Variational analysis of Nash equilibria for a model of traffic flow
Authors:
Alberto Bressan, Chen Jie Liu, Wen Shen and Fang Yu
Journal:
Quart. Appl. Math. 70 (2012), 495-515
MSC (2010):
Primary 35L65; Secondary 35Q93, 49J21, 49N70, 49N90
DOI:
https://doi.org/10.1090/S0033-569X-2012-01304-9
Published electronically:
May 2, 2012
MathSciNet review:
2986132
Full-text PDF Free Access
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Additional Information
Abstract: The paper is concerned with Nash equilibrium solutions for the Lighthill-Whitham model of traffic flow, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. Estimates are provided on how much the Nash solution may change, depending on the cost functions and on the flux function of the conservation law. It is shown that this equilibrium solution can also be determined as a global minimizer for a functional $\Phi$, measuring the maximum total cost among all drivers, in a given traffic pattern. The last section of the paper introduces two evolution models, describing how the traffic pattern can change, day after day. It is assumed that each driver adjusts his departure time based on previous experience, in order to lower his own cost. Numerical simulations are reported, indicating a possible instability of the Nash equilibrium.
References
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- Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779
References
- A. Bressan and K. Han, Optima and equilibria for a model of traffic flow. SIAM J. Math. Anal. 43 (2011), 2384-2417.
- 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 (2010).
- A. Fügenschuh, M. Herty, and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks. SIAM J. Optim. 16 (2006), 1155-1176. MR 2219137 (2007k:90009)
- M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models. AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences, Springfield, Mo., 2006. MR 2328174 (2008g:90023)
- M. Gugat, M. Herty, A. Klar, and G. Leugering, Optimal control for traffic flow networks. J. Optim. Theory Appl. 126 (2005), 589-616. MR 2164806 (2006m:49043)
- M. Herty, C. Kirchner, and A. Klar, Instantaneous control for traffic flow. Math. Methods Appl. Sci. 30 (2007), 153-169. MR 2285119 (2007k:93074)
- L. C. Evans, Partial Differential Equations. Second edition. American Mathematical Society, Providence, RI, 2010. MR 2597943 (2011c:35002)
- P. D. Lax, Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653 (20:176)
- M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society of London: Series A, 229 (1955), 317–345. MR 0072606 (17:310a)
- J. Smoller, Shock waves and reaction-diffusion equations. Second edition. Springer-Verlag, New York, 1994. MR 1301779 (95g:35002)
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Additional Information
Alberto Bressan
Affiliation:
Department of Mathematics, Penn State University, University Park, Pennsylvania 16802
Email:
bressan@math.psu.edu
Chen Jie Liu
Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China
Email:
cjliusjtu@gmail.com
Wen Shen
Affiliation:
Department of Mathematics, Penn State University, University Park, Pennsylvania 16802
MR Author ID:
613346
Email:
shen_w@psu.edu
Fang Yu
Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China
Email:
yufang0820@sjtu.edu.cn
Received by editor(s):
November 16, 2011
Published electronically:
May 2, 2012
Dedicated:
Dedicated to Constantine Dafermos in the occasion of his 70th birthday
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.
