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On the existence of a Lyapunov function for a family of switching systems

We consider a system in which the dynamics at any given time is governed by one of a fixed finite family of linear systems, and that switching between these is completely arbitrary. We show that asymptotic stability for all switching paths is equivalent to the existence of a common Lyapunov function for all systems in the family. Via an example it is shown that this Lyapunov function may fail to be quadratic.