Differential Games in $$L^{\infty }$$

Abstract

The Isaacs equations for differential games in \({L^{\infty }}\) first derived in Barron (Nonlinear Anal 14:971–989, 1990) are reformulated so that the Hamiltonians are continuous and result in a simpler problem to analyze numerically. Relaxed differential games in \({L^{\infty }}\) are considered. \(L^{\infty }\) differential games with time and state independent dynamics and convex or quasiconvex terminal data are solved explicitly using a type of Hopf–Lax formula. The stochastic differential game in \({L^{\infty }}\) connected to stochastic target problems is also discussed.