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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Optimal transportation under nonholonomic constraints


Authors: Andrei Agrachev and Paul Lee
Journal: Trans. Amer. Math. Soc. 361 (2009), 6019-6047
MSC (2000): Primary 49J20; Secondary 53C17
Published electronically: June 15, 2009
MathSciNet review: 2529923
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Abstract: 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.


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Additional Information

Andrei Agrachev
Affiliation: Scuola Internazionale Superiore di Studi Avanzat, International School for Advanced Studies, Trieste, Italy and Steklov Mathematical Institute, ul. Gubkina 8, Moscow, 119991 Russia
Email: agrachev@sissa.it

Paul Lee
Affiliation: Department of Mathematics, University of Toronto, Ontario, Canada M5S 2E4
Email: plee@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04813-2
PII: S 0002-9947(09)04813-2
Received by editor(s): November 27, 2007
Published electronically: June 15, 2009
Additional Notes: The authors were supported by PRIN (first author) and NSERC (second author) grants.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.