Synchronized traffic plans and stability of optima

Bernot, Marc; Figalli, Alessio

doi:10.1051/cocv:2008012

Bernot, Marc ; Figalli, Alessio

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 864-878.

Résumé

The irrigation problem is the problem of finding an efficient way to transport a measure $μ^{+}$ onto a measure $μ^{-}$ . 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.

MR Zbl

DOI : 10.1051/cocv:2008012

Classification : 49Q20, 90B10, 90B06, 90B20
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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/cocv:2008012/

Bibliographie
Cité par

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variations and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press (2000). | MR | Zbl

[2] M. Bernot, Irrigation and Optimal Transport. Ph.D. thesis, École Normale Supérieure de Cachan, France (2005). Available at http://www.umpa.ens-lyon.fr/ $\sim$ mbernot.

[3] M. Bernot, V. Caselles and J.-M. Morel, Traffic plans. Publ. Mat. 49 (2005) 417-451. | MR | Zbl

[4] M. Bernot, V. Caselles and J.-M. Morel, The structure of branched transportation networks. Calc. Var. Partial Differential Equations (online first). DOI: 10.1007/s00526-007-0139-0. | MR | Zbl

[5] A. Brancolini, G. Buttazzo and F. Santambrogio, Path functionals over Wasserstein spaces. J. EMS 8 (2006) 414-434. | MR | Zbl

[6] W. D'Arcy Thompson, On Growth and Form. Cambridge University Press (1942). | MR | Zbl

[7] R.M. Dudley, Real Analysis and Probability. Cambridge University Press (2002). | MR | Zbl

[8] E.N. Gilbert, Minimum cost communication networks. Bell System Tech. J. 46 (1967) 2209-2227.

[9] L. Kantorovich, On the transfer of masses. Dokl. Acad. Nauk. USSR 37 (1942) 7-8.

[10] F. Maddalena, S. Solimini and J.M. Morel, A variational model of irrigation patterns. Interfaces and Free Boundaries 5 (2003) 391-416. | MR | Zbl

[11] G. Monge, Mémoire sur la théorie des déblais et de remblais. Histoire de l'Académie Royale des Sciences de Paris (1781) 666-704.

[12] J.D. Murray, Mathematical Biology, Biomathematics Texts 19. Springer (1993). | MR | Zbl

[13] A.M. Turing, The chemical basis of morphogenesis. Phil. Trans. Soc. Lond. B237 (1952) 37-72.

[14] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence, RI (2003). | MR | Zbl

[15] Q. Xia, Optimal paths related to transport problems. Commun. Contemp. Math. 5 (2003) 251-279. | MR | Zbl

Cité par Sources :