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), 495515
MSC (2010):
Primary 35L65; Secondary 35Q93, 49J21, 49N70, 49N90
Published electronically:
May 2, 2012
MathSciNet review:
2986132
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The paper is concerned with Nash equilibrium solutions for the LighthillWhitham 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 , 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.
 1.
A. Bressan and K. Han, Optima and equilibria for a model of traffic flow. SIAM J. Math. Anal. 43 (2011), 23842417.
 2.
T. L. Friesz, T. Kim, C. Kwon, and M. A. Rigdon, Approximate network loading and dualtimescale dynamic user equilibrium. Transportation Research Part B (2010).
 3.
A.
Fügenschuh, M.
Herty, and A.
Martin, Combinatorial and continuous models for the optimization of
traffic flows on networks, SIAM J. Optim. 16 (2006),
no. 4, 1155–1176. MR 2219137
(2007k:90009), 10.1137/040605503
 4.
Mauro
Garavello and Benedetto
Piccoli, Traffic flow on networks, AIMS Series on Applied
Mathematics, vol. 1, American Institute of Mathematical Sciences
(AIMS), Springfield, MO, 2006. Conservation laws models. MR 2328174
(2008g:90023)
 5.
M.
Gugat, M.
Herty, A.
Klar, and G.
Leugering, Optimal control for traffic flow networks, J.
Optim. Theory Appl. 126 (2005), no. 3, 589–616.
MR
2164806 (2006m:49043), 10.1007/s109570055499z
 6.
M.
Herty, C.
Kirchner, and A.
Klar, Instantaneous control for traffic flow, Math. Methods
Appl. Sci. 30 (2007), no. 2, 153–169. MR 2285119
(2007k:93074), 10.1002/mma.779
 7.
Lawrence
C. Evans, Partial differential equations, 2nd ed., Graduate
Studies in Mathematics, vol. 19, American Mathematical Society,
Providence, RI, 2010. MR 2597943
(2011c:35002)
 8.
P.
D. Lax, Hyperbolic systems of conservation laws. II, Comm.
Pure Appl. Math. 10 (1957), 537–566. MR 0093653
(20 #176)
 9.
M.
J. Lighthill and G.
B. Whitham, On kinematic waves. II. A theory of traffic flow on
long crowded roads, Proc. Roy. Soc. London. Ser. A.
229 (1955), 317–345. MR 0072606
(17,310a)
 10.
Joel
Smoller, Shock waves and reactiondiffusion equations, 2nd
ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles
of Mathematical Sciences], vol. 258, SpringerVerlag, New York, 1994.
MR
1301779 (95g:35002)
 1.
 A. Bressan and K. Han, Optima and equilibria for a model of traffic flow. SIAM J. Math. Anal. 43 (2011), 23842417.
 2.
 T. L. Friesz, T. Kim, C. Kwon, and M. A. Rigdon, Approximate network loading and dualtimescale dynamic user equilibrium. Transportation Research Part B (2010).
 3.
 A. Fügenschuh, M. Herty, and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks. SIAM J. Optim. 16 (2006), 11551176. MR 2219137 (2007k:90009)
 4.
 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)
 5.
 M. Gugat, M. Herty, A. Klar, and G. Leugering, Optimal control for traffic flow networks. J. Optim. Theory Appl. 126 (2005), 589616. MR 2164806 (2006m:49043)
 6.
 M. Herty, C. Kirchner, and A. Klar, Instantaneous control for traffic flow. Math. Methods Appl. Sci. 30 (2007), 153169. MR 2285119 (2007k:93074)
 7.
 L. C. Evans, Partial Differential Equations. Second edition. American Mathematical Society, Providence, RI, 2010. MR 2597943 (2011c:35002)
 8.
 P. D. Lax, Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10 (1957), 537566. MR 0093653 (20:176)
 9.
 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), 317345. MR 0072606 (17:310a)
 10.
 J. Smoller, Shock waves and reactiondiffusion equations. Second edition. SpringerVerlag, New York, 1994. MR 1301779 (95g:35002)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35L65,
35Q93,
49J21,
49N70,
49N90
Retrieve articles in all journals
with MSC (2010):
35L65,
35Q93,
49J21,
49N70,
49N90
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
Email:
shen{\textunderscore}w@psu.edu
Fang Yu
Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China
Email:
yufang0820@sjtu.edu.cn
DOI:
http://dx.doi.org/10.1090/S0033569X2012013049
PII:
S 0033569X(2012)013049
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.
