An introduction to the theory of games by E. S. Venttsel'

By E. S. Venttsel'

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238 of 24 EXAMPLES R. I s s a c s : D i f f e r e n t i a l Games, Wiley, New York, e t c . , 1957 - This r e v e a l s another i n t e r p r e t a t i o n ( r e a d i l y v e r i f i e d by ret r a c i n g t h e successive s u b s t i t u t i o n s ) : ( 4 ) d e p i c t s t h e game on t h e pursuer's radar screen, i . e . , i n a movable coordinate system which has i t s o r i g i n a t t h e pursuer's c u r r e n t position, and with r e a l a x i s i n t h e d i r e c t i o n of h i s motion; s i m i l a r l y , ( 5 ) shows t h e game on t h e quarry's r a d a r screen.

Cockayne: Plane [ s i c ] pursuit with curvature cons t r a i n t s , SIAM J. Appl. Math. 15 (1967) 1511-1516. The a s s e r t i o n t h e r e i s t h a t pursuer can force capture from any i n i t i a l position i f , and only i f , In G. T . Rublein: On pursuit with curvature constraints, SIAM J. Control 10 (1972) 37-39 t h e method i s extended t o t h e case of dimension n = 35 sharp i n e q u a l i t i e s i n (6) ensure perfect capture from a= i n i t i a l positions. 3. 7 Unpleasant Examples The reason f o r t h e section heading i s t h a t t h e examples described here, simple instances of f a r more general situations, cannot be t r e a t e d by t h e methods t o be developed l a t e r i n t h e book (surely t h e r e i s an obligation t h a t the reader be made aware of t h i s circumstance).

35 The allono- PURSUIT GAMES mous version of (1) i s := (41 f(x,t,p,q); it i s customary t o i n t e r p r e t t h i s v i a ;r c = f(x,5,p,q), = 1 i . e . , an (n+l)-dimensional instance of (1)(thus, once l i n e a r - i t y i s abandoned, t h e r e i s l i t t l e formal d i f f e r e n c e between autonomous and allonomous systems). Second-order games have = f(x,i,p,q) (5) a s n-dimensional equation of motion; a s usual, t h i s i s i n t e r preted a s t h e 2n-dimensional case of (l), ;r = Y, i= f(x,y,p,q).

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