Characterization And Approximation Of Value Functions Of Differential Games With Maximum Cost In Infinite Horizon

Value functions of differential games withL∞criterion over infinite horizon are known to possess poor regularity. As an alternative to generalized solutions of the Isaacs equation, that usually requires some regularity properties, we propose a characterization of the value functions using the integral form of the Isaacs equation. We prove, without any regularity assumption, that value functions are the lowest super-solution and the largest element of a special set of sub-solutions, of the dynamic programming equation. We characterize also the limits of finite horizon value functions, and propose an approximation scheme in terms of iterations of an infinitesimal operator defined over the set of Lipschitz continuous functions. The images of this operator can be characterized by generalized solutions of a classical Isaacs equation.We illustrate these results on a example, whose value functions can be determined analytically.