Results on existence of smooth Lyapunov functions for (pre-)asymptotically stable hybrid systems with non-open basins of attraction

We show that (pre-)asymptotic stability, which is a natural generalization of asymptotic stability, of a compact set for a hybrid system satisfying mild regularity assumptions is equivalent to the existence of a smooth Lyapunov function. In this new converse Lyapunov theorem for (pre-)asymptotic stability, we no longer require the assumption that the basin of attraction is open (cf. [1, Theorem 1]). Furthermore, we demonstrate several applications of the converse Lyapunov theorem by establishing robustness of pre-asymptotic stability to various types of perturbations and achieving input-to-state stabilization for hybrid-feedback-control systems.