  We study path-complete Lyapunov functions, which are stability criteria for
switched systems, described by a combinatorial component (namely, an
automaton), and a functional component (a set of candidate Lyapunov functions,
called the template). We introduce a class of criteria based on what we call
memory-based Lyapunov functions, which generalize several techniques in the
literature. Our main result is an equivalence result: any path-complete
Lyapunov function is equivalent to a memory-based Lyapunov function, however
defined on another template. We show the usefulness of our result in terms of
numerical efficiency via an academic example.
