Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 21, pp. 1-18.

Title: Asymptotic stability of switching systems
       
Authors: Driss Boularas (Univ. de Limoges, Limoges, France)
         David Cheban (State Univ. of Moldova, Chisinau, Moldova)

Abstract:
 In this article, we study the uniform asymptotic
 stability of the switched system $u'=f_{\nu(t)}(u)$,
 $u\in \mathbb{R}^n$, where
 $\nu:\mathbb{R}_{+}\to \{1,2,\dots,m\}$ 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 $u'=f_p(u)$ ($p\in \{1,2,\dots,m\}$)
 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).
	 
Submitted December 21, 2009. Published February 02, 2010.
Math Subject Classifications: 34A37, 34D20, 34D23, 34D45, 37B55, 37C75, 93D20.
Key Words: Uniform asymptotic stability; cocycles; globalattractors;
           uniform exponential stability; switched systems.