The Dirichlet problem for first-order Hamilton-Jacobi equations arisin
g in differential games of pursuit and evasion is studied. Local and g
lobal sub- and superoptimality principles are stated for, respectively
, viscosity sub- and supersolutions. These results are applied to obta
in 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 approa
ch used in this paper allows Lyapunov functions satisfying the usual c
ondition in the weak sense of viscosity solutions.