Stability analysis of non-linear time-varying systems by Lyapunov functions with indefinite derivatives

This study is concerned with stability analysis of non-linear time-varying systems by using Lyapunov function-based approach. The classical Lyapunov stability theorems are generalised in the sense that the time-derivative of the Lyapunov functions are allowed to be indefinite. The stability analysis is accomplished with the help of the scalar stable functions introduced in the author's previous study. Both asymptotic stability and input-to-state stability are considered. Particularly, for asymptotic stability, several concepts such as uniform and non-uniform asymptotic stability, and uniform and non-uniform exponential stability are studied. The effectiveness of the proposed theorems is illustrated by several numerical examples.