Common Polynomial Lyapunov Functions for Linear Switched Systems

In this paper, we consider linear switched systems x(t)=Aᵤ₍ₜ₎ x(t), x∈ⁿ, u∈ U, {ᵤ: u∈ 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.