The Analysis Of Stability For Periodic Time-varying Systems

Stability theory plays an important role in the control theory and its application. The stability of equilibrium is studied in this paper based on Lyapunov second method. The criterions of stability for time-invariant systems are very rich in the past development. Thanks to LaSalle invariance principle, the derivative of a Lyapunov function, which is positive definite, along the trajectories of the systems can be reduced into the negative semi-definite from the negative definite. The latest research shows that the positive semi-definite function could be employed to examine the stability of equilibrium. But when the same idea is applied into the time-varying systems, we have much trouble to do so. This problem still remains open so far. This paper extends the idea to the periodic time-varying systems.In view of the properties of limit set and some invariance of the bounded orbits for the solution of semi-flow in the periodic time-varying systems, this paper studies various problems of stability and asymptotic stability for systems. In the presented criterions, we use a positive semi-definite function as a Lyapunov function instead of positive definite one for examining the stability of the given systems. Such a function is required only to be non-increased along the trajectories or to be non-increased on a particular sequence along the trajectories of the given system.This paper is organized in mainly three parts:The first part is an introduction. In this part, we first describe the background of the studied problem and then the main results are presented in this part.The second part is to establish the LaSalle invariant principle for the periodic time-varying systems.The last one is the detailed and rigorous proof of the main results.