Approximation of differential games of pursuit-evasion by discrete-time games

In this paper we consider the classical problem of pursuit and evasion for continuous-time and discrete-time systems. We prove the convergence, as the time step goes to 0, of the upper and lower value functions of the discrete-time game to the upper and lower values of the differential game. This is done assuming a capturability condition either on the differential game, or on the discrete-time game uniformly for small values of the time step. An application is the existence of the value in the sense of Fleming under rather general conditions.

[1]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[2]  W. Fleming.  The convergence problem for differential games , 1961 .

[3]  P. Varaiya.  On the Existence of Solutions to a Differential Game , 1967 .

[4]  E. Roxin.  Axiomatic approach in differential games , 1969 .

[5]  R. Elliott,et al.  The Existence Of Value In Differential Games , 1972 .

[6]  Robert J. Elliott,et al.  Values in differential games , 1972 .

[7]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[8]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[9]  I. Dolcetta.  On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming , 1983 .

[10]  P. Souganidis,et al.  Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations. , 1983 .

[11]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[12]  L. Evans,et al.  Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls , 1984 .

[13]  H. Ishii,et al.  Approximate solutions of the bellman equation of deterministic control theory , 1984 .

[14]  P. Souganidis.  Max-min representations and product formulas for the viscosity solutions of Hamilton-Jacobi equations with applications to differential games , 1985 .

[15]  H. Ishii.  Perron’s method for Hamilton-Jacobi equations , 1987 .

[16]  Alexander M. Tarasyev,et al.  A differential game of unlimited duration , 1987 .

[17]  G. Barles,et al.  Discontinuous solutions of deterministic optimal stopping time problems , 1987 .

[18]  M. Falcone.  A numerical approach to the infinite horizon problem of deterministic control theory , 1987 .

[19]  Tamer Basar,et al.  Differential Games and Applications , 1989 .

[20]  Hitoshi Ishii,et al.  A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations , 1989 .

[21]  M. Bardi,et al.  A PDE framework for games of pursuit-evasion type , 1989 .

[22]  M. Falcone,et al.  Discrete approximation of the minimal time function for systems with regular optimal trajectories , 1990 .

[23]  M. Falcone,et al.  An approximation scheme for the minimum time function , 1990 .

[24]  M. Bardi,et al.  Hamilton-Jacobi equations with singular boundary conditions on a free boundary and applications to differential games , 1991 .