Limit systems and attractivity for time-varying systems with applications to nonholonomic systems

The paper investigates the uniformly asymptotical stability (UAS) and the uniformly globally asymptotical stability for nonlinear time-varying systems consisting of asymptotically almost periodic (AAP) functions with an output-injection term. Several properties about AAP functions are given. The concept of reduced limit systems that describe the behavior of systems at infinity is introduced. By assuming an integrability condition of output function, the UAS of the origin can be guaranteed for uniformly Lyapunov-stable systems under a detectability condition relating to reduced limit systems. The proposed criterion can be viewed as a natural generalization of the so-called Krasovskii-LaSalle theorem to non-periodic systems whenever limit systems are used to study the asymptotic behavior of original system. The proposed criterion is also applied to study the tracking control problem of nonholonomic chained systems. From these examples, it can be seen that the proposed criterion is very suitable to analyze the stability of nonlinear time varying systems.