|
Optimal transportation under nonholonomic constraints
Author(s):
Andrei
Agrachev;
Paul
Lee
Journal:
Trans. Amer. Math. Soc.
361
(2009),
6019-6047.
MSC (2000):
Primary 49J20;
Secondary 53C17
Posted:
June 15, 2009
MathSciNet review:
2529923
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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  , where  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.
References:
-
- 1.
- L. Ambrosio, S. Rigot: Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208(2004), 261-301 MR 2035027 (2004m:49107)
- 2.
- A. A. Agrachev: Geometry of Optimal Control Problems and Hamiltonian Systems, Lecture Notes, 2004
- 3.
- A. A. Agrachev, J. P. Gauthier, On the subanalyticity of Carnot-Carathéodory distances, Ann. Inst. H. Poincaré Anal. Non Linéaire 18, (2001), 359-382 MR 1831660 (2002h:93031)
- 4.
- A. A. Agrachev, Y. L. Sachkov: Control Theory from the Geometric Viewpoint, Springer, 2004 MR 2062547 (2005b:93002)
- 5.
- A. A. Agrachev, A. V. Sarychev: Strong minimality of abnormal geodesics for 2-distributions, J. Dynam. Control Systems, 1995, v.1, 139-176 MR 1333769 (96d:58022)
- 6.
- A. A. Agrachev, A. V. Sarychev: Abnormal sub-Riemannian geodesics, Morse index and rigidity, Annales de l'Institut Henri Poincaré Analyse Non Linéaire, v.13, 1996, 635-690 MR 1420493 (98e:49047)
- 7.
- P. Bernard, B. Buffoni: Optimal mass transportation and Mather theory, J. Eur. Math. Soc., 9 (2007), no. 1, 85-121 MR 2283105 (2007j:49050)
- 8.
- Y. Brenier: Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44, 4(1991), 375-417 MR 1100809 (92d:46088)
- 9.
- P. Cannarsa, L. Rifford: Semiconcavity results for optimal control problems admitting no singular minimizing controls, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 773-802 MR 2436793
- 10.
- P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkhaüser Boston, 2004 MR 2041617 (2005e:49001)
- 11.
- A. Fathi, A. Figalli: Optimal transportation on non-compact manifolds, preprint
- 12.
- A. Figalli: Existence, uniqueness and regularity of optimal transport maps, SIAM J. Math. Anal. 39 (2007), 126-137 MR 2318378
- 13.
- R. V. Gamkrelidze: Principles of Optimal Control Theory, Plenum Publishing Corporation, New York, 1978 MR 0686793 (58:33350c)
- 14.
- V. Jurdjevic: Geometric control theory, Cambridge Studies in Advanced Mathematics, 52. Cambridge University Press, Cambridge, 1997 MR 1425878 (98a:93002)
- 15.
- L. Kantorovich: On the translocation of masses, C.R. (Doklady) Acad. Sci. URSS(N.S.), 37, 1942, 199-201 MR 0009619 (5:174d)
- 16.
- W. S. Liu, H. J. Sussmann: Shortest paths for sub-Riemannian metrics on rank-2 distributions, Memoirs Amer. Math. Soc., v.118, N. 564, 1995 MR 1303093 (96c:53061)
- 17.
- J. Marsden, T. Ratiu: Introduction to Mechanics and Symmetry, Springer, 1999 MR 1723696 (2000i:70002)
- 18.
- R. McCann: Polar factorization of maps in Riemannian manifolds, Geometric and Functional Analysis, Vol.11, 2001 MR 1844080 (2002g:58017)
- 19.
- R. Montgomery: Abnormal Minimizers , SIAM J. Control and Optimization, vol. 32, no. 6, 1994, 1605-1620. MR 1297101 (95g:49006)
- 20.
- R. Montgomery: A tour of subriemannian geometries, their geodesics and applications, Math. Surveys and Monographs, vol. 91, Amer. Math. Soc., 2002 MR 1867362 (2002m:53045)
- 21.
- A. Sarychev, D. Torres: Lipschitzian regularity conditions for the minimizing trajectories of optimal control problems, Nonlinear analysis and its applications to differential equations (Lisbon, 1998), 357-368, Progr. Nonlinear Differential Equations Appl., 43, Birkhäuser Boston, Boston, MA, 2001 MR 1800636 (2001j:49062)
- 22.
- H. J. Sussmann: A Cornucopia of Abnormal Sub-Riemannian Minimizers. Part I: The Four dimensional Case, IMA technical report no. 1073, December, 1992
- 23.
- C. Villani: Topics in Optimal Transportation, Graduate Studies in Mathematics 58, AMS, Providence, 2003 MR 1964483 (2004e:90003)
- 24.
- C. Villani: Optimal transport: Old and new, preprint, http://www.umpa.ens-lyon.fr/
 cvillani/Cedrif/B07B.StFlour.pdf
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2000):
49J20,
53C17
Retrieve articles in all Journals with
MSC (2000):
49J20,
53C17
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:
10.1090/S0002-9947-09-04813-2
PII:
S 0002-9947(09)04813-2
Received by editor(s):
November 27, 2007
Posted:
June 15, 2009
Additional Notes:
The authors were supported by PRIN (first author) and NSERC (second author) grants.
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
