PURSUIT-EVASION PROBLEMS AND VISCOSITY SOLUTIONS OF ISAACS EQUATIONS

The Dirichlet problem for first-order Hamilton-Jacobi equations arising in differential games of pursuit and evasion is studied. Local and global sub- and superoptimality principles are stated for, respectively, viscosity sub- and supersolutions. These results are applied to obtain a general existence theorem and to prove the existence of the value of the game. The main application concerns the problem of stability (terminability) of a dynamical system with two competitive controls and the opposite one of evadability from a general closed set. The approach used in this paper allows Lyapunov functions satisfying the usual condition in the weak sense of viscosity solutions.