Asymptotic stability of switching systems

Electronic Journal of Differential Equations

关键词: Uniform asymptotic stability;    cocycles;    globalattractors;    uniform exponential stability;    switched systems;   

DOI  :  

来源: DOAJ

【 摘 要 】

In this article, we study the uniform asymptoticstability of the switched system $u'=f_{ u(t)}(u)$,$uin mathbb{R}^n$, where$ u:mathbb{R}_{+}o {1,2,dots,m}$ is an arbitrarypiecewise constant function.We find criteria for the asymptotic stability of nonlinearsystems. In particular, for slow and homogeneous systems,we prove that the asymptotic stability of each individualequation $u'=f_p(u)$ ($pin {1,2,dots,m}$)implies the uniform asymptotic stability of the system(with respect to switched signals).For linear switched systems (i.e., $f_p(u)=A_pu$, where $A_p$is a linear mapping acting on $E^n$) we establish the followingresult: The linear switched system is uniformly asymptotically stableif it does not admit nontrivial bounded full trajectories andat least one of the equations $x'=A_px$ is asymptotically stable.We study this problem in the framework of linear non-autonomousdynamical systems (cocyles).

【 授权许可】

Unknown