Characterizations of input-to-state stability for hybrid systems

We study input-to-state stability (ISS) for a broad class of hybrid systems, which are combinations of a differential equation on a constraint set and a difference equation on another constraint set. For this class of hybrid systems, we establish the equivalence of ISS, nonuniform ISS, and the existence of a smooth ISS-Lyapunov function by “additionally” assuming that the right-hand side of the differential equation has a convex property with respect to inputs. Moreover, we demonstrate by examples that the equivalence may fail when such a convexity assumption is removed.