On the robustness of KL-stability for difference inclusions:: Smooth discrete-time Lyapunov functions

We consider stability with respect to two measures of a difference inclusion, i.e., of a discrete-time dynamical system with the push-forward map being set-valued. We demonstrate that robust stability is equivalent to the existence of a smooth Lyapunov function and that, in fact, a continuous Lyapunov function implies robust stability. We also present a sufficient condition for robust stability that is independent of a Lyapunov function. Toward this end, we develop several new results on the behavior of solutions of difference inclusions. In addition, we provide a novel result for generating a smooth function from one that is merely upper semicontinuous.