  This is a note on the extension of the Benamou-Brenier formulation of optimal
transport to nonlinear control affine systems on $\mathbb{R}^d$. They are the
non-compact version of the author and collaborators' previous result on compact
manifolds, stated here for the sake for completeness. Additionally, by using
Bernard's Young measure based weak formulation of optimal transport, the
results are established for cases not covered by previous works. 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
fluid formulation is established for general Sub-Riemmanian energy costs not
covered by literature previously. The results also provide controllability of
the continuity equation (using Borel measurable feedback laws) whenever the
corresponding Kantorovich problem has a feasible solution, due to the
established equivalence between the Kantorovich and Benamou-Brenier
formulation.
