From discrete to continuous Wardrop equilibria

Jean-Bernard Baillon; Guillaume Carlier; Jean-Bernard Baillon; Guillaume Carlier

doi:10.3934/nhm.2012.7.219

Networks and Heterogeneous Media

2012, Volume 7, Issue 2: 219-241. doi: 10.3934/nhm.2012.7.219

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From discrete to continuous Wardrop equilibria

Jean-Bernard Baillon ^{1
,},
Guillaume Carlier ^{2
,}

1.
Laboratoire Marin Mersenne, Université Paris I, 90 rue de Tolbiac, 75013, Paris
2.
CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16

Received: 01 November 2011 Revised: 01 March 2012
Primary: 90C46, 90C35; Secondary: 49N15.

The notion of Wardrop equilibrium in congested networks has been very popular in congested traffic modelling since its introduction in the early 50's, it is also well-known that Wardrop equilibria may be obtained by some convex minimization problem. In this paper, in the framework of $\Gamma$-convergence theory, we analyze what happens when a cartesian network becomes very dense. The continuous model we obtain this way is very similar to the continuous model of optimal transport with congestion of Carlier, Jimenez and Santambrogio [6] except that it keeps track of the anisotropy of the network.
- Wardrop equilibria,
- traffic congestion,
- $\Gamma$-convergence,
- eikonal equation
Citation: Jean-Bernard Baillon, Guillaume Carlier. From discrete to continuous Wardrop equilibria[J]. Networks and Heterogeneous Media, 2012, 7(2): 219-241. doi: 10.3934/nhm.2012.7.219

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Abstract

The notion of Wardrop equilibrium in congested networks has been very popular in congested traffic modelling since its introduction in the early 50's, it is also well-known that Wardrop equilibria may be obtained by some convex minimization problem. In this paper, in the framework of $\Gamma$-convergence theory, we analyze what happens when a cartesian network becomes very dense. The continuous model we obtain this way is very similar to the continuous model of optimal transport with congestion of Carlier, Jimenez and Santambrogio [6] except that it keeps track of the anisotropy of the network.

References

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[2]	M. Beckmann, C. McGuire and C. Winsten, "Studies in Economics of Transportation," Yale University Press, New Haven, 1956.
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[4]	A. Braides, "$\Gamma$-Convergence for Beginners," Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002.
[5]	L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 652-671.
[6]	G. Carlier, C. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and Wardrop equilibria, SIAM J. Control Optim., 47 (2008), 1330-1350. doi: 10.1137/060672832
[7]	G. Dal Maso, "An Introduction to $\Gamma-$Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993.
[8]	C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.
[9]	J. G. Wardrop, Some theoretical aspects of road traffic research, Proc. Inst. Civ. Eng., 2 (1952), 325-378.

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