Lyapunov Stability of Coalgebras

Stability of an equilibrium point is a central notion in the study of dynamical systems. In practice, it can be infeasible to check the stability of an equilibrium directly. Lyapunov’s insight was that stability can instead be certified by the existence of an auxiliary object, a function which is positive definite relative to the equilibrium and decreases along the dynamics. Such a function is called a Lyapunov function for the system.

In this talk, we present a categorical formulation of Lyapunov’s theory expressed in the language of endofunctor coalgebras. Working internal to a locally thin bicategory equipped with a monoid object representing time, we define equilibria, morphisms that are positive definite relative to equilibria, and stability of equilibria in purely categorical terms [1]. Within this setting, Lyapunov’s theorem says that the existence of a positive definite lax coalgebra homomorphism implies stability of the equilibrium, and admits a simple proof via a pasting diagram argument [2].

The framework is sufficiently general to recover the classical Lyapunov theory for continuous-time dynamical systems, as well as its discrete-time counterpart. It also yields new Lyapunov-type stability results for labeled transition systems and Markov kernels. The theory extends to hybrid systems, in which continuous evolution interacts with discrete transitions [3]. We give an endofunctor on a category of charts in the sense of Myers [4] of which hybrid systems can be encoded as coalgebras. The framework subsumes several Lyapunov-type results for hybrid systems from the literature.