  We study stability criteria for discrete-time switched systems and provide a
meta-theorem that characterizes all Lyapunov theorems of a certain canonical
type. For this purpose, we investigate the structure of sets of LMIs that
provide a sufficient condition for stability. Various such conditions have been
proposed in the literature in the past fifteen years. We prove in this note
that a family of languagetheoretic conditions recently provided by the authors
encapsulates all the possible LMI conditions, thus putting a conclusion to this
research effort. As a corollary, we show that it is PSPACE-complete to
recognize whether a particular set of LMIs implies stability of a switched
system. Finally, we provide a geometric interpretation of these conditions, in
terms of existence of an invariant set.
