Discontinuous differential games and control systems with supremum cost

Abstract

This paper concerns with the existence of a value for a zero sum two-player differential game with supremum cost of the form C t 0, x 0 ( u, v)=sup τ∈[ t 0, T] h( x( τ; t 0, x 0, u, v)) under Isaacs' condition. We characterize the value function as the unique solution—in a suitable sense—to a PDE, namely the Hamilton–Jacobi–Isaacs equation. As a byproduct, we obtain a PDE characterization of the value function for control system.