The irrigation problem is the problem of finding an efficient way to transport a measure onto a measure . By efficient, we mean that a structure that achieves the transport (which, following [Bernot, Caselles and Morel, Publ. Mat. 49 (2005) 417-451], we call traffic plan) is better if it carries the mass in a grouped way rather than in a separate way. This is formalized by considering costs functionals that favorize this property. The aim of this paper is to introduce a dynamical cost functional on traffic plans that we argue to be more realistic. The existence of minimizers is proved in two ways: in some cases, we can deduce it from a classical semicontinuity argument; the other cases are treated by studying the link between our cost and the one introduced in [Bernot, Caselles and Morel, Publ. Mat. 49 (2005) 417-451]. Finally, we discuss the stability of minimizers with respect to specific variations of the cost functional.
Mots-clés : irrigation problem, traffic plans, dynamical cost, stability
@article{COCV_2008__14_4_864_0, author = {Bernot, Marc and Figalli, Alessio}, title = {Synchronized traffic plans and stability of optima}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {864--878}, publisher = {EDP-Sciences}, volume = {14}, number = {4}, year = {2008}, doi = {10.1051/cocv:2008012}, mrnumber = {2451800}, zbl = {1148.49039}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2008012/} }
TY - JOUR AU - Bernot, Marc AU - Figalli, Alessio TI - Synchronized traffic plans and stability of optima JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 864 EP - 878 VL - 14 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008012/ DO - 10.1051/cocv:2008012 LA - en ID - COCV_2008__14_4_864_0 ER -
%0 Journal Article %A Bernot, Marc %A Figalli, Alessio %T Synchronized traffic plans and stability of optima %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 864-878 %V 14 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008012/ %R 10.1051/cocv:2008012 %G en %F COCV_2008__14_4_864_0
Bernot, Marc; Figalli, Alessio. Synchronized traffic plans and stability of optima. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 864-878. doi : 10.1051/cocv:2008012. http://www.numdam.org/articles/10.1051/cocv:2008012/
[1] Functions of bounded variations and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press (2000). | MR | Zbl
, and ,[2] Irrigation and Optimal Transport. Ph.D. thesis, École Normale Supérieure de Cachan, France (2005). Available at http://www.umpa.ens-lyon.fr/ mbernot.
,[3] Traffic plans. Publ. Mat. 49 (2005) 417-451. | MR | Zbl
, and ,[4] The structure of branched transportation networks. Calc. Var. Partial Differential Equations (online first). DOI: 10.1007/s00526-007-0139-0. | MR | Zbl
, and ,[5] Path functionals over Wasserstein spaces. J. EMS 8 (2006) 414-434. | MR | Zbl
, and ,[6] On Growth and Form. Cambridge University Press (1942). | MR | Zbl
,[7] Real Analysis and Probability. Cambridge University Press (2002). | MR | Zbl
,[8] Minimum cost communication networks. Bell System Tech. J. 46 (1967) 2209-2227.
,[9] On the transfer of masses. Dokl. Acad. Nauk. USSR 37 (1942) 7-8.
,[10] A variational model of irrigation patterns. Interfaces and Free Boundaries 5 (2003) 391-416. | MR | Zbl
, and ,[11] Mémoire sur la théorie des déblais et de remblais. Histoire de l'Académie Royale des Sciences de Paris (1781) 666-704.
,[12] Biomathematics Texts 19. Springer (1993). | MR | Zbl
, ,[13] The chemical basis of morphogenesis. Phil. Trans. Soc. Lond. B237 (1952) 37-72.
,[14] Topics in optimal transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence, RI (2003). | MR | Zbl
,[15] Optimal paths related to transport problems. Commun. Contemp. Math. 5 (2003) 251-279. | MR | Zbl
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