# 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 & 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.  doi: 10.1007/s10107-006-0076-2.  Google Scholar [2] M. Beckmann, C. McGuire and C. Winsten, "Studies in Economics of Transportation,", Yale University Press, (1956).   Google Scholar [3] F. Benmansour, G. Carlier, G. Peyré and F. Santambrogio, Numerical approximation of continuous traffic congestion equilibria,, Netw. Heterog. Media, 4 (2009), 605.   Google Scholar [4] A. Braides, "$\Gamma$-Convergence for Beginners,", Oxford Lecture Series in Mathematics and its Applications, 22 (2002).   Google Scholar [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.   Google Scholar [6] G. Carlier, C. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and Wardrop equilibria,, SIAM J. Control Optim., 47 (2008), 1330.  doi: 10.1137/060672832.  Google Scholar [7] G. Dal Maso, "An Introduction to $\Gamma-$Convergence,", Progress in Nonlinear Differential Equations and their Applications, 8 (1993).   Google Scholar [8] C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).   Google Scholar [9] J. G. Wardrop, Some theoretical aspects of road traffic research,, Proc. Inst. Civ. Eng., 2 (1952), 325.   Google Scholar

show all references

##### References:
 [1] J.-B. Baillon and R. Cominetti, Markovian traffic equilibrium,, Math. Prog., 111 (2008), 33.  doi: 10.1007/s10107-006-0076-2.  Google Scholar [2] M. Beckmann, C. McGuire and C. Winsten, "Studies in Economics of Transportation,", Yale University Press, (1956).   Google Scholar [3] F. Benmansour, G. Carlier, G. Peyré and F. Santambrogio, Numerical approximation of continuous traffic congestion equilibria,, Netw. Heterog. Media, 4 (2009), 605.   Google Scholar [4] A. Braides, "$\Gamma$-Convergence for Beginners,", Oxford Lecture Series in Mathematics and its Applications, 22 (2002).   Google Scholar [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.   Google Scholar [6] G. Carlier, C. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and Wardrop equilibria,, SIAM J. Control Optim., 47 (2008), 1330.  doi: 10.1137/060672832.  Google Scholar [7] G. Dal Maso, "An Introduction to $\Gamma-$Convergence,", Progress in Nonlinear Differential Equations and their Applications, 8 (1993).   Google Scholar [8] C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).   Google Scholar [9] J. G. Wardrop, Some theoretical aspects of road traffic research,, Proc. Inst. Civ. Eng., 2 (1952), 325.   Google Scholar
 [1] Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465 [2] Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170 [3] George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 [4] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 [5] Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 [6] Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303 [7] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [8] Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345 [9] Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105 [10] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [11] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [12] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [13] Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 [14] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [15] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [16] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [17] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [18] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [19] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454 [20] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

2019 Impact Factor: 1.053