Hybrid Systems as Coalgebras: Lyapunov Morphisms for Zeno Stability

TL;DR

This paper unifies various stability results for hybrid systems through a categorical framework, introducing Lyapunov morphisms as coalgebra homomorphisms to analyze Zeno stability.

Contribution

It presents a novel categorical approach to hybrid systems, deriving new Lyapunov-like conditions for Zeno stability via coalgebra morphisms.

Findings

Unified Lyapunov framework for hybrid systems

New conditions for Zeno equilibrium stability

Expressed hybrid systems as coalgebras of an endofunctor

Abstract

Hybrid dynamical systems exhibit a diverse array of stability phenomena, each currently addressed by separate Lyapunov-like results. We show that these results are all instances of a single theorem: a Lyapunov function is a morphism from a hybrid system into a simple stable target system $\sigma$, and different stability notions such as Lyapunov stability, asymptotic stability, exponential stability, and Zeno stability correspond to different choices of $\sigma$. This unification is achieved by expressing hybrid systems as coalgebras of an endofunctor $\mathcal H$ on a category $\mathsf{Chart}$ that naturally blends continuous and discrete dynamics. Instantiating a general categorical Lyapunov theorem for coalgebras to this setting results in new Lypaunov-like conditions for the stability of Zeno equilibria and the existence of Zeno behavior in hybrid systems.