# American Institute of Mathematical Sciences

September  2013, 8(3): 627-648. doi: 10.3934/nhm.2013.8.627

## Existence of optima and equilibria for traffic flow on networks

 1 Department of Mathematics, Penn State University, University Park, Pa.16802 2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802

Received  August 2012 Revised  February 2013 Published  October 2013

This paper is concerned with a conservation law model of traffic flow on a network of roads, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. The model includes various groups of drivers, with different origins and destinations and having different cost functions. Under a natural set of assumptions, two main results are proved: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. In the case of Nash solutions, all departure rates are uniformly bounded and have compact support.
Citation: Alberto Bressan, Ke Han. Existence of optima and equilibria for traffic flow on networks. Networks & Heterogeneous Media, 2013, 8 (3) : 627-648. doi: 10.3934/nhm.2013.8.627
##### References:
 [1] N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow. I. Fluid dynamic and kinetic modeling,, Mathematical Models and Methods in Applied Sciences, 12 (2002), 1801.  doi: 10.1142/S0218202502002343.  Google Scholar [2] A. Bressan and K. Han, Optima and equilibria for a model of traffic flow,, SIAM Journal on Mathematical Analysis, 43 (2011), 2384.  doi: 10.1137/110825145.  Google Scholar [3] A. Bressan and K. Han, Nash equilibria for a model of traffic flow with several groups of drivers,, ESAIM, 18 (2012), 969.  doi: 10.1051/cocv/2011198.  Google Scholar [4] A. Bressan, C. J. Liu, W. Shen and F. Yu, Variational analysis of Nash equilibria for a model of traffic flow,, Quarterly of Applied Mathematics, 70 (2012), 495.  doi: 10.1090/S0033-569X-2012-01304-9.  Google Scholar [5] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM Journal on Mathematical Analysis, 36 (2005), 1862.  doi: 10.1137/S0036141004402683.  Google Scholar [6] C. Daganzo, "Fundamentals of Transportation and Traffic Operations,", Pergamon-Elsevier, (1997).   Google Scholar [7] L. C. Evans, "Partial Differential Equations,", $2^{nd}$ edition, (2010).   Google Scholar [8] T. L. Friesz, "Dynamic Optimization and Differential Games,", Springer, (2010).  doi: 10.1007/978-0-387-72778-3.  Google Scholar [9] T. L. Friesz, T. Kim, C. Kwon and M. A. Rigdon, Approximate network loading and dual-time-scale dynamic user equilibrium,, Transportation Research Part B, 45 (2011), 176.  doi: 10.1016/j.trb.2010.05.003.  Google Scholar [10] M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models,", AIMS Series on Applied Mathematics, (2006).   Google Scholar [11] M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks,, Journal of Optimization Theory and Applications, 126 (2005), 589.  doi: 10.1007/s10957-005-5499-z.  Google Scholar [12] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,'', Reprint of the 1980 original. Classics in Applied Mathematics, (1980).  doi: 10.1137/1.9780898719451.  Google Scholar [13] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317.  doi: 10.1098/rspa.1955.0089.  Google Scholar [14] P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar [15] J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1994).   Google Scholar [16] J. Wardrop, Some theoretical aspects of road traffic research,, in, 1 (1952), 325.  doi: 10.1680/ipeds.1952.11362.  Google Scholar

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##### References:
 [1] N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow. I. Fluid dynamic and kinetic modeling,, Mathematical Models and Methods in Applied Sciences, 12 (2002), 1801.  doi: 10.1142/S0218202502002343.  Google Scholar [2] A. Bressan and K. Han, Optima and equilibria for a model of traffic flow,, SIAM Journal on Mathematical Analysis, 43 (2011), 2384.  doi: 10.1137/110825145.  Google Scholar [3] A. Bressan and K. Han, Nash equilibria for a model of traffic flow with several groups of drivers,, ESAIM, 18 (2012), 969.  doi: 10.1051/cocv/2011198.  Google Scholar [4] A. Bressan, C. J. Liu, W. Shen and F. Yu, Variational analysis of Nash equilibria for a model of traffic flow,, Quarterly of Applied Mathematics, 70 (2012), 495.  doi: 10.1090/S0033-569X-2012-01304-9.  Google Scholar [5] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM Journal on Mathematical Analysis, 36 (2005), 1862.  doi: 10.1137/S0036141004402683.  Google Scholar [6] C. Daganzo, "Fundamentals of Transportation and Traffic Operations,", Pergamon-Elsevier, (1997).   Google Scholar [7] L. C. Evans, "Partial Differential Equations,", $2^{nd}$ edition, (2010).   Google Scholar [8] T. L. Friesz, "Dynamic Optimization and Differential Games,", Springer, (2010).  doi: 10.1007/978-0-387-72778-3.  Google Scholar [9] T. L. Friesz, T. Kim, C. Kwon and M. A. Rigdon, Approximate network loading and dual-time-scale dynamic user equilibrium,, Transportation Research Part B, 45 (2011), 176.  doi: 10.1016/j.trb.2010.05.003.  Google Scholar [10] M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models,", AIMS Series on Applied Mathematics, (2006).   Google Scholar [11] M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks,, Journal of Optimization Theory and Applications, 126 (2005), 589.  doi: 10.1007/s10957-005-5499-z.  Google Scholar [12] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,'', Reprint of the 1980 original. Classics in Applied Mathematics, (1980).  doi: 10.1137/1.9780898719451.  Google Scholar [13] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317.  doi: 10.1098/rspa.1955.0089.  Google Scholar [14] P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar [15] J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1994).   Google Scholar [16] J. Wardrop, Some theoretical aspects of road traffic research,, in, 1 (1952), 325.  doi: 10.1680/ipeds.1952.11362.  Google Scholar
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