Hostname: page-component-7479d7b7d-qs9v7 Total loading time: 0 Render date: 2024-07-10T22:17:41.811Z Has data issue: false hasContentIssue false
  • English
  • Français

Nash equilibria for a model of traffic flow with several groups of drivers

Published online by Cambridge University Press:  16 January 2012

Alberto Bressan  and
Ke Han
Alberto Bressan
Affiliation:
Department of Mathematics, Penn State University University Park, 16802 Pa, USA. bressan@math.psu.edu; kxh323@psu.edu
Ke Han
Affiliation:
Department of Mathematics, Penn State University University Park, 16802 Pa, USA. bressan@math.psu.edu; kxh323@psu.edu
Get access

Abstract

Traffic flow is modeled by a conservation law describing the density of cars. It is assumed that each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost. There are N groups of drivers, The i-th group consists of κi drivers, sharing the same departure and arrival costs ϕi(t),ψi(t). For any given population sizes κ1,...,κn, we prove the existence of a Nash equilibrium solution, where no driver can lower his own total cost by choosing a different departure time. The possible non-uniqueness, and a characterization of this Nash equilibrium solution, are also discussed.

Keywords

Scalar conservation lawHamilton-Jacobi equationNash equilibrium
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 969 - 986
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.P. Aubin and A. Cellina, Differential inclusions. Set-Valued Maps and Viability Theory. Springer-Verlag, Berlin (1984).
Bressan, A. and Han, K., Optima and equilibria for a model of traffic flow. SIAM J. Math. Anal. 43 (2011) 2384–2417. Google Scholar
Cellina, A., Approximation of set valued functions and fixed point theorems. Ann. Mat. Pura Appl. 82 (1969) 17–24. Google Scholar
F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998).
T.L. Friesz, Dynamic Optimization and Differential Games, Springer, New York (2010).
T.L. Friesz, T. Kim, C. Kwon and M.A. Rigdon, Approximate network loading and dual-time-scale dynamic user equilibrium. Transp. Res. Part B (2010).
Fügenschuh, A., Herty, M. and Martin, A., Combinatorial and continuous models for the optimization of traffic flows on networks. SIAM J. Optim. 16 (2006) 1155–1176.Google Scholar
M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models. AIMS Series on Applied Mathematics, Springfield, Mo. (2006).
Gugat, M., Herty, M., Klar, A. and Leugering, G., Optimal control for traffic flow networks. J. Optim. Theory Appl. 126 (2005) 589–616. Google Scholar
Herty, M., Kirchner, C. and Klar, A., Instantaneous control for traffic flow. Math. Methods Appl. Sci. 30 (2007) 153–169. Google Scholar
L.C. Evans, Partial Differential Equations, 2nd edition. American Mathematical Society, Providence, RI (2010).
Lax, P.D., Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10 (1957) 537–566. Google Scholar
Lighthill, M. and Whitham, G., On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A 229 (1955) 317–345. Google Scholar
Richards, P.I., Shock waves on the highway. Oper. Res. 4 (1956), 42–51. Google Scholar
J. Smoller, Shock waves and reaction-diffusion equations, 2nd edition. Springer-Verlag, New York (1994).