From discrete to continuous Wardrop equilibria

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June 2012, 7(2): 219-241. doi: 10.3934/nhm.2012.7.219

From discrete to continuous Wardrop equilibria

Jean-Bernard Baillon ^1, and Guillaume Carlier ^2,

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

Received November 2011 Revised March 2012 Published June 2012

Abstract
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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.

Keywords: $\Gamma$-convergence, eikonal equation., Wardrop equilibria, traffic congestion.

Mathematics Subject Classification: Primary: 90C46, 90C35; Secondary: 49N1.

Citation: Jean-Bernard Baillon, Guillaume Carlier. From discrete to continuous Wardrop equilibria. Networks and Heterogeneous Media, 2012, 7 (2) : 219-241. doi: 10.3934/nhm.2012.7.219

References:

[1]	J.-B. Baillon and R. Cominetti, Markovian traffic equilibrium, Math. Prog., 111 (2008), 33-56. doi: 10.1007/s10107-006-0076-2.
[2]	M. Beckmann, C. McGuire and C. Winsten, "Studies in Economics of Transportation," Yale University Press, New Haven, 1956.
[3]	F. Benmansour, G. Carlier, G. Peyré and F. Santambrogio, Numerical approximation of continuous traffic congestion equilibria, Netw. Heterog. Media, 4 (2009), 605-623.
[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.

show all references

References:

[1]	J.-B. Baillon and R. Cominetti, Markovian traffic equilibrium, Math. Prog., 111 (2008), 33-56. doi: 10.1007/s10107-006-0076-2.
[2]	M. Beckmann, C. McGuire and C. Winsten, "Studies in Economics of Transportation," Yale University Press, New Haven, 1956.
[3]	F. Benmansour, G. Carlier, G. Peyré and F. Santambrogio, Numerical approximation of continuous traffic congestion equilibria, Netw. Heterog. Media, 4 (2009), 605-623.
[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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