A nested Matrosov theorem for hybrid systems

We show that for time-invariant hybrid systems given by a flow map, flow set, jump map, and jump set, uniform global stability of a compact set plus the existence of Lyapunov-like functions and continuous functions satisfying a nested condition imply uniform global asymptotic stability of the compact set ("uniform" in the sense that bounds on the solutions and on the convergence time depend only on the distance to the compact set of interest). The required nested condition is a combination of the conditions in nested Matrosov theorems for time-varying continuous-time and discrete-time systems available in the literature. Our result also shows that Matrosov's theorem is a reasonable alternative to LaSalle's invariance principle for time-invariant hybrid systems to conclude attractivity to a compact set. We illustrate the application of our main result by examples.