Asymptotic Stability of Switching Systems

International audienceIn this article, we study the uniform asymptotic stability of the switched system u′ = fν(t)(u), u ∈ Rn, where ν : R+ → {1,2,...,m} is an arbitrary piecewise constant function. We find criteria for the asymptotic sta- bility of nonlinear systems. In particular, for slow and homogeneous systems, we prove that the asymptotic stability of each individual equation u′ = fp(u) (p ∈ {1, 2, . . . , m}) implies the uniform asymptotic stability of the system (with respect to switched signals). For linear switched systems (i.e., fp(u) = Apu, where Ap is a linear mapping acting on En) 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′ = Apx is asymptotically stable. We study this problem in the framework of linear non-autonomous dynamical systems (cocyles)