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

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.  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

[1]

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

[2]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041

[3]

Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077

[4]

Elvio Accinelli, Humberto Muñiz. A dynamic for production economies with multiple equilibria. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021002

[5]

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

[6]

Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020034

[7]

Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021004

[8]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[9]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[10]

Qiang Fu, Yanlong Zhang, Yushu Zhu, Ting Li. Network centralities, demographic disparities, and voluntary participation. Mathematical Foundations of Computing, 2020, 3 (4) : 249-262. doi: 10.3934/mfc.2020011

[11]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097

[12]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129

[13]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049

[14]

Kengo Nakai, Yoshitaka Saiki. Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1079-1092. doi: 10.3934/dcdss.2020352

[15]

Yicheng Liu, Yipeng Chen, Jun Wu, Xiao Wang. Periodic consensus in network systems with general distributed processing delays. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2021002

[16]

Rajendra K C Khatri, Brendan J Caseria, Yifei Lou, Guanghua Xiao, Yan Cao. Automatic extraction of cell nuclei using dilated convolutional network. Inverse Problems & Imaging, 2021, 15 (1) : 27-40. doi: 10.3934/ipi.2020049

[17]

Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164

[18]

Jan Březina, Eduard Feireisl, Antonín Novotný. On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021009

[19]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093

[20]

Editorial Office. Retraction: Honggang Yu, An efficient face recognition algorithm using the improved convolutional neural network. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 901-901. doi: 10.3934/dcdss.2019060

2019 Impact Factor: 1.053

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]