Transport maps as flows of control-affine systems

TL;DR

This paper explores how control-affine systems can be used to realize optimal transport maps between probability measures through time-varying feedback controls, under certain controllability conditions.

Contribution

It demonstrates that controllability of driftless control-affine systems guarantees the existence of feedback controls that produce optimal transport maps.

Findings

Controllability implies existence of optimal transport flows.

Feedback controls can realize optimal transport maps.

Conditions for regularity and controllability are established.

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.