Computation of Lyapunov Functions under State Constraints using Semidefinite Programming Hierarchies

Marianne Souaiby (1) , Aneel Tanwani (1) , Didier Henrion (1)

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande

We provide algorithms for computing a Lyapunov function for a class of systems where the state trajectories are constrained to evolve within a closed convex set. The dynamical systems that we consider comprise a differential equation which ensures continuous evolution within the domain, and a normal cone inclusion which ensures that the state trajectory remains within a prespecified set at all times. Finding a Lyapunov function for such a system boils down to finding a function which satisfies certain inequalities on the admissible set of state constraints. It is well-known that this problem, despite being convex, is computationally difficult. For conic constraints, we provide a discretization algorithm based on simplicial partitioning of a sim-plex, so that the search of desired function is addressed by constructing a hierarchy (associated with the diameter of the cells in the partition) of linear programs. Our second algorithm is tailored to semi-algebraic sets, where a hierarchy of semidefinite programs is constructed to compute Lyapunov functions as a sum-of-squares polynomial.

Mots clés: convex optimization; complementarity systems; copositive Lyapunov functions; Constrained systems

Domaines: Optimisation et contrôle [math.OC]

Marianne Souaiby, Aneel Tanwani, Didier Henrion. Computation of Lyapunov Functions under State Constraints using Semidefinite Programming Hierarchies. 21st IFAC World Congress, Jul 2020, Berlin virtual), Germany. ⟨10.1016/j.ifacol.2020.12.1746⟩. ⟨hal-02937714v2⟩