On the modelling and management of traffic
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 5, pp. 853-872.

Several realistic situations in vehicular traffic that give rise to queues can be modeled through conservation laws with boundary and unilateral constraints on the flux. This paper provides a rigorous analytical framework for these descriptions, comprising stability with respect to the initial data, to the boundary inflow and to the constraint. We present a framework to rigorously state optimal management problems and prove the existence of the corresponding optimal controls. Specific cases are dealt with in detail through ad hoc numerical integrations. These are here obtained implementing the wave front tracking algorithm, which appears to be very precise in computing, for instance, the exit times.

DOI : 10.1051/m2an/2010105
Classification : 35L65, 90B20
Mots clés : optimal control of conservation laws, constrained hyperbolic pdes, traffic modelling
@article{M2AN_2011__45_5_853_0,
     author = {Colombo, Rinaldo M. and Goatin, Paola and Rosini, Massimiliano D.},
     title = {On the modelling and management of traffic},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {853--872},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {5},
     year = {2011},
     doi = {10.1051/m2an/2010105},
     mrnumber = {2817547},
     zbl = {1267.90032},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2010105/}
}
TY  - JOUR
AU  - Colombo, Rinaldo M.
AU  - Goatin, Paola
AU  - Rosini, Massimiliano D.
TI  - On the modelling and management of traffic
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2011
SP  - 853
EP  - 872
VL  - 45
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2010105/
DO  - 10.1051/m2an/2010105
LA  - en
ID  - M2AN_2011__45_5_853_0
ER  - 
%0 Journal Article
%A Colombo, Rinaldo M.
%A Goatin, Paola
%A Rosini, Massimiliano D.
%T On the modelling and management of traffic
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2011
%P 853-872
%V 45
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2010105/
%R 10.1051/m2an/2010105
%G en
%F M2AN_2011__45_5_853_0
Colombo, Rinaldo M.; Goatin, Paola; Rosini, Massimiliano D. On the modelling and management of traffic. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 5, pp. 853-872. doi : 10.1051/m2an/2010105. http://www.numdam.org/articles/10.1051/m2an/2010105/

[1] D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws. NoDEA 4 (1997) 1-42. | MR | Zbl

[2] D. Amadori and R.M. Colombo, Continuous dependence for 2×2 conservation laws with boundary. J. Differ. Equ. 138 (1997) 229-266. | MR | Zbl

[3] F. Ancona and A. Marson, Scalar non-linear conservation laws with integrable boundary data. Nonlinear Anal. 35 (1999) 687-710. | MR | Zbl

[4] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws. Numer. Math. 115 (2010) 609-645. | MR | Zbl

[5] A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60 (2000) 916-938. | MR | Zbl

[6] C. Bardos, A.Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions. Comm. Partial Differential Equations 4 (1979) 1017-1034. | MR | Zbl

[7] S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic. SIAM J. Appl. Math. (to appear). | MR | Zbl

[8] A. Bressan, Hyperbolic systems of conservation laws - The one-dimensional Cauchy problem,Oxford Lecture Series in Mathematics and its Applications 20. Oxford University Press, Oxford (2000). | MR | Zbl

[9] W. Chen, S.C. Wong, C.W. Shu and P. Zhang, Front tracking algorithm for the Lighthill-Whitham-Richards traffic flow model with a piecewise quadratic, continuous, non-smooth and non-concave fundamental diagram. Int. J. Numer. Anal. Model. 6 (2009) 562-585. | MR

[10] R.M. Colombo, Hyperbolic phase transitions in traffic flow. SIAM J. Appl. Math. 63 (2002) 708-721. | MR | Zbl

[11] R.M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint. J. Differ. Equ. 234 (2007) 654-675. | MR | Zbl

[12] R.M. Colombo and A. Groli, Minimising stop and go waves to optimise traffic flow. Appl. Math. Lett. 17 (2004) 697-701. | MR | Zbl

[13] R.M. Colombo, P. Goatin, G. Maternini and M.D. Rosini, Conservation laws with unilateral constraints in traffic modeling, in Transport Management and Land-Use Effects in Presence of Unusual Demand, L. Mussone and U. Crisalli Eds., Atti del convegno SIDT 2009 (2009). | Zbl

[14] R.M. Colombo, P. Goatin and B. Piccoli, Road networks with phase transitions. J. Hyperbolic Differ. Equ. 7 (2010) 85-106. | MR | Zbl

[15] R.M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound. SIAM J. Appl. Math. 70 (2010) 2652-2666. | MR | Zbl

[16] C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38 (1972) 33-41. | MR | Zbl

[17] C. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory. Transp. Res. B 28B (1994) 269-287.

[18] F. Dubois and P. Lefloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71 (1988) 93-122. | MR | Zbl

[19] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions. Math. Comput. Model. 44 (2006) 287-303. | MR | Zbl

[20] J. Goodman, Initial Boundary Value Problems for Hyperbolic Systems of Conservation Laws. Ph.D. thesis, California University (1982). | MR

[21] H. Greenberg, An analysis of traffic flow. Oper. Res. 7 (1959) 79-85. | MR

[22] B. Greenshields, A study of traffic capacity. Proceedings of the Highway Research Board 14 (1935) 448-477.

[23] B. Haut, G. Bastin and Y. Chitour, A macroscopic traffic model for road networks with a representation of the capacity drop phenomenon at the junctions, in Proceedings 16th IFAC World Congress, Prague, Czech Republic, July (2005) Tu-M01-TP/3.

[24] D. Helbing, S. Lämmer and J.-P. Lebacque, Self-Organized Control of Irregular or Perturbed Network Traffic, in Optimal Control and Dynamic Games, Advances in Computational Management Science 7, Springer (2005) 239-274. | Zbl

[25] J.C. Herrera and A.M. Bayen, Incorporation of lagrangian measurements in freeway traffic state estimation. Transp. Res. Part B: Methodol. 44 (2010) 460-481.

[26] H. Holden and N.H. Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences 152. Springer-Verlag, New York (2002). | MR | Zbl

[27] W.-L. Jin, Continuous kinematic wave models of merging traffic flow. Transp. Res. Part B: Methodol. 44 (2010) 1084-1103.

[28] W.L. Jin and H.M. Zhang, The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model. Transp. Res. B 37 (2003) 207-223.

[29] W.L. Jin and H.M. Zhang, On the distribution schemes for determining flows through a merge. Transp. Res. Part B: Methodol. 37 (2003) 521-540.

[30] B.S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow. Phys. Rev. E 48 (1993) R2335-R2338.

[31] B.S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow. Phys. Rev. E 50 (1994) 54-83.

[32] B.S. Kerner and H. Rehborn, Experimental features and characteristics of traffic jams. Phys. Rev. E 53 (1996) R1297-R1300.

[33] A. Klar, Kinetic and Macroscopic Traffic Flow Models. School of Computational Mathematics: Computational aspects in kinetic models, XXth edition (2002).

[34] S.N. Kružhkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228-255. | Zbl

[35] L. Leclercq, Bounded acceleration close to fixed and moving bottlenecks. Transp. Res. Part B: Methodol. 41 (2007) 309-319.

[36] L. Leclercq, Hybrid approaches to the solutions of the Lighthill-Whitham-Richards model. Transp. Res. Part B: Methodol. 41 (2007) 701-709.

[37] H. Lee, H.-W. Lee and D. Kim, Empirical phase diagram of traffic flow on highways with on-ramps, in Traffic and Granular Flow '99, M.S.D.W.D. Helbing and H.J. Herrmann Eds. (2000). | Zbl

[38] R.J. Leveque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2002). | MR | Zbl

[39] J. Li, Q. Chen, H. Wang and D. Ni, Analysis of LWR model with fundamental diagram subject to uncertainties, in TRB 88th Annual Meeting Compendium of Papers, number 09-1189 in TRB (2009) 14.

[40] 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 | Zbl

[41] H.X. Liu, X. Wu, W. Ma and H. Hu, Real-time queue length estimation for congested signalized intersections. Transp. Res. Part C 17 (2009) 412-427.

[42] S. Mammar, J.-P. Lebacque and H.H. Salem, Riemann problem resolution and Godunov scheme for the Aw-Rascle-Zhang model. Transp. Sci. 43 (2009) 531-545.

[43] G. Newell, A simplified theory of kinematic waves in highway traffic, part II. Transp. Res. B 27 B (1993) 289-303.

[44] E.Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws. J. Hyperbolic Differ. Equ. 4 (2007) 729-770. | MR | Zbl

[45] B. Piccoli and M. Garavello, Traffic flow on networks - Conservation laws models, AIMS Series on Applied Mathematics 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006). | MR | Zbl

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

[47] D. Serre, Systems of conservation laws 1 & 2. Cambridge University Press, Cambridge (1999). | MR | Zbl

[48] C. Tampere, S. Hoogendoorn and B. Van Arem, A behavioural approach to instability, stop & go waves, wide jams and capacity drop, in Proceedings of 16th International Symposium on Transportation and Traffic Theory (ISTTT), Maryland (2005).

[49] B. Temple, Global solution of the Cauchy problem for a class of 2×2 nonstrictly hyperbolic conservation laws. Adv. Appl. Math. 3 (1982) 335-375. | MR | Zbl

[50] E. Tomer, L. Safonov, N. Madar and S. Havlin, Optimization of congested traffic by controlling stop-and-go waves. Phys. Rev. E 65 (2002) 4. | MR

[51] M. Treiber, A. Hennecke and D. Helbing, Congested traffic states in empirical observations and microscopic simulation. Phys. Rev. E 62 (2000) 1805-1824.

Cité par Sources :