Categorical Lyapunov Theory I: Stability of Flows

TL;DR

This paper introduces a categorical framework for Lyapunov stability analysis, generalizing classical methods to abstract categorical settings and demonstrating its applicability to various types of dynamical systems.

Contribution

It develops a minimal axiomatic categorical approach to Lyapunov stability, unifying classical and enriched stability theories within a single framework.

Findings

Defines stability of equilibria categorically

Shows the necessity and sufficiency of Lyapunov morphisms for stability

Applies the framework to classical and enriched categories

Abstract

Lyapunov's theorem provides a fundamental characterization of the stability of dynamical systems. This paper presents a categorical framework for Lyapunov theory, generalizing stability analysis with Lyapunov functions categorically. Core to our approach is the set of axioms underlying a setting for stability, which give the necessary ingredients for ``doing Lyapunov theory'' in a category of interest. With these minimal assumptions, we define the stability of equilibria, formulate Lyapunov morphisms, and demonstrate that the existence of Lyapunov morphisms is necessary and sufficient for establishing the stability of flows. To illustrate these constructions, we show how classical notions of stability, e.g., for continuous and discrete time dynamical systems, are captured by this categorical framework for Lyapunov theory. Finally, to demonstrate the extensibility of our framework, we illustrate how enriched categories, e.g., Lawvere metric spaces, yield settings for stability enabling one to ``do Lyapunov theory'' in enriched categories.