On the minimal degree of a common Lyapunov function for planar switched systems

In this paper, we consider linear switched systems x(t) = A_u(t)x(t), x ε Rⁿ, u ε 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 polynomial common Lyapunov function. Then our main result is that the degree of that the common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin.