Stabilizability of Time-Varying Switched Systems Based on Piecewise Continuous Scalar Functions

Inspired by the idea of multiple Lyapunov functions (<inline-formula><tex-math notation="LaTeX">$\mathtt {MLFs}$</tex-math></inline-formula>), we use piecewise continuous scalar functions to investigate the stabilizability of time-varying switched systems. Starting with time-varying switched linear systems, we first combine the idea of <inline-formula><tex-math notation="LaTeX">$\mathtt {MLFs}$</tex-math></inline-formula> with the existence of asymptotically (exponentially, uniformly exponentially) stable functions to provide necessary and sufficient conditions for their asymptotic (exponential, uniform exponential) stabilizability. Compared to traditional differential Lyapunov inequalities, we release the requirement on negative definiteness of the derivatives of <inline-formula><tex-math notation="LaTeX">$\mathtt {MLFs}$</tex-math></inline-formula>. Successively, the above results are extended to time-varying switched nonlinear systems. Then, two illustrative examples are given to show the applicability of our theoretical results. In the end, we consider the computation issue of our current results for a special class of nonautonomous switched systems, i.e., rational nonautonomous switched systems.