Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls

Abstract

It is demonstrated that the upper and lower values of a two-person, zero-sum differential game solve the respective upper and lower Isaacs' equations in the viscosity sense (introduced by Crandall and Lions (Trans. Amer. Math. Soc. 277 (1983), 1-42). Since such solutions are unique, this yields a fairly simple proof that the game has value should the minimax condition hold. As a further application of viscosity techniques, a new and simpler proof that the upper and lower values can be approximated by the values of certain games with Lipschitz controls is given. © 1984.