In the beginning of the 2000's, J. D. Benamou and Y. Brenier have proposed a dynamical formulation of the optimal transport problem, corresponding to the time-space search of a density and a momentum minimizing a transport energy between two densities. The authors then proposed to solve this problem by using an augmented Lagrangian algorithm. We will study the convergence of this algorithm, in the most general conditions, particularly in cases where initial and final densities cancel on some areas of the transportation domain. The principal difficulty of our work consist in the proof, under these conditions, of the existence of a saddle point, and especially in the uniqueness of the density-momentum component. Indeed, these conditions imply to have to deal with non-regular optimal transportation maps, what requires a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space is required.