# American Institute of Mathematical Sciences

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

## From discrete to continuous Wardrop equilibria

 1 Laboratoire Marin Mersenne, Université 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

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.
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.

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##### 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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