Hybrid Systems as Coalgebras: Lyapunov Morphisms for Zeno Stability
This work addresses the theoretical unification of stability analysis for hybrid systems, which is incremental in providing a categorical framework but does not solve a broad practical problem.
The paper tackles the problem of unifying stability analysis for hybrid dynamical systems by showing that various Lyapunov-like stability notions can be expressed as morphisms into a target system, with different choices corresponding to stability types like Zeno stability. It results in new Lyapunov-like conditions for Zeno equilibria and behavior, though no concrete numerical results are provided.
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 $Ï$, and different stability notions such as Lyapunov stability, asymptotic stability, exponential stability, and Zeno stability correspond to different choices of $Ï$. 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.