Converse Lyapunov theorems and robust asymptotic stability for hybrid systems

We state results on the existence of smooth Lyapunov functions for hybrid systems whose solutions satisfy a class-ICCE estimate with respect to two measures. The class-KLL estimate, a natural extension of class-KL estimate, is in terms of the elapsed time and the number of jumps that have occurred. The main result is that a smooth Lyapunov function exists if and only if the class-ICCE estimate is robust. In turn, sufficient conditions for robustness are given. Special cases include systems with compact attractors. Most of the results parallel, and unify, what has been developed previously for differential inclusions and difference inclusions.