Discontinuous differential games and control systems with supremum cost

This paper concerns with the existence of a value for a zero sum two-player differential game with supremum cost of the form C_t0,x0(u,v)=sup_τ[t0,T]h(x(τ;t₀,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.