Zero sum differential games with stopping times: Some results about its numerical resolution. (English) Zbl 0827.90148

Tidball, Mabel M.; González, R. L. V.

Başar, Tamer (ed.) et al., Advances in dynamic games and applications: refereed and edited papers from biannual symposium of ISDG, held in July 1992, Geneva, Switzerland. Boston, MA: Birkhäuser. Ann. Int. Soc. Dyn. Games. 1, 106-124 (1994).

The authors consider a zero-sum differential game with uncontrolled dynamics and a Bolza type payoff. Players choose the stopping times of the game. The value function in this problem is the unique viscosity solution of the system of two Isaacs inequalities and two inequalities on its values. The aim of the paper is to describe numerical solutions of the system of inequalities.
The authors give a discretization procedure and the optimal rate of convergence for the discrete solutions. They propose an accelerated algorithm and prove that the algorithm converges in a finite number of steps. There is an analysis of modifications of the algorithm for different stopping rules. To compare the classical discretization procedure and the accelerated one several numerical examples are presented.
For the entire collection see [Zbl 0792.00004].

Reviewer: N.Subbotina (Sverdlovsk)

MSC:

91A23	Differential games (aspects of game theory)
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
91A05	2-person games
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Keywords:

zero-sum differential game; uncontrolled dynamics; Bolza type payoff; viscosity solution; Isaacs inequalities; optimal rate of convergence; discrete solutions; stopping rules