Converse Lyapunov Theorem for Switched Nonlinear Systems with Constrained Switching Signals

This paper investigates converse Lyapunov theorems for switched nonlinear systems comprising both stable and unstable subsystems, uniformly over a constrained set of switching signals. A novel hybrid timer is introduced to quantify switching behavior, and the considered class of signals--characterized by a uniformly bounded hybrid timer--encompasses known signal classes defined by mixed average dwell-time and average activation-time conditions. The main result is a necessary and sufficient condition, expressed via the existence of multiple Lyapunov functions with prescribed decay or growth rates at flows and jumps, ensuring global uniform asymptotic stability uniformly over this set of switching signals. The proof employs extended classes of comparison functions and a constructive analysis tailored to hybrid-timer-based signal models. This work extends previous results, offering a deeper understanding of stability in switched systems with both stabilizing and destabilizing dynamics.