o u n d e d d o m a i n A C G , t h e r e i s a f u n c t i o n w ~ : c l F --~ g +

w i t h t h e f o l l o w i n g p r o p e r t i e s :

604



( a ) t h e f u n c t i o n w e s a t is f ie s t h e L i p s c h i t z c o n d i t i o n o n c l F a n d i s c o n t i n u o u s l y d i f f e r e n t i a b t e o n F ;

( b ) f o r a l l ( x , y ) E F , t h e i n e q u a l i t y

w , ( z , y ) + H ( z , - D : , w , ( z , y ) ) - H ( y , D ~ ( x , y ) ) >_ 0

h o l d s ;

( c ) t h e b o u n d s

( 4 . 1 3 )

w e( z , x ) <_ e fo r x 9 (7, (4 .14)l i m i n f { w , ( x , y ) : [ [ x - y ] [ > r } = 0 f o r 0 < r _ < 1 ( 4. 15 )

el0

h o l d ;

( H 4 ) . T h e b o u n d a r y f u n c t i o n o r(x ) i s c o n t i n u o u s o n O G a n d s a t i s f i e s t h e b o u n d

[o '(x)[ < b for a l l x E OG. (4 .16)

L e t u s n o t e t h a t c o n d i t i o n ( H 3 ) is b o r r o w e d f r o m [ 29 , 3 7 ] . T h i s c o n d i t i o n is f o r m u l a t e d i n a g e n e r a l f o r m ,

w h i c h is c o n v e n i e n t f o r u s e i n t h e p r o o f o f t h e u n i q u e n e s s t h e o r e m . L e t u s s h o w t h a t c o n d i t i o n ( H 3 ) h o l d s if

t h e f u n c t i o n H s a t is f i es t h e f o l lo w i n g L i p s c h i t z c o n d i t i o n w i t h r e s p e c t t o t h e v a r i a b l e x :

( H 3* ). F o r a n y b o u n d e d d o m a i n A C G , t h e re is a n u m b e r A s u c h t h a t

I H ( z l , p ) - n ( z 2 , p ) l < A I I z l - ~11(1 + Ilpll), p 9 R ~ , z , e A ( i = 1 ,2 ) . (4 .17)

C o n s i d e r t h e f u n c t i o n

w , ( ~ , y ) : = ( e ~ / ~ + I I~ - y l I 2 ) ~ , ( 4 . 1 8 )

w h e r e 0 < 2 v < m i n { 1, l / A } , e A < 1 - 2 A g . L e t u s v e r if y t h a t t h e f u n c t i o n w ~ o f t h e t y p e ( 4 . 1 8 ) s a t is f i e s th e

i n e q u a l i t y ( 4 .1 3 ) . A s s u m e

s : = 2 , , ( ~ ~ - /" + I I z - y l l ~ ) " - ' ( z - y ) / ~ .

W e h a v e

D = w ~ ( = , y ) = - D , w ~ ( = , y ) = s .

N o t e t h a t

I l sl l I Ix - yl l < 2~ w ,( x, y) , w , ( x , y ) > l l z - y l l ~ V ~ , I l x - y l l ~ > I l x - y ll , V ( x , y ) e F ; ( 4. 19 )

t h e la s t i n e q u a l i t y f o ll o w s f r o m t h e b o u n d s I lx - y [[ < 1 a n d 2 u < 1 . F r o m c o n d i t i o n ( 4 . 1 7 ) a n d t h e b o u n d s

( 4 .1 9 ) w e o b t a i n

w e (x , y ) + H ( z , - D = w , ( x , y ) ) - H ( y , D y w , ( x , y )) = w e (z , y ) + H ( x , - s ) - H ( y , - s ) > w e ( x, Y ) - ~ t l z - Y l l ( l + l l s t l )

_ > ( 1 - 2 ~ ) w , ( ~ , y ) - A I I ~ - y l l > ( 1 - 2 A v ) l l z - y l 1 2 ~ l ~ - A I I z - y l t - > ( 1 - 2 A ~ - ~ ) l l z - y l l l ~ > -- o .

T h e r e f o r e , w e g e t t h a t t h e f u n c t i o n w , o f t h e t y p e ( 4 .1 8 ) s a t is f ie s c o n d i t i o n ( 4 . 1 3 ). I t is n o t d i f f i c u l t t o s e e

t h a t t h i s f u n c t i o n a l s o s a t i s f i e s c o n d i t i o n s ( 4 . 1 4 ) a n d ( 4 . 1 5 ) .

4 . 4 . L e t u s d e f in e t h e u p p e r , l ow e r , a n d m i n i m a x s o l u t i o n s o f t h e p r o b l e m ( 4 . 1 ) , (4 . 2 ).

D e f i n i t i o n . W e s h a l l d e f in e a n upper solution o f t h e p r o b l e m ( 4 .1 ) , ( 4 .2 ) a s a n u p p e r m i n i m a x s o l u t io n o f

E q . ( 4 .1 ) , s a t i s fy i n g t h e b o u n d a r y c o n d i t i o n ( 4 .2 ) a n d t h e b o u n d

] u ( z ) ] < _ c : = m a x { a , b } , V z e c l G . ( 4 .2 0 )

A lower so lu t ion o f t h e p r o b l e m ( 4 .1 ) , ( 4. 2 ) is d e f i n e d a s a lo w e r m i n i m a x s o l u t i o n o f E q . ( 4 . 1 ), w h i c h is

c o n t i n u o u s a t e a c h p o i n t x E 0 G w h i c h s a ti sf ie s t h e b o u n d a r y c o n d i t i o n ( 4. 2) a n d t h e b o u n d ( 4 .2 0 ). A

min ima z so lu t io n o f t h e p r o b l e m ( 4 . 1 ) , (4 .2 ) is d e f i n e d a s a f u n c t i o n u : c l G ~ R s u c h t h a t t h e r e e x i s t a

s e q u e n c e o f u p p e r s o l u t i o n s u (k ) a n d a s e q u e n c e o f l o w e r s o l u t i o n s u k ( k = 1 , 2 , 3 , . . . ) w h i c h p o i n t w i s e c o n v e r g e

t o t h e f u n c t i o n u .

60 5



T h e f o l l o w i n g a s s e r t i o n s h o l d .

T h e o r e m 4 .1 . L e t t h e c o n d i t i o n s ( H 1 ) - ( H 4 ) h o l d. T h e n , f o r a n y u p p e r s o l u ti o n u + o f t h e p r o b l e m (4 .1 ) ,

(4 .2) a n d f o r a n y l o w e r s o lu t i o n u - o f t h i s p r o b l em , t h e i n e q u a li t y

~ ,+ (z ) > u - ( z ) , V z e c l O ( 4. 21 )

holds.

T h e o r e m 4 .2 . L e t t h e c o n d i t i o n s ( H 1 ) - ( H 4 ) ho ld and le t there ex i s t a t l eas t one low er so lu t ion o f the

prob lem (4 .1 ) , (4 .2 ) ; t h e n t h e r e e x i s t a n u p p e r s o l u ti o n U o o f t h i s p r o b l e m a n d a p o i n t w i s e c o n v e r g e n t s e q u e n c e

o f lo w e r s o l u t io n s u k ( k = 1 , 2 , 3 , . . . ) s u c h t h a t

l i ra Uk(X) > Uo(X) , V x e c lG . (4 .22 )k--*oo

T h e s e a s s e r t i o n s i m p l y t h e f o l l o w i n g :

T h e o r e m 4 .3 . I f a l l c o n d i t i o n s o f T h e o r e m 4 .2 are va l id , then there ex i s t s one and on ly one min imax

s o l u ti o n o f th e p r o b l e m (4 .1 ) , (4 .2 ) . T h i s m i n i m a x s o l u t i o n i s l o w e r s e m i c o n t i n u o u s a n d c o i n c i d e s w i t h th e

u p p e r m i n i m a x s o l u t i o n .

5 . T h e U n i q u e n e s s o f a S o l u t i o n o f t h e B o u n d a r y - V a l u e P r o b l e m

5 . 1 . L e t u s c o n s i d e r a n a u x i l i a r y a s s e r t i o n , w h i c h w e sh a l l u s e i n Se c . 5 .2 in p r o v i n g T h e o r e m 4 . 1 .

L e t F b e a n o p e n d o m a i n o f t h e t y p e ( 4 . 1 2 ). L e t t h e c o n t i n u o u s f u n c t i o n s

( ~ , y ) ~ p ( x , y ) : r ~ x " ,

b e g i v e n , w h i c h a r e b o u n d e d o n F , i . e . ,

s u p I I p ( ~ , y ) l l < ~ ,(~,~)er

( z , y ) ~ - ~ q ( z , y ) : F - - - ~ R ~ ,

s u p I I q ( x , y ) l l < ~ .( x , ~ ) e r '

C o n s i d e r t h e f o l lo w i n g s y s t e m o f d i f f er e n t ia l in e q u a l i t i e s a n d d i f f er e n t ia l e q u a t i o n s :

II~(t) l l < (1 + IIz (t ) l l ) g ,

~ ( t ) = ( p ( z ( t ) , y ( t ) ) , $ ( t ) ) - H ( z ( t ) , p ( z ( t ) , y ( t ) ) ) + ~ ( t ) , (5 .1 )

I I # ( t ) l l _ < ( 1 - 4 - I l y ( t ) l l ) ~ , ,i l ( t) = (q ( x ( t ) , y ( t ) ) , ~ ) (t )) - H (y ( t ) , q ( x C t ), y ( t ) ) ) + ~ l( t) .

L e t u s d e f i n e a s o l u t i o n o f t h i s s y s t e m . L e t 0 < t l < t ~ < c r T : = [ q , t ~] ,

s ( . ) : t ~ s C t ) : = ( ~ ( t ) , @ , y ( t ) , , ( t ) ) : x + - ~ x " • ~ • r • x

b e a c o n t i n u o u s f u n c t i o n , s T ( - ) : = ( z r ( - ) , ~ r ( ' ) , y r ( ' ) , 7 / T ( ' ) ) b e a r e s t / i c t i o n o f t h i s f u n c t i o n t o t h e d o s e d

i n t e rv a l T . W e s h a h d e f i n e a s o l u t io n o f t h e s y s t e m ( 5 .1 ) a s a c o n t i n u o u s f u n c t i o n s ( - ) s a t i s fy i n g t h e f o l lo w i n g

c o n d i t i o n : i f

{ ( x ( t ) , y ( t ) ) : t e T } C F ,

t h e n t h e f u n c t i o n ( X T ( ' ) , ~ r ( ' ) , Y T (' ), 7 / r (' ) ) i s a b s o l u t e l y c o n t i n u o u s a n d , f o r a l m o s t a l l t e T , c o n d i t i o n s ( 5 .1 )

h o l d .

L e t ( z ( . ) , ~ ( - ) , y ( - ) , ~ /( -) ) b e s o m e s o l u t i o n o f t h e s y s t e m ( 5 . 1) . L e t u s a s s u m e

{ ~ r i r ( x C t ) , y ( t ) ) e r f o r a l l t > O ,

t o ( X ( . ) , u ( . ) , r ) : = m i n { t >_ 0 : ( x ( t ) , y ( t ) ) r F } , o t h e r w i s e . (5 .2 )

L e t u s c h o s e a n a r b i t r a r y p o i n t ( z 0 , y 0 ) E F , a n d l e t ~0 : = u(zo), ~lo : = v (y o ) , w h e r e u i s s o m e u p p e r

s o l u t i o n o f E q . ( 4 . 1 ) , v is a l o w e r s o l u t i o n o f t h i s e q u a t i o n .

60 6



P r o p o s i t i o n .

c o n d i t i o n

There i s the so lu t ion s ( . ) : ( x ( . ) , ~ ( . ) , y ( . ) , g ( . ) ) o f t h e s y s t e m ( 5 . 1 ) , sa t i s fy ing the in i t ia l

x ( o ) = ~ o , y ( 0 ) = y o , ~ ( o ) = ~ o : = ~ ( ~ o ) , g ( o ) = g o : = ~ ( y o )

a n d t h e i n e q u a l i t i e s

~ ( t ) _ > u ( ~ ( t ) ) , g ( t ) _ < , ( v ( t ) ) , v t e [O , t o ( z ( . ) , y ( . ) , r ) ] . ( 5 . 3 )

P r o o f .

e q u a t i o n s

L e t u s c h o s e a n a r b i t r a r y 5 > 0 . C o n s i d e r t h e s y s t e m o f d if f e r e n ti a l i n e q u a l i t i e s a n d d i f f e r e n t i a l

{ l :~ , ( t) lt < ( 1 + ] [ ~ 6 ( t ) l l ) # ,

( , ( t ) = ( p 6 ( t ) , k 6 ( t ) ) - H ( z s ( t ) , p 6 ( t ) ) + ~ 6 ( t ),

l i y 6 ( t ) l [ < ( 1 + I l y n ( t ) l [ ) ~ ,r~(t ) = (q~(t ) , ~6(t )) - H C y , ( t ) , q s ( t ) ) + g s ( t ) ,

w h e r e

r e ( t ) : = p ( ~ ( i 6 ) , y ~ ( i 6 ) ) ,

q 6 ( t ) : = q ( x s ( i S ) , y 6 ( i S ) ) fo r i 5 < t < ( i + 1)5,

L e t u s d e f i n e t h e s o l u t i o n ~ ( . ) : = ( ~ ( . ) , ~ ( . ) , y ~ ( - ) , g ~ ( . ) )

(5 .4 )

i = 0 , 1 , 2 , . . . .

o f t h e s y s t e m ( 5 . 4 ), a n d a l s o t h e v a l u e t o ( x 6 ( ' ) , y 6 ( ' ) , F ) , i n t h e s a m e w a y a s w a s a l r e a d y d o n e f o r t h e s y s t e m

( 5 . 1 ) .

S i n c e t h e f u n c t i o n s p 6 ( t) a n d q 6 (t) a r e p i e c e w i s e c o n s t a n t , t h e n t h e d e f i n i t i o n s o f u p p e r a n d l o w e r s o l u t io n s

o f E q . ( 4 .1 ) i m p l y t h e e x i s t e n c e o f t h e s o l u t i o n s 6 (- ) o f t h e s y s t e m ( 5 .4 ) , w h i c h s a ti s fi e s i n e q u a l i t i e s o f

t h e t y p e ( 5 .3 ) a n d i n i t i a l c o n d i t i o n ( z 6 ( 0 ) , ~ s ( 0 ) , y 6 ( 0 ) , y 6 ( 0 ) ) = (X o ,~ 0, y o, g o ). A s s u m e s 6 ( t ) = c o n s t f o r

t _> t o ( ~ ( . ) , v ~ ( - ) , r ) .

L e t u s c h o o s e a z e r o - c o n v e r g i n g s e q u e n c e 6 k ( k = 1 , 2 , . . . ) i n s u c h a w a y t h a t t h e s e q u e n c e ( x 6 k ( ' ) , f 6 ~( ') ,

Y ~ k(') , g 6 ~ (') ) c o r r e s p o n d i n g t o i t i s c o n v e r g e n t t o t h e l i m i t m o t i o n s ( . ) = ( x ( . ) , f ( . ) , y ( .) ,7 7 ( .) ) u n i f o r m l y o n

a n y c l o s e d i n t e r v a l [ 0, 8 ] a n d s a t i s fi e s t h e c o n d i t i o n s

( . , . ( o ) , ~ ( o ) , y ~ k ( o ) , g s ~ ( o ) ) = ( : o , ~ o , v o , g o ) , ( , k ( t ) > ~ ( : . ~ ( t ) ) ,

w h e r e t k : = t o ( x 6 k ( ' ) , Y ' k ( ' ) , r ) "

N o t e t h a t t h e l i m i t m o t i o n s ( . ) s a t i s f i e s t h e s y s t e m ( 5 . 1 ) and

v t e [ o , t 4 ,

( 5 . 5 )

t o ( x ( - ) , y ( . ) , F ) < l i m t = .k-*oo

L e t u s p as s t o a l i m i t i n ( 5 .5 ) . F r o m t h e l o w e r s e m i c o n t i n u i t y o f t h e f u n c t i o n u a n d f r o m t h e u p p e r s e m i c o n -

t i n u i t y o f t h e f u n c t i o n v , i t f ol lo w s t h a t i n e q u a l i ti e s ( 5 . 3 ) h o l d f o r t h e l i m i t m o t i o n .

5 . 2 . P r o o f o f T h e o r e m 4 . 1 . L e t u be a n u p p e r m i n i m a x s o lu t io n o f t h e p r o b l e m ( 4. 1) , (4 .2 ), v b e a

l o w er m i n i m a x s o l u t i o n o f t h i s p r o b l e m . A s s u m e t h e c o n t r a r y , i .e ., t h a t t h e r e i s a p o i n t x 0 6 G s u c h t h a t

: = , ( x o ) - ~ ( x 0 ) > 0 .

C h o o s e a n u m b e r 8 > 0 s u c h t h a t t h e i n e q u a li t y

Ot

c e x p ( - 8 ) <

( 5 . 6 )

( 5 . 7 )

h o l d s, w h e r e c i s a c o n s t a n t f r o m t h e b o u n d ( 4 .2 0 ). D e n o t e b y X ( z o ) t h e s e t o f a ll a b s o l u te l y c o n t i n u o u s

f u n c t i o n s x ( . ) : [ 0 ,8 ] ~ R " s u c h t h a t x ( 0 ) = X o a n d [ [ ~( t) [ [ < ( 1 + [ [ x ( t) ] [) # fo r a l m o s t a l l t e [ 0 , 8 ] . L e t D b e

507



a n o p e n d o m a i n i n g " s u c h t h a t D ~ { z ( t ) : t e [ 0 , 0 ], x ( . ) e X ( z o ) } . A s s u m e A : = G f ) D i n t h e c o n d i t i o n

( H 3 ) ( i n e q u a l i t y ( 4 . 1 2 ) ) .

C o n s i d e r t h e s y s t e m o f d i ff e r e n ti a l e q u a t io n s a n d d i f f e r e n ti a l in e q u a l i t i e s

{ I [ ~ l] < _ ( 1 + H z l l ) / ~ , ~ = ( p ( d , ~ > _ H ( x , p ( ' ) ) - ~ ,

I1 ~ 11 < ( 1 + I l y l t ) ~ , , i = ( q ( d , y > - g ( y , q ( d ) _ ~ ? , ( 5 . 8 )p (~ ) = - D ~ w ~ , q ( d = D ~ w ~ ,

w h e r e w ~ i s a f u n c t i o n d e f i n e d i n t h e c o n d i t i o n ( H 3 ) ( s e e S e c . 4 . 3 ) .

D e n o t e b y ( x ~ ( . ) , ~ ( . ) , y ~ ( -) , ~ /~ (. )) a s o l u t i o n o f t h e s y s t e m ( 5 . 8 ) , s a t i s f y i n g t h e c o n d i t i o n s

. ( o ) = y . ( o ) = ~ o , ~ ( o ) = ~ ( ~ o ) , ~ . ( o ) = ~ ( ~ o ) ,( 5 . 9 )

w h e r e t~ = t 0 ( z ~ ( - ), y ~ ( . ) , F ) ( l e t u s r e c a l l t h a t t h e v a l u e t o ( x ( . ) , y ( . ) , F ) i s d e f i n e d b y t h e e q u a l i t y ( 5 . 2 )) . T h e

e x i s te n c e o f a s o l u t i o n ( z ~ ( .) , ~ ( . ) , y , ( . ) , ~ /~ (.) ) w i t h t h e p r o p e r t i e s p o i n t e d o u t h a s a l r e a d y b e e n s h o w n i n S e c .

5.1.

C o n s i d e r t h e f u n c t i o n ( t h e L y a p u n o v f u n c t io n )

L ~ (t, z , y , ~ , , ) : = ~ - ' ( ~ , ( ~ , y ) + ~ - , ) , ( t e a + , ( ~ , y ) e r , ( ~ , , ) e z • R ) . (5 .1 0)

A s s u m e

~( ~) := L , ( ~ , ~ , ( t ) , y ~ ( t ) , ~ ~ ( 5 . 1 1 )

F r o m ( 4 . 1 3 ) , ( 5 . 1 0 ), a n d ( 5 . 8 ) i t f o l lo w s t h a t ; / ~( t) < 0 f o r 0 < t < t ~ . T a k i n g ( 4 . 1 4 ) i n t o a c c o u n t , w e o b t a i n

~ ( ~ ) < ~ , ( 0 ) < ~ , v ~ e [ 0 , ~ , ) . ( 5 . 1 2 )

L e t u s c h o o s e a ( c o n v e r g i n g t o z e r o ) s e q u e n c e e k . > 0 , k = 1 , 2 , . . . . L e t u s i n t r o d u c e t h e f o l lo w i n g

n o t a t i o n s :

t c k ) = t , ~ , ~ c ~ ) ( . ) = ~ , ( . ) , y c ~ ) ( .) = y , ~ ( . ) , . . . , ~ c ~ ) ( . ) = ~ . ( . ) .

W e c a n a s s u m e , w i t h o u t l os s o f g e n e r a l i t y , t h a t t h e r e e x i s ts t h e l i m i t ] i m k - . ~ t (k) E ( 0 , c o ] . T h e f o l l o w i n g t w o

c a s e s a r e p o s s i b l e :

l im t (k) > 0 , (5 .13)k - - * ~

l i m i ( ~ ) < 0 . ( 5 . 1 4 )k--~oo

C o n s i d e r t h e c a s e ( 5 . 1 3 ) . F r o m ( 5 . 9 ) , ( 5 . 1 0 ) , a n d ( 5 . 1 2 ) w e h a v e

~ C ~ ) ( 0 ) = ~ C ~ ) ( ~ o , ~ o ) + u ( ~ o ) - V ( ~ o ) _ > " r C ~ ) ( 0 )

= ~ - ~ ( ~ C ~ ) ( z ( ~ ) ( 0 ) , y C ~ ) (O ) ) + ~ C ~ ) ( O ) _ ~ c ~ ) ( 0 ) ) _ > ~ - ~ ( ~ c ~ ) ( e ) _ ~ c ~ ) ( o ) ) .

T h e n f r o m ( 5 . 9 ) a n d ( 4 . 2 0 ) , w e g e t ~ (k ) (0 ) - ~ 7( k) (0 ) _> - 2 c . T a k i n g t h e b o u n d ( 4 . 14 ) i n t o a c c o u n t , w e

hav e w(~)(zo , Xo) _< e (~) an d u ( z o ) v ( z o ) >_ - 2 c e - ~ - e (~ ) . P a s s i n g t o t h e l i m i t a s k + c o a n d t a k i n g ( 5 . 7 )i n t o a c c o u n t , w e o b t a i n U ( X o ) - v ( z o ) > - a . T h e r e f o r e , i n t h e c a s e ( 5 .1 3 ), w e c o m e to a c o n t r a d i c t i o n w i t h

( 5 . ~ ) .

C o n s i d e r t h e c a s e (5 . 1 4) . W i t h o u t l os s o f g e n e r a l i t y , w e c a n a s s u m e t h a t t h e r e e x i s t t h e l i m i t s

l im~--.oo z(~)(t (~)) an d l i ra~. . .oo y(~)(t(~)) . Le t us show th at

l im z(~)(t (~)) = l im yC~)(t ~)) e O G . (5 .15 )k - - * ~ k - - - *~

F r o m t h e i n e q u a l i t i e s ( 5 . 1 2 ), ( 4 .2 0 ) , a n d ( 5 .3 ) w e h a v e

~ c ~ ) ( ~ c ~ ) ( t c ~ ) ) , u c ~ ) ( t c ~ ) ) ) < ~ , ( ~ ) ( ~ c ~ ) ( ~ o , ~ 0 ) + ~ ( ~ 0 ) - ~ ( ~ 0 ) ) - ~ ( ~ ) ( tc ~ ) ) + , c ~ ) ( t c ~ ) ) _ < ~ 0 ( ~ c ~ ) + 2 c ) + 2 c .

50 8



The r e f o r e , by ( 4 . 15 ) , we c onc lude tha t

l i m f l x ( k ) ( tC k l ) - - y ( k l ( t ( ~ ) ) l f = 0 . ( 5 . 1 6 )k - - - * o o

Rec a l l tha t t (k) = to(x(k) ( . ) , y(k) ( .) , F ) ; h en ce f rom (4 .12) an d (5 .2) we hav e

( ~ ( k / ( t ( ~ ) ) , y ( ~ l ( t c ~ l ) ) e 0 r = 0 A • 0 a u { ( x , y ) e ~ x x ~ x : I Ix - y l l = 1 } .

F r o m t h e d e f i n i t i o n o f t h e s e t A i t f o ll o w s t h a t

Thus , u s ing ( 5 . 16 ) we c om e to ( 5 . 15 ) .

F r om ( 5 . 12 ) a nd ( 5 .9 ) i t fo l l ows tha t

~ + u ( , o ) - v ( ~ 0 ) > e - ' c ' ~ [ u ( x r - v ( y ~ ( t ( k ~ ) ) ] .

R e c a l l t h a t t h e f u n c t i o n s u a n d v a r e lo w e r a n d u p p e r s e m i c o n t i n u o u s o n c l G r e s p e c t i v e l y a n d s a t i s fy t h e

b o u n d a r y c o n d i t i o n u(x ) = v ( z ) = r r (z ) for z E OF; h e n c e , p a s s i n g to t h e l i m i t a s k ~ o o in t h e l a s t b o u n d

a n d t a k i n g ( 5 .1 5 ) i n t o a c c o u n t , w e o b t a i n t h e i n e q u a l i t y U(Xo) - v( z a ) _> 0 , wh ic h c on t r a d i c t s ( 5 . 6 ) .T h e o r e m 4 .1 i s p ro v e d .

L I T E R A T U R E C I T E D

1. R. Isaacs, Dif ferent ial Fames [ R uss i a n t r a ns l a t i on ] i M i r , M osc ow ( 1967) .

2 . F . C la rke , Opt i mi za t ion and Non-Smoot h A na l y s i s [ R u s s ia n t r a n s l a t i o n ] , N a u k a , M o s c o w ( 1 9 8 8 ) .

3 . N . N . Kr a sovsk i i , Gam e-Theore t ic Problems on the Encoun ter of Mo t ions [ In R u s s ia n ] , N a u k a , M o s c o w

( 1 9 7 0 ) .

4 . N . N . Kr a sovs k i i , " D i f f e r e n t i a l ga m e o f pu r su i t - - e va s ion , " Izv . Akad . N auk SS SR , Tekh. Kibern. , 2 ,

3- 18 ( 1973) .

5 . N . N . Kr a so vsk i i a nd A . I. S ub bo t in , Posi t ional Di f f erent ial Fames [ I n R uss i a n ] , N a ukn , M osc ow ( 1974) .

6 . S . N . K r u z h k o v , " G e n e r a l i z e d s o l u t io n s o f H a m i l t o n - J a c o b i e q u a t i o n s o f E i k o n a l ty p e , " Mat . Sb. , 9 8 ,

No . 3 , 450 - 493 ( 1975) .

7 . A . B . K u r z h a n s k i i a n d T . F . F i l i p p o v a , " O n a d e s c r i p t i o n o f t h e s e t o f v i a b l e t r a j e c t o r i e s f o r a d i ff e r e n t ia l

i nc lus ion , " Dokl. Akad. N auk SSS R, 289, No. 1 , 38-41 (1986) .

8 . E . F . M i s h c h e n k o , " T h e p r o b l e m s o f p u r s u i t- - e v as i o n i n t h e t h e o r y o f d i f fe r e n t i a l g a m e s , " Izv. Akad.

Nau k SSS R, Tekh. Kibern., 5, 3-9 (1971) .

9 . E . F . M i s h c h e n k o , " O n s o m e g a m e - t h e o r e t i c p r o b l e m s o f p u r s u i t - e v a s i o n , " A v t omat . Te l e me k h . , No. 9 ,

24 - 30 ( 1972) .

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