Benamou-Brenier and Kantorovich on sub-Riemannian manifolds with no abnormal geodesics

We prove that the Benamou-Brenier formulation of the Optimal Transport problem and the Kantorovich formulation are equivalent on a sub-Riemannian connected and complete manifold $M$ without boundary and with no non-trivial abnormal geodesics, when the problems are considered between two measures with finite $2$-momentum. Furthermore, we prove the existence of a minimizer for the Benamou-Brenier formulation and link it to the optimal transport plan.