8 AVNER FRIEDMA N where p = p(R)0 i f R i s sufficiently larg e an d p(R)T°° i f R T °°. I f ty*(t, x)/dx i (1 / m) exis t an d ar e continuous, then w e shall sa y tha t T * i s a continuously differen- tiable strategy corresponding t o y*(t, JC) . Similarly on e defines th e concep t o f a continuous (or continuousl y differentiable ) strat - egy A * correspondin g t o a continuous (or continuousl y differentiable ) functio n z*(f , x) with values in Z . EXAMPLE 2 . Le t y*(t, JC) , z*(r, JC ) b e continuou s an d continuousl y jc-differentiabl e functions i n [t 0, T 0] X Rm, wit h values in Y an d Z respectively . Le t f(t, x, y, z) b e uniformly Lipschit z continuou s in(JC , y, z) . Fo r an y contro l functio n z (t), i f we substitut e z = z(t), y = y*(t, x) int o (1.1), we get a unique solutio n x(t) (wit h JC(/ 0) = x0). Denot e the correspondin g payof f b y P(y* 9 z) . Similarl y w e define P(y, z* ) (fo r an y contro l func - tion y(t)), an d P(y*,z*). Suppos e now tha t P(y, z* ) / (y*, z* ) P(y*, z ) fo r an y control function s j , z. The n on e can easily prov e tha t V ~ P(y*, z*) an d tha t th e strat - egies A* , T * correspondin g t o z*,y* for m a saddle point it i s called a continuously dif- ferentiable saddle point.
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