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.