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Jeux différentiels et approximation numérique de fonctions valeur. I: Étude théorique. (Differential games and numerical approximation of value functions. I: Theoretical analysis). (French) Zbl 0734.90131

An antagonistic pursuit-evasion game on the plane is studied (as a model, the lion-antilope game is presented). It is a simple pursuit game.
The dynamic programming approach is used for the study of the value function for both players in the game under consideration. It is proved that the capture time is finite and satisfies some estimation conditions. The value functions for both players are bounded, Lipschitzian and are unique viscosity solutions of the corresponding Isaacs equations.
[For part II see the author, ibid. 25, No.5, 535-560 (1991; Zbl 0734.90132).]

MSC:

91A24 Positional games (pursuit and evasion, etc.)
49L20 Dynamic programming in optimal control and differential games
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
91A23 Differential games (aspects of game theory)

Citations:

Zbl 0734.90132
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References:

[1] M. BARDI & P. SORAVIA, A P.D.E. framework for games of pursuit evasion type, to appear in differential games and application pp. 62-71. Zbl0679.90109 MR1230190 · Zbl 0679.90109
[2] P. BERNHARD, Differential Games : Isaacs’ equation, Encyclopedia of Systems and control, Editor : Madan Singh, Pergamon Press, 1987.
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[4] I. CAPUZZO-DOLCETTA & P.-L. LIONS, Hamilton-Jacobi equation and state constraints problem, IMA Preprint Ser. 342, Minneapolis, September 1987. MR951880
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[6] L. C. EVANS & P. E. SOUGANIDIS, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana, 33, n^\circ 5, 1984. Zbl1169.91317 MR756158 · Zbl 1169.91317
[7] M. G. CRANDALL & P.-L. LIONS, Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations, Illinois J. Math. 31, n^\circ 4, winter 1987. Zbl0678.35009 MR909790 · Zbl 0678.35009
[8] M. G. CRANDALL, L. C. EVANS & P.-L. LIONS, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282, April 1984. Zbl0543.35011 MR732102 · Zbl 0543.35011
[9] M. H. SONER, Optimal control with state-space constraint, I. SIAM J. Control Optim. 24, n^\circ 3, 1986. Zbl0597.49023 MR838056 · Zbl 0597.49023
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