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Numerical methods for pursuit-evasion games via viscosity solutions. (English) Zbl 1018.49027

Bardi, Martino (ed.) et al., Stochastic and differential games. Theory and numerical methods. Dedicated to Prof. A. I. Subbotin. Boston: Birkhäuser. Ann. Int. Soc. Dyn. Games. 4, 105-175 (1999).
This paper concerns with numerical approximation of viscosity solutions of Hamilton-Jacobi-Isaacs equations. These equations are related to differential games.
The authors present an overview of uniqueness results for Hamilton-Jacobi equations including some discontinuous cases where viscosity solutions are replaced by Subbotin solutions (cf. for, instance, [A. I. Subbotin, “Generalized solutions of first-order PDEs. The dynamical optimization perspective” (1994; Zbl 0820.35003)]), called “envelope solutions” in the present paper.
The minimum time problem to reach a target with constraints is studied when both target and set of constraints are regular enough (Lipschitz) (for a more general result see [P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, C. R. Acad. Sci., Paris, Sér. I 321, No. 12, 1543-1548 (1995; Zbl 0842.90138)]).
Various numerical experiments are discussed for classical examples of differential games.
A very interesting section discusses tricks and algorithms for obtaining efficient computation of the value function; among these tricks we can notice the refinement procedure introduced by P. Saint-Pierre in [P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, RAIRO, Modélisation Math. Anal. Numér. 28, No. 4, 441-461 (1994; Zbl 0821.90145); Ann. Int. Soc. Dyn. Games 4, 177-247 (1999; Zbl 0982.91014)] and [P. Saint-Pierre, Appl. Math. Optimization 29, No. 2, 187-209 (1994; Zbl 0790.65081)].
For the entire collection see [Zbl 0919.00032].

MSC:

49N75 Pursuit and evasion games
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
91A24 Positional games (pursuit and evasion, etc.)
49N70 Differential games and control
91A23 Differential games (aspects of game theory)
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