Section: New Results

Applications of control methods to dynamical systems

Participants : Gonzalo Contreras, Alessio Figalli, Ayadi Lazrag, Ludovic Rifford, Raffael Ruggiero.

Ludovic Rifford and collaborators have been applying with success, techniques from geometric control theory to open problems in dynamical systems, mostly on genericity properties and using controllability methods to build suitable perturbations.

This has been applied to closing geodesics [64] . Ayadi Lazrag's PhD also deals with such problems; applying techniques close to these in [65] , one goal is to establish a version of Francks' lemma for geodesic flows and to apply this to persitence problems. The approach relies on control theory results, with order 2 conditions. See [18] and another preprint (“Franks' lemma for $C^2$-Mañé perturbations of Riemannian metrics and applications to persistence” by Lazrag, Rifford and Ruggiero, available from the authors).

In [17] , a non trivial conjecture on generic hyperbolicity of the so-called Aubry set of a Hamiltonian is solved on compact surfaces and in the $C^2$ topology (for genericity).