Finite-time stability of switched nonlinear time-varying systems via indefinite Lyapunov functions

This note considers the problem of finite-time stability (FTS) for switched nonlinear time-varying systems. First, a relaxed condition is proposed to verify the FTS of nonlinear time-varying systems by using an indefinite Lyapunov function. Then, the result obtained is extended to study the FTS of switched nonlinear time-varying systems. Several relaxed conditions are given by using a common indefinite Lyapunov function and multiple indefinite Lyapunov functions. Moreover, the corresponding estimates on convergence regions and times of systems are also given. Comparing with the existing results, the conditions obtained allow the time derivative of Lyapunov functions of subsystems (or systems) to be indefinite and all subsystems to be not finite-time stable or even unstable. Finally, a numerical example is given to illustrate the theoretical results.