Synchronized traffic plans and stability of optima

Marc Bernot; Alessio Figalli

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

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

Abstract

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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.

How to cite

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Bernot, 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

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  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).  
  2. 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/
  3. M. Bernot, V. Caselles and J.-M. Morel, Traffic plans. Publ. Mat.49 (2005) 417–451.  
  4. 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
  5. A. Brancolini, G. Buttazzo and F. Santambrogio, Path functionals over Wasserstein spaces. J. EMS8 (2006) 414–434.  
  6. W. D'Arcy Thompson, On Growth and Form. Cambridge University Press (1942).  
  7. R.M. Dudley, Real Analysis and Probability. Cambridge University Press (2002).  
  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. USSR37 (1942) 7–8.  
  10. F. Maddalena, S. Solimini and J.M. Morel, A variational model of irrigation patterns. Interfaces and Free Boundaries5 (2003) 391–416.  
  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 Texts19. Springer (1993).  
  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 Mathematics58. American Mathematical Society, Providence, RI (2003).  
  15. Q. Xia, Optimal paths related to transport problems. Commun. Contemp. Math.5 (2003) 251–279.  

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