Hybrid Systems as Coalgebras: Lyapunov Morphisms for Zeno Stability
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914450854903808 |
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| author | Moeller, Joe Ames, Aaron D. |
| author_facet | Moeller, Joe Ames, Aaron D. |
| contents | 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 $σ$, and different stability notions such as Lyapunov stability, asymptotic stability, exponential stability, and Zeno stability correspond to different choices of $σ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05255 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hybrid Systems as Coalgebras: Lyapunov Morphisms for Zeno Stability Moeller, Joe Ames, Aaron D. Category Theory Systems and Control Dynamical Systems 18M35, 18B20, 93C30, 34A38 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 $σ$, and different stability notions such as Lyapunov stability, asymptotic stability, exponential stability, and Zeno stability correspond to different choices of $σ$. 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. |
| title | Hybrid Systems as Coalgebras: Lyapunov Morphisms for Zeno Stability |
| topic | Category Theory Systems and Control Dynamical Systems 18M35, 18B20, 93C30, 34A38 |
| url | https://arxiv.org/abs/2604.05255 |