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Optimal Control for Traffic Flow Networks

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Abstract

We consider traffic flow models for road networks where the flow is controlled at the nodes of the network. For the analytical and numerical optimization of the control, the knowledge of the gradient of the objective functional is useful. The adjoint calculus introduced below determines the gradient in two ways. We derive the adjoint equations for the continuous traffic flow network model and derive also the adjoint equations for a discretized model. Numerical examples for the solution of problems of optimal control for traffic flow networks are presented.

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References

  1. G. B. Whitham (1974) Linear and Nonlinear Waves Wiley New York, NY

    Google Scholar 

  2. H. J. Payne (1979) ArticleTitleFREFLO: A Macroscopic Simulation Model of Freeway Traffic Transportation Research Record 722 68–75

    Google Scholar 

  3. Kühne, R.D. Macroscopic Freeway Model for Dense Traffic. 9th International Symposium on Transportation and Traffic Theory, Edited by N. Vollmuller. VNU Science Press, Utrecht, Netherlands: 21–42, 1984.

  4. A. Aw M. Rascle (2000) ArticleTitleResurrection of Second-Order Models of Traffic Flow? SIAM Journal on Applied Mathematics 60 916–938 Occurrence Handle10.1137/S0036139997332099

    Article  Google Scholar 

  5. Günther, M., Klar, A., Materne, T., Wegener R. Multivalued Fundamental Diagrams and Stop-and-Go Waves for Continuum Traffic Flow Equations. Preprint, TU Darmstadt, Dermstodt, Germany.

  6. Herty, M., Klar, A. Modelling and Optimization of Traffic Flow Networks. SIAM Journal on Statistical and Scientific Computing, 2003 (to appear).

  7. H. Holden N. H. Risebro (1995) ArticleTitleA Mathematical Model of Traffic Flow on a Network of Unidirectional Roads SIAM Journal on Mathematical Analysis 4 999–1017 Occurrence Handle10.1137/S0036141093243289

    Article  Google Scholar 

  8. Coclite, G. M., Piccoli, B. Traffic Flow on a Road Network. SISSA, Preprint, 2002.

  9. C. F. Daganzo (1997) ArticleTitleA Continuum Theory of Traffic Dynamics for Freeways with Special Lanes Transportation Research 31B 83–102

    Google Scholar 

  10. Lebacque, J. P., Khoshyaran, M. M. First-Order Macroscopic Traffic Flow Models for Networks in the Context of Dynamic Assignment. Transportation Planning: State of the Art, Edited by M. Patriksson an M. Labbe, Kluwer Academic Publishers, Dordrecht, Netherlands, 2002.

  11. C. Bardos O. Pironneau (2002) ArticleTitleA Formalism for the Differentation of Conservation Laws Comptes Rendus de l’Academie des Sciences Paris Serie I 335 839–845

    Google Scholar 

  12. S. Ulbrich (2002) ArticleTitleA Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms SIAM Journal on Control and Optimization 41 740–797 Occurrence Handle10.1137/S0363012900370764

    Article  Google Scholar 

  13. Ulbrich, S. On the Existence and Approximation of Solutions for the Optimal Control of Nonlinear Hyperbolic Conservation Laws. Optimal Control of Partial Differential Equations, Edited by K. H. Hoffmann et al., Birkhäuser, Basel, Switzerland: 287–299, 1999.

  14. J. M. Coron (2002) ArticleTitleLocal Controllability of a 1-D Tank Containing a Fluid Modeled by the Shallow Water Equations ESAIM: Control Optimization and Calculus of Variations 8 513–554 Occurrence Handle10.1051/cocv:2002050

    Article  Google Scholar 

  15. Gugat, M. Problems of Optimal Boundary Control in Flood Management. Computational Optimization and Applications, 2004 (submitted).

  16. Gugat, M. Nodal Control of Conservation Laws on Networks. Control and Boundary Analysis, Edited by J. Cagnol and J. P. Zolesio, Marcel Dekker, New York, NY, 2004.

  17. Gugat, M., Leugering, G., Schittkowski, K., Schmidt, E. J. P. G. Modelling, Stabilization, and Control of Flow in Networks of Open Channels Online Optimization of Large-Scale Systems, Edited by M. Groetschel, et al., Springer, Berlin, Germany: 251–270, 2001.

  18. G. Leugering E. J. P. G. Schmidt (2002) ArticleTitleOn the Modeling and Stabilization of Flows in Networks of Open Canals SIAM Journal on Control and Optimization 41 164–180 Occurrence Handle10.1137/S0363012900375664

    Article  Google Scholar 

  19. M. Gugat G. Leugering (2003) ArticleTitleGlobal Boundary Controllability of the de St Venant Equations between Steady States Annales de l’Institut Henri Poincaré: Analysise Non Lineaire 20 1–11 Occurrence Handle10.1016/S0294-1449(02)00004-5

    Article  Google Scholar 

  20. M. Gugat G. Leugering E. J. P. G. Schmidt (2004) ArticleTitleGlobal Controllability between Steady Supercritical Flows in Channel Networks Mathematical Methods in the Applied Sciences 27 781–802 Occurrence Handle10.1002/mma.471

    Article  Google Scholar 

  21. Lighthill, M. J., Whitham, J. B. On Kinematic Waves Proceedings of the Royal Society of Edinburgh, 229A: 281–345, 1983.

  22. Klar, A., Kuehne R. D., Wegener R. Mathematical Models for Vehicular Traffic. Surveys on Mathematics for Industry, 6, 1996.

  23. C. Bardos A. Y. LeRoux J. C. Nedelec (1979) ArticleTitleFirst-Order Quasilinear Equations with Boundary Conditions Communications in Partial Differential Equations 4 1017–1034

    Google Scholar 

  24. A. Bressan (2000) Hyperbolic Systems of Conservation Laws Oxford University Press Oxford, UK

    Google Scholar 

  25. H. Holden N. H. Risebro (2002) Front Tracking for Hyperbolic Conservation Laws Springer New York, NY

    Google Scholar 

  26. M. Herty A. Klar (2004) ArticleTitleSimplified Dynamics and Optimization of Large-Scale Traffic Networks M3AS 14 579–601

    Google Scholar 

  27. H. Byrd R. P. Lu J. Nocedal C. Zhu (1995) ArticleTitleA Limited-Memory Algorithm for Bound-Constrained Optimization SIAM Journal on Statistical and Scientific Computing 16 1190–1208 Occurrence Handle10.1137/0916069

    Article  Google Scholar 

  28. Zhu, C., Byrd, R. H., Lu, J., Nocedal, J. L-BFGS-B: Fortran Subroutines for Large-Scale Bound-Constrained Optimization. Technical Report NAM-11. EECS Department, Northwestern University, 1994.

  29. R. Byrd J. Nocedal R. Schinabel (1994) ArticleTitleRepresentations of Quasi-Newton Matrices and Their Use in Limited-Memory Methods Mathematical Programming 63 129–156 Occurrence Handle10.1007/BF01582063

    Article  Google Scholar 

  30. C. T. Kelley (1999) Iterative Methods for Optimization SIAM Philadelphia, Pennsylvania

    Google Scholar 

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H.J. Pesch

This author was supported by Deutsche Forschungsgemeinschaft (DFG), Grant KL 1105/5.

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Gugat, M., Herty, M., Klar, A. et al. Optimal Control for Traffic Flow Networks. J Optim Theory Appl 126, 589–616 (2005). https://doi.org/10.1007/s10957-005-5499-z

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