Bardi, Martino; Falcone, Maurizio; Soravia, Pierpaolo 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]. Reviewer: Marc Quincampoix (Brest) Cited in 25 Documents 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) Keywords:envelope solutions; numerical approximation; viscosity solutions; Hamilton-Jacobi-Isaacs equations; Subbotin solutions; differential games; value function Citations:Zbl 0820.35003; Zbl 0842.90138; Zbl 0821.90145; Zbl 0982.91014; Zbl 0790.65081 PDFBibTeX XMLCite \textit{M. Bardi} et al., in: Stochastic and differential games. Theory and numerical methods. Dedicated to Prof. A. I. Subbotin. Boston: Birkhäuser. 105--175 (1999; Zbl 1018.49027)