Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls (Q799508)

Differential games of prescribed duration are considered. The authors prove that the corresponding Hamilton-Jacobi-Isaacs equation has a unique so-called ''viscosity'' solution V under the minimax condition, and that V is the value of the game. This concept of solution for nonlinear first- order partial differential equations was introduced previously by \textit{M.G. Crandall} and \textit{P.-L. Lions} [Trans. Am. Math. Soc. 277, 1-42 (1983)]. It is proved that if the players are constrained to use only Lipschitzean functions as controls then the game has a value \(V_{M,L}\), where M, L are the Lipschitz constraints. It is proved that \(V_{M,L}\) converges to V as M, L approach infinity.