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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Optimal transportation under nonholonomic constraints
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by Andrei Agrachev and Paul Lee PDF
Trans. Amer. Math. Soc. 361 (2009), 6019-6047 Request permission

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
  • MR Author ID: 190426
  • Email: agrachev@sissa.it
  • Paul Lee
  • Affiliation: Department of Mathematics, University of Toronto, Ontario, Canada M5S 2E4
  • Email: plee@math.toronto.edu
  • 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.
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 6019-6047
  • MSC (2000): Primary 49J20; Secondary 53C17
  • DOI: https://doi.org/10.1090/S0002-9947-09-04813-2
  • MathSciNet review: 2529923