Almost Everywhere Conditions for Hybrid Lipschitz Lyapunov Functions

Matteo Della Rossa (1) , Rafal Goebel (2) , Aneel Tanwani (1) , Luca Zaccarian (3, 1)

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
2 Loyola University of Chicago
3 Dipartimento di Ingegneria Industriale [Trento]

We introduce a class of locally Lipschitz continuous functions to establish stability of hybrid dynamical systems. Under certain regularity assumptions on system dynamics, we provide sufficient conditions for asymptotic stability on the candidate Lyapunov function. In contrast to the existing literature, these conditions need to be checked only on a dense set using the conventional gradient of certain functions, without the necessity of relying on Clarke's generalized gradient. We discuss the relevance of the stated assumptions with the help of some counterexamples, underlining the subtlety of the proposed relaxation. As an application of our result, we study the stability of a classical example from the reset control literature: the Clegg integrator model, with convex and nonconvex Lyapunov functions, which are almost everywhere differentiable.

Domaines: Optimisation et contrôle [math.OC]; Automatique / Robotique

Matteo Della Rossa, Rafal Goebel, Aneel Tanwani, Luca Zaccarian. Almost Everywhere Conditions for Hybrid Lipschitz Lyapunov Functions. 58th IEEE Conference on Decision and Control (CDC 2019), Dec 2019, Nice, France. ⟨hal-02417585⟩