1. Basi c Definitions an d Preliminary Result s We denote th e m-dimensiona l Euclidea n spac e b y R m. A point (x x , '",x m ) i n Rm i s denoted als o by x. A n m-vecto r functio n (f lt ••• , f m ) wil l be denote d als o by f. Consider a system o f m differentia l equation s (1.1) dx/dt = f(t, x, y, z) (t 0t T 0) with initia l conditio n (1.2) x(t 0) = x 0. Let Y, Z b e compac t subset s of som e Euclidea n space s R p an d R q respectively . W e assume (A) f(t, x, y, z ) i s continuous i n (t, x, y, z ) G [f 0 , T 0 ] X R m X Y X Z x f(t, x, y, z ) fc(f)(l + |JC| 2) wher e tf° k(t)dt ~, and , fo r an y R 0 , |/(f, x , j / , z ) -/(?, 3f :K, z)| k R(t)\x -x | , for al l r G [r 0, r j j e y . z E Z , \x\ R, \x\ R, wher e ^ ( 0 i s a function depend - ing on R an d satisfyin g / f ° k R {t)dt °° . The se t F (Z ) i s called th e control set for th e playe r j (z). A measurable functio n j/(f) (z(0 ) wit h value s in F (Z ) fo r almos t al l r i s called a control function fo r ^ (z) . If we substitute int o (1.1) an y contro l function s y = y(t), z = z(t) (t 0 ? T 0), the n w e obtain a differential syste m (1.3) dx/dt = /(r, x, 7(0, *(0)C o ' r 0 ). The conditio n (A ) ensures that th e syste m (1.3), (1.2) has a unique solution . Here, a solution i s understood t o b e a n absolutel y continuou s functio n satisfyin g (1.3) almost everywhere in othe r words , x(t) = x0 +fi 0f(s, x(s),y(s), z(s))ds. We call x(t) th e trajectory corresponding t o y(t), z(t). W e denote b y X t T th e space o f al l trajectories. W e denote b y C m [t 0 , TQ] th e spac e o f al l continuous m-vector s u(t) = (ux(t), •• , um(t)) define d o n [/ 0, r o ] , wit h th e nor m m I|H|| = X ma x |ii y .(r)l. /=0 ^O^'^^O 2 http://dx.doi.org/10.1090/cbms/018/02
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