On robustness of stability and Lyapunov functions for discontinuous difference equations

We demonstrate that strong global asymptotic stability (GAS) of the origin for an upper semicontinuous difference inclusion is equivalent to the existence of a smooth Lyapunov function. This result is of interest in discrete-time because the robustness of the stability property is dependent on the existence of such a smooth Lyapunov function. We also propose a regularization that allows us to state when GAS of the origin is robust for difference equations.