Smooth Lyapunov Functions for Hybrid Systems—Part I: Existence Is Equivalent to Robustness

Hybrid systems are dynamical systems where the state is allowed to either evolve continuously (flow) on certain subsets of the state space or evolve discontinuously (jump) from other subsets of the state space. For a broad class of such systems, characterized by mild regularity conditions on the data, we establish the equivalence between the robustness of stability with respect to two measures and a characterization of such stability in terms of a smooth Lyapunov function. This result unifies and generalizes previous results for differential and difference inclusions with outer semicontinuous and locally bounded right-hand sides. Furthermore, we give a description of forward completeness of a hybrid system in terms of a smooth Lyapunov-like function.