Common Polynomial Lyapunov Functions for Linear Switched Systems

Abstract

In this paper, we consider linear switched systems $\dot x(t)=A_{u(t)} x(t)$, $x\in\R^n$, $u\in U$, {$\{A_u: u\in U \}$ compact,} and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS). {Given a UAS system, it is always possible to build a common polynomial Lyapunov function. Our main result is that} the degree of that common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin. A generalization to a class of piecewise-polynomial Lyapunov functions is given.