Categorical Lyapunov Theory II: Stability of Systems
Aaron D. Ames, Sébastien Mattenet, Joe Moeller
TL;DR
This work generalizes Lyapunov stability to coalgebraic systems by formulating two stability notions for $\mathcal{F}$-systems and proving Lyapunov and Converse Lyapunov theorems within a unified categorical framework. It builds on CLT1 by recasting dynamics as $\mathcal{F}$-coalgebras, develops a trajectory-based Lyapunov theory for $\mathcal{F}$-systems, and then enhances the theory with monoidal settings to obtain a richer, point-free perspective. A central contribution is the existence of an integral functor $\int$ that pairs with a derivative functor $\mathrm D$ to provide an equivalence between $T$-complete systems and $T$-flows, under suitable completeness assumptions; this yields existence/uniqueness results and a monadic structure. The paper culminates with a Converse Lyapunov result under additional axioms, establishing necessary and sufficient conditions for stability in the coalgebraic setting and connecting trajectory stability to system stability in a broad, abstract context. These results unify point-based and generalized-element stability notions across varied concrete instances, including flows on manifolds, graphs, and Markov kernels, highlighting the practical impact for systematic stability analysis in heterogeneous dynamical models.
Abstract
Lyapunov's theorem provides a foundational characterization of stable equilibrium points in dynamical systems. In this paper, we develop a framework for stability for F-coalgebras. We give two definitions for a categorical setting in which we can study the stability of a coalgebra for an endofunctor F. One is minimal and better suited for concrete settings, while the other is more intricate and provides a richer theory. We prove a Lyapunov theorem for both notions of setting for stability, and a converse Lyapunov theorem for the second.