By A. V. Balakrishnan (auth.), G. Geymonat (eds.)

A. Balakrishnan: A optimistic method of optimum control.- R. Glowinski: Méthodes itératives duales pour los angeles minimisation de fonctionnelles convexes.- J.L. Lions: Approximation numérique des inéquations d’évolution.- G. Marchuk: creation to the equipment of numerical analysis.- U. Mosco: An creation to the approximate resolution of variational inequalities.- I. Singer: top approximation in normed linear spaces.- G. Strang: A Fourier research of the finite aspect variational method.- M. Zerner: Caractéristiques d’approximation des compacts dans les espaces fonctionnels et problèmes aux limites elliptiques.

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**Example text**

B a s i s functions i n C[O,T]. Let 1. 1 b (t) denote a sequence of F o r each n , l e t A n denote the s e t of functions spanned by b k ( ), k = I , . n, such that A. V. Balakrishnan F o r each n, vJe now consider the epsilon p r o b l e m over the c l a s s of s t a t e functions (denoted S ) of the form: n w h e r e the 1 ak} 1 m u s t satisfy ( 3 . l ) , corresponding to condition (F'), and over controls u ( t ) a s before. ) L e t u s denote the corresponding infimum by hn(c). Clearly any admissible s t a t e function can be approximated uniformly in t by functions i n S a s closely a s n d e s i r e d for l a r g e enough n, and of c o u r s e S n i s a l s o conditionally compact.

And since we may take z n ( t ) converging weakly (in the L (0,T) sense) 2 to s o ( t ) , we have also: Hence + h ( c ) = ~ ~ l j i ( e ; t ; ; ; ~ ( t ) :d~t ~ ( t ~) )J ( Z ~ ( T ) ) 0 Letting Y(c;t) = (golt) - f (t;zO(t);Go(t)))/c N we s e e that the optimal control Go(*) (which we r e p e a t i s now, in general, relaxed) i s characterized by the Maximum Principle: A. V. Balakrishnan where the maximum i s now taken over the c l a s s of control m e a s u r e s . Finally a routine f i r s t variation analysis shows tkiat Y(c;t) m u s t satisfy.

This i s b e s t s e e n by examining a R i t e approximation, o r of : rersion of i t f o r the p r o b l e m . ,us s t a t e functions under the sup n o r m . b a s i s functions i n C[O,T]. Let 1. 1 b (t) denote a sequence of F o r each n , l e t A n denote the s e t of functions spanned by b k ( ), k = I , . n, such that A. V. Balakrishnan F o r each n, vJe now consider the epsilon p r o b l e m over the c l a s s of s t a t e functions (denoted S ) of the form: n w h e r e the 1 ak} 1 m u s t satisfy ( 3 .