Nonsmooth analysis and sufficient conditions for a saddle in a differential games

One of the major difficulties in the theory of differential games is the lack of differentiability of the value function. One way around this difficulty is to use viscosity solutions (see Lions and Souganidis [5]) of the Isaacs-Bellman equations. In this paper we demonstrate the use of nonsmooth analysis (see Clarke [2]) as a natural tool for obtaining the Isaacs-Bellman equations when the value function W satisfies a Lipschitz condition.