On the stability and existence of common Lyapunov functions for stable linear switching systems

A sufficient condition for the existence of a common quadratic Lyapunov function (CQLF) for the linear systems x˙=A_ix, A _i∈{A₁,A₂,…,A_m}, A_i∈IR^n×n is that the matrices can be simultaneously triangularized using a non-singular transformation T. In this paper, we show that this result follows trivially from the structure of the matrices in the set A, and that the switching system, constructed by switching between the matrices in this set, is benign from a stability viewpoint. Finally, we then discuss several conditions under which a transformation T exists