# 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

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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

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- F. Maddalena, S. Solimini and J.M. Morel, A variational model of irrigation patterns. Interfaces and Free Boundaries5 (2003) 391–416.
- 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).
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