Asymptotic stability of switching systems - Driss Boularas; David Cheban (Electronic Journal of Differential Equations, 2010)
In this article, we study the uniform asymptotic stability of the switched system $u'=f_{ u(t)}(u)$, $uin mathbbR^n$, where $ u:mathbbR_{+}o {1,2,dots,m}$ is an arbitrary piecewise constant function. We find criteria for the asymptotic stability of nonlinear systems. In particular, for slow and homogeneous systems, we prove that the asymptotic stability of each individual equation $u'=f_p(u)$ ($pin {1,2,dots,m}$) 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 following result: The linear switched system is uniformly asymptotically stable if it does not admit nontrivial bounded full trajectories and at least one of the equations $x'=A_px$ is asymptotically stable. We study this problem in the framework of linear non-autonomous dynamical systems (cocyles).