Benamou-Brenier Formulation of Optimal Transport for Nonlinear Control Systems on Rd

In this paper we consider the Benamou-Brenier formulation of optimal transport for nonlinear control affine systems on $\Rd$, removing the compactness assumption of the underlying manifold in previous work by the author. By using Bernard's Young measure based weak formulation of optimal transport, the results are established for cases not covered by previous treatments using the Monge problem. Particularly, no assumptions are made on the non-existence of singular minimizing controls or the cost function being Lipschitz. Therefore, the existence of solutions to dynamical formulation is established for general Sub-Riemmanian energy costs not covered by literature previously. The results also establish controllability of the continuity equation whenever the corresponding Kantorovich problem admits a feasible solution, leveraging the equivalence between the Kantorovich and Benamou-Brenier formulations. Furthermore, when the cost function does not admit singular minimizing curves, we demonstrate that the Benamou-Brenier problem is equivalent to its convexified formulation in momentum and measure coordinates. In this regular setting, we further show that the constructed transport solutions possess sufficient regularity: for the feedback control laws that achieve transport, the associated continuity equation admits a unique weak solution. These findings apply in particular to linear-quadratic costs for controllable linear time-invariant (LTI) systems, as well as to certain classes of driftless nonlinear systems. Thus, in these cases, controllability of the continuity equation is achieved with control laws regular enough to guarantee uniqueness of solutions.