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 and weak-KAM theory [39] , [38] .

Ayadi Lazrag's PhD also deals with such problems; applying techniques close to these in [61] , he established a version of Francks' lemma for geodesic flows; one goal is to apply this to persistence problems. The approach relies on control theory results, with order 2 conditions. See [14] and [15] , where 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).