*(English)*Zbl 1193.34014

This paper addresses the stability problem for a switched system formed by a finite family of linear subsystems

The authors assume that for each $i$, there exists a stabilizing linear feedback $u={K}_{i}x$ for the $i$th subsystem, and that the resulting family of closed-loop subsystems admits a common quadratic Lyapunov function.

The control law for the switched system is defined by means of a sequence of switching times ${\tau}_{0}=0<{\tau}_{1}<{\tau}_{2}<\cdots $ . On each interval $[{\tau}_{j},{\tau}_{j+1})$ the control law may be indifferently defined as either $u\left(t\right)={K}_{i}x\left(t\right)$ (where $i$ is the active index for the $j$th interval) or the sampled value $u\left(t\right)\equiv {K}_{i}x\left({\tau}_{j}\right)$.

The authors prove that asymptotic stability is guaranteed provided that the mesh size $max\{{\tau}_{j+1}-{\tau}_{j}\}$ is sufficiently small.

The paper contains also a sufficient condition for the existence of a common Lyapunov function, and extensions of the result to systems with static output feedback and/or centralized/decentralized control.