Indefinite derivative for stability of time-varying nonlinear systems

This paper is concerned with stability analysis of nonlinear time-varying systems by using Lyapunov function based approach. The classical Lyapunov stability theorems are generalized in the sense that the time-derivative of the Lyapunov functions are allowed to be indefinite. Then, under quite general assumptions, we first present a new converse stability theorem for a large class of time-varying systems which will be used to prove certain stability properties of nonlinear systems with perturbation. Therefore, a new Lyapunov function is presented that guarantees global asymptotic stability under some restrictions on the perturbed system. Furthermore, some illustrative examples are presented.