Converse Lyapunov Results for Switched Systems with Lower and Upper Bounds on Switching Intervals

The topic of this manuscript is the stability analysis of continuous-time switched nonlinear systems with constraints on the admissible switching signals. Our particular focus lies in considering signals characterized by upper and lower bounds on the length of the switching intervals. We adapt and extend the existing theory of multiple Lyapunov functions, providing converse results and thus a complete characterization of uniform stability for this class of systems. We specify our results in the context of switched linear systems, providing the equivalence of exponential stability and the existence of multiple Lyapunov norms. By restricting the class of candidate Lyapunov functions to the set of quadratic functions, we are able to provide semidefinite-optimization-based numerical schemes to check the proposed conditions. We provide numerical examples to illustrate our approach and highlight its advantages over existing methods.