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Globally optimal departure rates for several groups of drivers

1 Department of Mathematics, Penn State University, University Park, PA 16802, USA
2 Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK

† This contribution is part of the Special Issue: Nonlinear models in applied mathematics
   Guest Editor: Giuseppe Maria Coclite
   Link: https://www.aimspress.com/newsinfo/1213.html

Special Issues: Nonlinear models in applied mathematics

The first part of this paper contains a brief introduction to conservation law models of traffic flow on a network of roads. Globally optimal solutions and Nash equilibrium solutions are reviewed, with several groups of drivers sharing different cost functions. In the second part we consider a globally optimal set of departure rates, for different groups of drivers but on a single road. Necessary conditions are proved, which lead to a practical algorithm for computing the optimal solution.
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Keywords conservation law; traffic flow; globally optimal solution

Citation: Alberto Bressan, Yucong Huang. Globally optimal departure rates for several groups of drivers. Mathematics in Engineering, 2019, 1(3): 583-613. doi: 10.3934/mine.2019.3.583


  • 1. Bellomo N, Delitala M, Coscia V (2002) On the mathematical theory of vehicular traffic flow I: Fluid dynamic and kinetic modeling. Math Models Methods Appl Sci 12: 1801–1843.    
  • 2. Bellomo N, Dogbe C (2011) On the modeling of traffic and crowds: A survey of models, speculations, and perspectives. SIAM Rev 53: 409–463.    
  • 3. Bressan A (2000) Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem. Oxford University Press.
  • 4. Bressan A, Canic S, Garavello M, et al. (2014) Flow on networks: Recent results and perspectives. EMS Surv Math Sci 1: 47–111.    
  • 5. Bressan A, Han K (2011) Optima and equilibria for a model of traffic flow. SIAM J Math Anal 43: 2384–2417.    
  • 6. Bressan A, Han K (2012) Nash equilibria for a model of traffic flow with several groups of drivers. ESAIM Control Optim Calc Var 18: 969–986.    
  • 7. Bressan A, Liu CJ, Shen W, et al. (2012) Variational analysis of Nash equilibria for a model of traffic flow. Quarterly Appl Math 70: 495–515.    
  • 8. Bressan A, Marson A (1995) A variational calculus for discontinuous solutions of conservative systems. Commun Part Diff Eq 20: 1491–1552.    
  • 9. Bressan A, Marson A (1995) A maximum principle for optimally controlled systems of conservation laws. Rend Sem Mat Univ Padova 94: 79–94.
  • 10. Bressan A, Nguyen K (2015) Conservation law models for traffic flow on a network of roads. Netw Heter Media 10: 255–293.    
  • 11. Bressan A, Nguyen K (2015) Optima and equilibria for traffic flow on networks with backward propagating queues. Netw Heter Media 10: 717–748.    
  • 12. Bressan A, Nordli A (2017) The Riemann Solver for traffic flow at an intersection with buffer of vanishing size. Netw Heter Media 12: 173–189.    
  • 13. Bressan A, Shen W (2007) Optimality conditions for solutions to hyperbolic balance laws, In: Ancona, F., Lasieka, I., Littman, W., et al. Editors. Control Methods in PDE - Dynamical Systems, AMS Contemporary Mathematics 426: 129–152.    
  • 14. Bressan A, Yu F (2015) Continuous Riemann solvers for traffic flow at a junction. Discr Cont Dyn Syst 35: 4149–4171.    
  • 15. Chitour Y, Piccoli B (2005) Traffic circles and timing of traffic lights for cars flow. Discrete Contin Dyn Syst B 5: 599–630.    
  • 16. Coclite GM, Garavello M, Piccoli B (2005) Traffic flow on a road network. SIAM J Math Anal 36: 1862–1886.    
  • 17. Dafermos C (1972) Polygonal approximations of solutions of the initial value problem for a conservation law. J Math Anal Appl 38: 33–41.    
  • 18. Daganzo C (1997) Fundamentals of Transportation and Traffic Operations. Oxford, UK: Pergamon-Elsevier.
  • 19. Evans LC (2010) Partial Differential Equations. 2 Eds., Providence, RI: American Mathematical Society.
  • 20. Garavello M, Han K, Piccoli B (2016) Models for Vehicular Traffic on Networks. Missouri: AIMS Series on Applied Mathematics, Springfield.
  • 21. Garavello M, Piccoli B (2006) Traffic Flow on Networks. Conservation Laws Models. Missouri: AIMS Series on Applied Mathematics, Springfield.
  • 22. Garavello M, Piccoli B (2009) Traffic flow on complex networks. Ann Inst H Poincaré Anal Nonlinear 26: 1925–1951.    
  • 23. Herty M, Moutari S, Rascle M (2006) Optimization criteria for modeling intersections of vehicular traffic flow. Netw Heterog Media 1: 275–294.    
  • 24. Holden H, Risebro NH (1995) A mathematical model of traffic flow on a network of unidirectional roads. SIAM J Math Anal 26: 999–1017.    
  • 25. Lax PD (1957) Hyperbolic systems of conservation laws. Comm Pure Appl Math 10: 537–556.    
  • 26. Lighthill M, Whitham G (1955) On kinematic waves. II. A theory of traffic flow on long crowded roads. P Roy Soc A Math Phys Eng Sci 229: 317–345.
  • 27. Pfaff S, Ulbrich S (2015) Optimal boundary control of nonlinear hyperbolic conservation laws with switched boundary data. SIAM J Control Optim 53: 1250–1277.    
  • 28. Richards PI (1956) Shock waves on the highway. Oper Res 4: 42–51.    
  • 29. Smoller J (1994) Shock Waves and Reaction-Diffusion Equations. 2 Eds., New York: Springer-Verlag.
  • 30. Ulbrich S (2002) A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J Control Optim 41: 740–797.    


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