# Synchronized traffic plans and stability of optima

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 14, Issue: 4, page 864-878
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topBernot, Marc, and Figalli, Alessio. "Synchronized traffic plans and stability of optima." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 864-878. <http://eudml.org/doc/250321>.

@article{Bernot2008,

abstract = {
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.
},

author = {Bernot, Marc, Figalli, Alessio},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Irrigation problem; traffic plans; dynamical cost; stability; irrigation problem},

language = {eng},

month = {1},

number = {4},

pages = {864-878},

publisher = {EDP Sciences},

title = {Synchronized traffic plans and stability of optima},

url = {http://eudml.org/doc/250321},

volume = {14},

year = {2008},

}

TY - JOUR

AU - Bernot, Marc

AU - Figalli, Alessio

TI - Synchronized traffic plans and stability of optima

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/1//

PB - EDP Sciences

VL - 14

IS - 4

SP - 864

EP - 878

AB -
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.

LA - eng

KW - Irrigation problem; traffic plans; dynamical cost; stability; irrigation problem

UR - http://eudml.org/doc/250321

ER -

## References

top- 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). Zbl0957.49001
- M. Bernot, Irrigation and Optimal Transport. Ph.D. thesis, École Normale Supérieure de Cachan, France (2005). Available at ~mbernot. URIhttp://www.umpa.ens-lyon.fr/
- M. Bernot, V. Caselles and J.-M. Morel, Traffic plans. Publ. Mat.49 (2005) 417–451. Zbl1086.49029
- M. Bernot, V. Caselles and J.-M. Morel, The structure of branched transportation networks. Calc. Var. Partial Differential Equations (online first). DOI: . DOI10.1007/s00526-007-0139-0
- A. Brancolini, G. Buttazzo and F. Santambrogio, Path functionals over Wasserstein spaces. J. EMS8 (2006) 414–434. Zbl1130.49036
- W. D'Arcy Thompson, On Growth and Form. Cambridge University Press (1942).
- R.M. Dudley, Real Analysis and Probability. Cambridge University Press (2002). Zbl1023.60001
- E.N. Gilbert, Minimum cost communication networks. Bell System Tech. J.46 (1967) 2209–2227.
- L. Kantorovich, On the transfer of masses. Dokl. Acad. Nauk. USSR37 (1942) 7–8.
- F. Maddalena, S. Solimini and J.M. Morel, A variational model of irrigation patterns. Interfaces and Free Boundaries5 (2003) 391–416. Zbl1057.35076
- G. Monge, Mémoire sur la théorie des déblais et de remblais. Histoire de l'Académie Royale des Sciences de Paris (1781) 666–704.
- J.D. Murray, Mathematical Biology, Biomathematics Texts19. Springer (1993).
- A.M. Turing, The chemical basis of morphogenesis. Phil. Trans. Soc. Lond. B237 (1952) 37–72.
- C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics58. American Mathematical Society, Providence, RI (2003). Zbl1106.90001
- Q. Xia, Optimal paths related to transport problems. Commun. Contemp. Math.5 (2003) 251–279. Zbl1032.90003

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.