Kleene Algebra to Compute Invariant Sets of Dynamical Systems

Thomas Le Mézo (1, 2) , Luc Jaulin (1, 2) , Damien Massé (2, 3) , Benoit Zerr (1, 2)

1 ENSTA Bretagne - École Nationale Supérieure de Techniques Avancées Bretagne
2 Lab-STICC_ROBEX - Equipe ROBotics for EXploration
3 UBO - Université de Brest

In this paper, we show that a basic fixed point method used to enclose the greatest fixed point in a Kleene algebra will allow us to compute inner and outer approximations of invariant-based sets for continuous-time nonlinear dynamical systems. Our contribution is to provide the definitions and theorems that will allow us to make the link between the theory of invariant sets and the Kleene algebra. This link has never be done before and will allow us to compute rigorously sets that can be defined as a combination of positive invariant sets. Some illustrating examples show the nice properties of the approach.

Mots clés: invariant sets; Kleene algebra; nonlinear dynamical systems; path planning

Domaines: Automatique / Robotique; Géométrie algébrique [math.AG]; Systèmes dynamiques [math.DS]

Thomas Le Mézo, Luc Jaulin, Damien Massé, Benoit Zerr. Kleene Algebra to Compute Invariant Sets of Dynamical Systems. Algorithms, 2022, 15 (3), pp.90. ⟨10.3390/a15030090⟩. ⟨hal-03648084⟩