Relaxed Conditions for the Input-to-State Stability of Switched Nonlinear Time-Varying Systems

This technical note considers the problem of input-to-state stability (ISS) for switched nonlinear time-varying systems. First, a sufficient condition is proposed to verify the ISS of nonlinear time-varying systems by using an improved Lyapunov function. Then, the results obtained are extended to study the ISS of switched nonlinear time-varying systems. Three relaxed conditions are given by using the methods of multiple Lyapunov functions, minimum dwell time, and infinite switchings, respectively. Comparing with the existing results, the conditions obtained have two relaxations, i.e., the derivative of Lyapunov functions of subsystems are allowed to be indefinite, and all subsystems are allowed to be unstable in the case of infinite switchings. All results obtained also be used to study the uniformly asymptotic stability (UAS) of systems. Finally, a numerical example is given to illustrate the theoretical results.