Optimal transportation under nonholonomic constraints

Agrachev, Andrei; Lee, Paul

doi:10.1090/S0002-9947-09-04813-2

Optimal transportation under nonholonomic constraints
HTML articles powered by AMS MathViewer

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.

References

L. Ambrosio and S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208 (2004), no. 2, 261–301. MR 2035027, DOI 10.1016/S0022-1236(03)00019-3
A. A. Agrachev: Geometry of Optimal Control Problems and Hamiltonian Systems, Lecture Notes, 2004
Andrei Agrachev and Jean-Paul Gauthier, On the subanalyticity of Carnot-Caratheodory distances, Ann. Inst. H. Poincaré C Anal. Non Linéaire 18 (2001), no. 3, 359–382. MR 1831660, DOI 10.1016/S0294-1449(00)00064-0
Andrei A. Agrachev and Yuri L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences, vol. 87, Springer-Verlag, Berlin, 2004. Control Theory and Optimization, II. MR 2062547, DOI 10.1007/978-3-662-06404-7
A. A. Agrachëv and A. V. Sarychev, Strong minimality of abnormal geodesics for $2$-distributions, J. Dynam. Control Systems 1 (1995), no. 2, 139–176. MR 1333769, DOI 10.1007/BF02254637
A. A. Agrachev and A. V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 6, 635–690 (English, with English and French summaries). MR 1420493, DOI 10.1016/S0294-1449(16)30118-4
Patrick Bernard and Boris Buffoni, Optimal mass transportation and Mather theory, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 1, 85–121. MR 2283105, DOI 10.4171/JEMS/74
Yann Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375–417. MR 1100809, DOI 10.1002/cpa.3160440402
P. Cannarsa and L. Rifford, Semiconcavity results for optimal control problems admitting no singular minimizing controls, Ann. Inst. H. Poincaré C Anal. Non Linéaire 25 (2008), no. 4, 773–802. MR 2436793, DOI 10.1016/j.anihpc.2007.07.005
Piermarco Cannarsa and Carlo Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications, vol. 58, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2041617
A. Fathi, A. Figalli: Optimal transportation on non-compact manifolds, preprint
Alessio Figalli, Existence, uniqueness, and regularity of optimal transport maps, SIAM J. Math. Anal. 39 (2007), no. 1, 126–137. MR 2318378, DOI 10.1137/060665555
R. V. Gamkrelidze, Osnovy optimal′nogo upravleniya, Izdat. Tbilis. Univ., Tbilisi, 1975 (Russian). MR 0686791
Velimir Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics, vol. 52, Cambridge University Press, Cambridge, 1997. MR 1425878
L. Kantorovitch, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942), 199–201. MR 0009619
Wensheng Liu and Héctor J. Sussman, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564, x+104. MR 1303093, DOI 10.1090/memo/0564
Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, 2nd ed., Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1999. A basic exposition of classical mechanical systems. MR 1723696, DOI 10.1007/978-0-387-21792-5
Robert J. McCann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal. 11 (2001), no. 3, 589–608. MR 1844080, DOI 10.1007/PL00001679
Richard Montgomery, Abnormal minimizers, SIAM J. Control Optim. 32 (1994), no. 6, 1605–1620. MR 1297101, DOI 10.1137/S0363012993244945
Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
Andrei V. Sarychev and Delfim F. Marado Torres, Lipschitzian regularity conditions for the minimizing trajectories of optimal control problems, Nonlinear analysis and its applications to differential equations (Lisbon, 1998) Progr. Nonlinear Differential Equations Appl., vol. 43, Birkhäuser Boston, Boston, MA, 2001, pp. 357–368. MR 1800636
H. J. Sussmann: A Cornucopia of Abnormal Sub-Riemannian Minimizers. Part I: The Four dimensional Case, IMA technical report no. 1073, December, 1992
Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483, DOI 10.1090/gsm/058
C. Villani: Optimal transport: Old and new, preprint, http://www.umpa.ens-lyon.fr/ $^\sim$cvillani/Cedrif/B07B.StFlour.pdf