Optimal transportation under nonholonomic constraints

By Andrei A. Agrachev, Paul W. Y. Lee. 2007

92 citations recorded by the source corpus.

We study Monge's optimal transportation problem, where the cost is given by an 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 the Grushin plane.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/63d627fdadb225115a0e1c982c22e079e67e22f0

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Optimal transportation, topology and uniqueness (2010, background): “Our discussion is predicated on global differentiability of the cost, since a wide variety of existence and uniqueness results concerning optimal solutions to the Monge–Kantorovich problem have been established for costs with singular sets—including distances in Riemannian [28,46,77], sub-Riemannian”
Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds (2009, background): “To do this, we first recall the recent results on the theory of optimal transportation in [ 3 ] and [10].”