We study Monge optimal transportation problem where the cost function is
given by the cost of an optimal control problem and try to understand what are natural regularity conditions which would allow to extend well-known
results of Brenier and McCann on the existence, uniqueness and
characterization of solutions. It happens, in particular, that optimal control approach works well for a wide class of Carnot-Caratheodory spaces with the cost given by the square of the distance. I'll also discuss further progress made by Figalli and Rifford for this case.