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On the modelling and management of traffic

Published online by Cambridge University Press:  23 February 2011

Rinaldo M. Colombo ,
Paola Goatin  and
Massimiliano D. Rosini
Rinaldo M. Colombo
Affiliation:
Dipartimento di Matematica, Università degli studi di Brescia, Italia. rinaldo@ing.unibs.it
Paola Goatin
Affiliation:
INRIA Sophia Antipolis – Méditerranée, EPI OPALE, France. paola.goatin@inria.fr
Massimiliano D. Rosini
Affiliation:
ICM, Uniwersytet Warszawski, Polska. mrosini@icm.edu.pl
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Abstract

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.

Keywords

Optimal control of conservation lawsconstrained hyperbolic PDEstraffic modelling
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 5 , September 2011 , pp. 853 - 872
Copyright
© EDP Sciences, SMAI, 2011

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References

Amadori, D., Initial-boundary value problems for nonlinear systems of conservation laws. NoDEA 4 (1997) 1–42. CrossRef
Amadori, D. and Colombo, R.M., Continuous dependence for 2×2 conservation laws with boundary. J. Differ. Equ. 138 (1997) 229–266. CrossRef
Ancona, F. and Marson, A., Scalar non-linear conservation laws with integrable boundary data. Nonlinear Anal. 35 (1999) 687–710. CrossRef
Andreianov, B., Goatin, P. and Seguin, N., Finite volume schemes for locally constrained conservation laws. Numer. Math. 115 (2010) 609–645. CrossRef
Aw, A. and Rascle, M., Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60 (2000) 916–938. CrossRef
Bardos, C., le Roux, A.Y. and Nédélec, J.-C., First order quasilinear equations with boundary conditions. Comm. Partial Differential Equations 4 (1979) 1017–1034. CrossRef
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).
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).
Chen, W., Wong, S.C., Shu, C.W. and Zhang, P., 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.
Colombo, R.M., Hyperbolic phase transitions in traffic flow. SIAM J. Appl. Math. 63 (2002) 708–721. CrossRef
Colombo, R.M. and Goatin, P., A well posed conservation law with a variable unilateral constraint. J. Differ. Equ. 234 (2007) 654–675. CrossRef
Colombo, R.M. and Groli, A., Minimising stop and go waves to optimise traffic flow. Appl. Math. Lett. 17 (2004) 697–701. CrossRef
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).
Colombo, R.M., Goatin, P. and Piccoli, B., Road networks with phase transitions. J. Hyperbolic Differ. Equ. 7 (2010) 85–106. CrossRef
Colombo, R.M., Marcellini, F. and Rascle, M., A 2-phase traffic model based on a speed bound. SIAM J. Appl. Math. 70 (2010) 2652–2666. CrossRef
Dafermos, C.M., Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38 (1972) 33–41. CrossRef
Daganzo, C., The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory. Transp. Res. B 28B (1994) 269–287. CrossRef
Dubois, F. and LeFloch, P., Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71 (1988) 93–122. CrossRef
Goatin, P., The Aw-Rascle vehicular traffic flow model with phase transitions. Math. Comput. Model. 44 (2006) 287–303. CrossRef
J. Goodman, Initial Boundary Value Problems for Hyperbolic Systems of Conservation Laws. Ph.D. thesis, California University (1982).
Greenberg, H., An analysis of traffic flow. Oper. Res. 7 (1959) 79–85. CrossRef
Greenshields, B., A study of traffic capacity. Proceedings of the Highway Research Board 14 (1935) 448–477.
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.
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.
Herrera, J.C. and Bayen, A.M., Incorporation of lagrangian measurements in freeway traffic state estimation. Transp. Res. Part B: Methodol. 44 (2010) 460–481. CrossRef
H. Holden and N.H. Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences 152. Springer-Verlag, New York (2002).
Jin, W.-L., Continuous kinematic wave models of merging traffic flow. Transp. Res. Part B: Methodol. 44 (2010) 1084–1103. CrossRef
Jin, W.L. and Zhang, H.M., The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model. Transp. Res. B 37 (2003) 207–223. CrossRef
Jin, W.L. and Zhang, H.M., On the distribution schemes for determining flows through a merge. Transp. Res. Part B: Methodol. 37 (2003) 521–540. CrossRef
Kerner, B.S. and Konhäuser, P., Cluster effect in initially homogeneous traffic flow. Phys. Rev. E 48 (1993) R2335–R2338. CrossRef
Kerner, B.S. and Konhäuser, P., Structure and parameters of clusters in traffic flow. Phys. Rev. E 50 (1994) 54–83. CrossRef
Kerner, B.S. and Rehborn, H., Experimental features and characteristics of traffic jams. Phys. Rev. E 53 (1996) R1297–R1300. CrossRef
A. Klar, Kinetic and Macroscopic Traffic Flow Models. School of Computational Mathematics: Computational aspects in kinetic models, XXth edition (2002).
Kružhkov, S.N., First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228–255.
Leclercq, L., Bounded acceleration close to fixed and moving bottlenecks. Transp. Res. Part B: Methodol. 41 (2007) 309–319. CrossRef
Leclercq, L., Hybrid approaches to the solutions of the Lighthill-Whitham-Richards model. Transp. Res. Part B: Methodol. 41 (2007) 701–709. CrossRef
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).
R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2002).
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.
Lighthill, M.J. and Whitham, G.B., On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A. 229 (1955) 317–345. CrossRef
Liu, H.X., Wu, X., Ma, W. and Real-time, H. Hu queue length estimation for congested signalized intersections. Transp. Res. Part C 17 (2009) 412–427. CrossRef
Mammar, S., Lebacque, J.-P. and Salem, H.H., Riemann problem resolution and Godunov scheme for the Aw-Rascle-Zhang model. Transp. Sci. 43 (2009) 531–545. CrossRef
G. Newell, A simplified theory of kinematic waves in highway traffic, part II. Transp. Res. B 27 B (1993) 289–303.
Panov, E.Y., Existence of strong traces for quasi-solutions of multidimensional conservation laws. J. Hyperbolic Differ. Equ. 4 (2007) 729–770. CrossRef
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).
Richards, P.I., Shock waves on the highway. Oper. Res. 4 (1956) 42–51. CrossRef
D. Serre, Systems of conservation laws 1 & 2. Cambridge University Press, Cambridge (1999).
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).
Temple, B., Global solution of the Cauchy problem for a class of 2×2 nonstrictly hyperbolic conservation laws. Adv. Appl. Math. 3 (1982) 335–375. CrossRef
Tomer, E., Safonov, L., Madar, N. and Havlin, S., Optimization of congested traffic by controlling stop-and-go waves. Phys. Rev. E 65 (2002) 4.
Treiber, M., Hennecke, A. and Helbing, D., Congested traffic states in empirical observations and microscopic simulation. Phys. Rev. E 62 (2000) 1805–1824. CrossRef