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 and 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), 1801-1843. doi: 10.1142/S0218202502002343.

[2]

A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM Journal on Mathematical Analysis, 43 (2011), 2384-2417. doi: 10.1137/110825145.

[3]

A. Bressan and K. Han, Nash equilibria for a model of traffic flow with several groups of drivers, ESAIM, Control, Optimization and Calculus of Variations, 18 (2012), 969-986. doi: 10.1051/cocv/2011198.

[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), 495-515. doi: 10.1090/S0033-569X-2012-01304-9.

[5]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM Journal on Mathematical Analysis, 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.

[6]

C. Daganzo, "Fundamentals of Transportation and Traffic Operations," Pergamon-Elsevier, Oxford, U. K., 1997.

[7]

L. C. Evans, "Partial Differential Equations," $2^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.

[8]

T. L. Friesz, "Dynamic Optimization and Differential Games," Springer, New York, 2010. doi: 10.1007/978-0-387-72778-3.

[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), 176-207. doi: 10.1016/j.trb.2010.05.003.

[10]

M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models," AIMS Series on Applied Mathematics, 1. American Institute of Mathematical Sciences (AIMS), Springfield, Mo., 2006.

[11]

M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, Journal of Optimization Theory and Applications, 126 (2005), 589-616. doi: 10.1007/s10957-005-5499-z.

[12]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,'' Reprint of the 1980 original. Classics in Applied Mathematics, 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.

[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), 317-345. doi: 10.1098/rspa.1955.0089.

[14]

P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.

[15]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258. Springer-Verlag, New York, 1994.

[16]

J. Wardrop, Some theoretical aspects of road traffic research, in "ICE Proceedings: Part II, Engineering Divisions," 1 (1952), 325-362. doi: 10.1680/ipeds.1952.11362.

show all references

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), 1801-1843. doi: 10.1142/S0218202502002343.

[2]

A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM Journal on Mathematical Analysis, 43 (2011), 2384-2417. doi: 10.1137/110825145.

[3]

A. Bressan and K. Han, Nash equilibria for a model of traffic flow with several groups of drivers, ESAIM, Control, Optimization and Calculus of Variations, 18 (2012), 969-986. doi: 10.1051/cocv/2011198.

[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), 495-515. doi: 10.1090/S0033-569X-2012-01304-9.

[5]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM Journal on Mathematical Analysis, 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.

[6]

C. Daganzo, "Fundamentals of Transportation and Traffic Operations," Pergamon-Elsevier, Oxford, U. K., 1997.

[7]

L. C. Evans, "Partial Differential Equations," $2^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.

[8]

T. L. Friesz, "Dynamic Optimization and Differential Games," Springer, New York, 2010. doi: 10.1007/978-0-387-72778-3.

[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), 176-207. doi: 10.1016/j.trb.2010.05.003.

[10]

M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models," AIMS Series on Applied Mathematics, 1. American Institute of Mathematical Sciences (AIMS), Springfield, Mo., 2006.

[11]

M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, Journal of Optimization Theory and Applications, 126 (2005), 589-616. doi: 10.1007/s10957-005-5499-z.

[12]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,'' Reprint of the 1980 original. Classics in Applied Mathematics, 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.

[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), 317-345. doi: 10.1098/rspa.1955.0089.

[14]

P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.

[15]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258. Springer-Verlag, New York, 1994.

[16]

J. Wardrop, Some theoretical aspects of road traffic research, in "ICE Proceedings: Part II, Engineering Divisions," 1 (1952), 325-362. doi: 10.1680/ipeds.1952.11362.

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