Section: Scientific Foundations

Optimal Transport

The study of optimal mass transport problems in the Euclidean or Riemannian setting has a long history which goes from the pioneer works of Monge  [65] and Kantorovitch  [57] to the recent revival initiated by fundamental contributions due to Brenier  [38] , [39] and McCann  [64] . However, the study of the same transportation problems in the presence of non-holonomic constraints –(like being an admissible trajectory for a control system– is quite new. The first contributors were Ambrosio and Rigot  [25] who proved the existence and uniqueness of an optimal transport map for the Monge problem associated with the squared canonical sub-Riemannian distance on the Heisenberg groups. This result was extended later by Agrachev and Lee  [22] , then by Figalli and Rifford  [46] who showed that the Ambrosio-Rigot theorem holds indeed true on many sub-Riemannian manifolds satisfying reasonable assumptions. The problem of existence and uniqueness of an optimal transport map for the squared sub-Riemannian distance on a general complete sub-Riemannian manifold remains open; it is strictly related to the regularity of the sub-Riemannian distance in the product space, and remains a formidable challenge. Generalized notions of Ricci curvatures (bounded from below) in metric spaces have been developed recently by Lott and Villani  [61] and Sturm  [79] , [80] . A pioneer work has been work in the Heisenberg group by Juillet  [54] who captured the right notion of curvature in this setting. Agrachev and Lee [23] have elaborated on this work to define new notions of curvatures in three dimensional sub-Riemannian structures. The optimal transport approach happened to be very fruitful in this context. Many things remain to do in a more general context.

One of the results of A. Hindawi's PhD under the supervision of L. Rifford and J.-B. Pomet was to extend regularity theory established in the Euclidean case to the more general quadratic costs associated with linear optimal control problems (LQR), see  [50] . This successful result opens a new range of optimal transport problems associated with cost coming from optimal control problems. We can nowadays expect regularity properties for optimal transport maps associated with reasonable optimal control problems with constraints on the state or on the velocities.

We believe that matching optimal transport with geometric control theory is one originality of our team. We expect interactions in both ways.