Transport maps as flows of control-affine systems

Abstract: We consider the problem of transporting \nota{one probability measure into another through} the flow of a given driftless control-affine system. Under suitable regularity conditions, the controllability of the system by means of open-loop controls is a sufficient condition for the existence of time-varying feedback controls such that the $1$-time flow of the system is the optimal transport map for the quadratic cost.