This paper discusses the mathematical descript i o n of propogation and growth of wave-type phenomena which may o c c u r i n many physical systems, such as, magnetohydrodynamic system, fluid flow system, electromagnetic field system. This work also discusses the statement of a maximum princ i p l e f o r t h i s wave generating system. One dimens i o n a l v e r s i o n of t h i s s y s t e m w i t h a q u a d r a t i c cost function has been taken, and t h e method of g r a d i e n t s a n d q u a s i l i n e a r i z a t i o n t e c h n i q u e s h a v e been applied to derive the optimal control funct i o n . I n t r o d u c t i o n Very r e c e n t l y , t h e o p t i m a l c o n t r o l o f d i s t r ibuted parameter systems has received the attention of many c o n t r o l e n g i n e e r s . The f i r s t s e r i o u s work i n t h i s d i r e c t i o n was introduced by B u t k o v~k y~~~*~. Wang4 and Lions5 attempted to present a g e n e r a l d i s c u s s i o n of various problems a s s o c i a t e d w i t h t h e c o n t r o l o f d i s t r i b u t e d p a r ameter systems. Chaudhuri6,7, Sage and Chaudhuri8 s t u d i e d o p t i m a l c o n t r o l o f a v a r i e t y o f l i n e a r and nonlinear distributed parameter systems. In t h i s work, optimal control laws for distributed parameter systems described by nonlinear hyperbol i c 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 h a v e b e e n obt a i n e d t h r o u g h t h e d i s c r e t i z a t i o n schemes and v i a t h e method of g r a d i e n t s and q u a s i l i n e a r i z a t i o n technique. System Description The a t t e n t i o n w i l l be focused on svstems which may be described by a g e n e r a l n o n l i n e a r v e c t 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 o f t h e form a x ( y , t ) 2 aKzcy, t ) azcyy t ) -2 k ( y . t ) , K 3 a t 2 ,U(Y,t) , Y , t l av a t -4-The i n i t i a l c o n d i t i o n s a r e : x ( y , t O ) = & , ( y ) and ax(y, (1) a t = & ( y ) , The boundary conditions are: f i f [ I f 2 [ ] . ..f [ 1. y d e n o t e s t h e s p a t i a l coo r d i h t e v e c t o r El [ y y . . .yM] belonging to t h e s p a c e S l and an denotei $he boundary of n. x ( y , t ) is t h e s t a t e o?f'ti?e system d e f i n e d a t any t stands f o r time. a ...aK-l are c o n s t a n t s . i n s t a n t o f time t and space ~€ 0 . A t any i n s t a n t of time t , g ( y , t ) i s t h e c o n t r o l v e c t o r f u n c t i o n which may run over the whole of t h e s p a t i a l domain 52 o r c e r t a i n s u b s e t s o f a. It is d e s i r e d t o f i n d the optimal control g(y,t) which minimizes the f u n c t i o n a l form defined by + P a r t i a l l y s u p p o r t e d by the University of E v a n s v i l l e u n d e r t h e f a c u l t y r e s e a r c h g r a n t . T --where 8 and 4 a r e s c a l a r f u n c t i o n s o f t h e a r g uments shown i n (2). reduced into two v e c t o r p a r t i a l d i f f e r e n t i a l equat i o n s a s shown below. If The p a r t i a l d i f f e r e n t i a l...
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