Common Polynomial Lyapunov Functions for Linear Switched Systems

In this paper, we consider linear switched systems ẋ(t) = Au(t)x(t), x ∈ R, u ∈ U , and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS for short). We first prove that, given a UAS system, it is always possible to build a common polynomial Lyapunov function. Then 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.