Nash equilibria for a model of traffic flow with several groups of drivers
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 969-986.

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.

DOI : 10.1051/cocv/2011198
Classification : 35E15, 49K20, 91A12
Mots clés : scalar conservation law, Hamilton-Jacobi equation, Nash equilibrium
@article{COCV_2012__18_4_969_0,
     author = {Bressan, Alberto and Han, Ke},
     title = {Nash equilibria for a model of traffic flow with several groups of drivers},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {969--986},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {4},
     year = {2012},
     doi = {10.1051/cocv/2011198},
     mrnumber = {3019468},
     zbl = {1262.35199},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2011198/}
}
TY  - JOUR
AU  - Bressan, Alberto
AU  - Han, Ke
TI  - Nash equilibria for a model of traffic flow with several groups of drivers
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2012
SP  - 969
EP  - 986
VL  - 18
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2011198/
DO  - 10.1051/cocv/2011198
LA  - en
ID  - COCV_2012__18_4_969_0
ER  - 
%0 Journal Article
%A Bressan, Alberto
%A Han, Ke
%T Nash equilibria for a model of traffic flow with several groups of drivers
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 969-986
%V 18
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2011198/
%R 10.1051/cocv/2011198
%G en
%F COCV_2012__18_4_969_0
Bressan, Alberto; Han, Ke. Nash equilibria for a model of traffic flow with several groups of drivers. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 969-986. doi : 10.1051/cocv/2011198. http://archive.numdam.org/articles/10.1051/cocv/2011198/

[1] J.P. Aubin and A. Cellina, Differential inclusions. Set-Valued Maps and Viability Theory. Springer-Verlag, Berlin (1984). | MR | Zbl

[2] A. Bressan and K. Han, Optima and equilibria for a model of traffic flow. SIAM J. Math. Anal. 43 (2011) 2384-2417. | MR | Zbl

[3] A. Cellina, Approximation of set valued functions and fixed point theorems. Ann. Mat. Pura Appl. 82 (1969) 17-24. | MR | Zbl

[4] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998). | MR | Zbl

[5] T.L. Friesz, Dynamic Optimization and Differential Games, Springer, New York (2010). | Zbl

[6] 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).

[7] 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 | Zbl

[8] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models. AIMS Series on Applied Mathematics, Springfield, Mo. (2006). | MR | Zbl

[9] M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks. J. Optim. Theory Appl. 126 (2005) 589-616. | MR | Zbl

[10] M. Herty, C. Kirchner and A. Klar, Instantaneous control for traffic flow. Math. Methods Appl. Sci. 30 (2007) 153-169. | MR | Zbl

[11] L.C. Evans, Partial Differential Equations, 2nd edition. American Mathematical Society, Providence, RI (2010). | Zbl

[12] P.D. Lax, Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10 (1957) 537-566. | MR | Zbl

[13] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A 229 (1955) 317-345. | MR | Zbl

[14] P.I. Richards, Shock waves on the highway. Oper. Res. 4 (1956), 42-51. | MR

[15] J. Smoller, Shock waves and reaction-diffusion equations, 2nd edition. Springer-Verlag, New York (1994). | MR | Zbl

Cité par Sources :