Memory-Based Lyapunov Functions and Path-complete Framework: Equivalence and Properties

In this paper, in the framework of stability analysis of switched systems, we review and analyze multiple Lyapunov functions structures. We formalize and study a class of Lyapunov functions that do not only depend on the state, but also on the past switching sequence, the 'memory', in a general language-theory setting. We recall and extend an equivalence result between these stability criteria and a class of combinatorial Lyapunov techniques, also known as path-complete Lyapunov functions. We provide the dual results based on the knowledge/prediction of the future values of the switching signals and we illustrate our techniques via numerical examples.