Information, Geometry, and Physics Seminar

Lyapunov functions characterize stability for dynamical systems, and Lyapunov analysis forms the foundations for modern nonlinear control. We present a categorical framework for Lyapunov theory, generalizing stability analysis through Lyapunov functions. Core to our approach is the 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 stability, formulate Lyapunov morphisms, and demonstrate that the existence of Lyapunov morphisms imply stability. This is applied to systems, framed as coalgebras, wherein we recover the classic Lyapunov conditions for stability and asymptotic stability.