 
Summary: OPTIMAL TRANSPORTATION UNDER
NONHOLONOMIC CONSTRAINTS
ANDREI AGRACHEV AND PAUL LEE
Abstract. We study Monge's optimal transportation problem,
where the cost is given by optimal control cost. We prove the ex
istence 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 2generating distri
bution and cost given by d2
, where d is the subriemannian distance.
Also, we discuss some properties of the optimal plan when abnor
mal minimizers are present. Finally, we consider some examples of
displacement interpolation in the case of Grushin plane.
1. Introduction
Let (X, µ), (Y, ) be probability spaces and let c : X × Y R
{+} be a fixed measurable function. The Monge's optimal trans
portation problem is the minimization of the following functional
X
