A Stability Criterion Of Positive Semi-definite Functions

Stability theory plays an important role in the control theory and its application. We have much trouble when we study stability theory for the time-varying systems. This paper presents stability analysis for general nonlinear time-varying systems. It uses a positive semi-definite time-varying function as a candidate Lyapunov function to provide sufficient conditions for the uniformly asymptotic stability of the equilibrium of the given time-varying systems. This method extends the range of the Lyapunov function, from positive definite functions, to positive semi-definite functions. Since there is no systematic approach to construct a Lyapunov function in general, this method on the extension of the range has important significance in both control theory and its applications in industries.This paper is organized in mainly three parts:The first part is an introduction. In this part, we first introduce the background of the studied problem and present some notions and results in the previous literatures.The second part is the main theorems and their detailed proof. Meanwhile, an example is given to show the validity of the results.The third part is a summery on this paper.