Optimal Transportation under Nonholonomic Constraints

TL;DR

This paper investigates Monge's optimal transportation problem with costs derived from optimal control, establishing existence and uniqueness of solutions under specific conditions, especially in subriemannian geometries.

Contribution

It proves the existence and uniqueness of optimal maps for control-based costs under regularity and non-abnormal minimizer conditions, extending to subriemannian manifolds.

Findings

Existence and uniqueness of optimal maps under certain conditions

Application to subriemannian manifolds with 2-generating distributions

Analysis of properties of optimal plans with abnormal minimizers

Abstract

We study Monge's optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantly the absence of sharp abnormal minimizers. In particular, this result is applicable in the case of subriemannian manifolds with a 2-generating distribution and cost given by $d^2$, where $d$ is the subriemannian distance. Also, we discuss some properties of the optimal plan when abnormal minimizers are present. Finally, we consider some examples of displacement interpolation in the case of Grushin plane.