Categorical Lyapunov Theory II: Stability of Systems

TL;DR

This paper extends Lyapunov stability theory to categorical frameworks for F-coalgebras, providing new definitions and theorems that generalize classical stability concepts to a broader mathematical setting.

Contribution

It introduces two categorical definitions of stability for F-coalgebras and proves Lyapunov and converse Lyapunov theorems within this abstract framework.

Findings

Established Lyapunov theorems for both categorical stability notions.

Developed a richer stability theory with a converse Lyapunov theorem.

Bridged classical stability analysis with categorical coalgebra frameworks.

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.