Common polynomial Lyapunov functions for linear switched systems

In this paper, we consider linear switched systems. x( t) = A(u( t))x( t), x is an element of R-n, u is an element of U, {Au : u is an element of 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.