A categorical view on the converse Lyapunov theorem

Abstract

In 1892, Lyapunov provided a fundamental contribution to stability theory by introducing so-called Lyapunov functions and Lyapunov equilibria. He subsequently showed that, for linear systems, the two concepts are equivalent. These concepts have since been extended to diverse types of dynamical systems, and in all settings the equivalence remains valid. However, this involves an often technical proof in each new setting where the concepts are introduced. In this article, we investigate a categorical framework where these results can be unified, exposing a single underlying reason for the equivalence to hold in all cases. First we define what is a dynamical system. Then we introduce the notion of a level-set morphism, which in turn allows us to define the concepts of a Lyapunov equilibrium and a Lyapunov function in a categorical setting. We conclude by a proof of their equivalence.