Skip to main content
SpringerLink
Log in

Mixed Initial-Boundary Value Problems for Scalar Conservation Laws: Application to the Modeling of Transportation Networks

  • Conference paper
Book cover Hybrid Systems: Computation and Control (HSCC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3927))

Included in the following conference series:

Abstract

This article proves the existence and uniqueness of a weak solution to a scalar conservation law on a bounded domain. A weak formulation of hybrid boundary conditions is needed for the problem to be well posed. The boundary conditions are represented by a hybrid automaton with switches between the modes determined by the direction of characteristics of the system at the boundary. The existence of the solution results from the convergence of a Godunov scheme derived in this article. This weak formulation is written explicitly in the context of a strictly concave flux function (relevant for highway traffic). The numerical scheme is then applied to a highway scenario with data from the I210 highway obtained from the California PeMS system. Finally, the existence of a minimizer of travel time is obtained, with the corresponding optimal boundary control.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)

    MATH  Google Scholar 

  2. Ancona, F., Marson, A.: On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Control Optim. 36(1), 290–312 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ancona, F., Marson, A.: Scalar non-linear conservation laws with integrable boundary data. Nonlinear Analysis 35, 687–710 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ansorge, R.: What does the entropy condition mean in traffic flow theory? Transportation Research 24B(2), 133–143 (1990)

    Article  MathSciNet  Google Scholar 

  5. Atwell, J.A., Borggaard, J.T., King, B.B.: Reduced order controllers for Burgers’ equation with a nonlinear observer. Applied Mathematics and Computational Science 11(6), 1311–1330 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Baker, J., Armaou, A., Christofides, P.D.: Nonlinear control of incompressible fluid flow: Application to burgers’ equation and 2d channel flow. Journal of Mathematical Analysis and Applications 252, 230–255 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Balogh, A., Krstic, M.: Burgers’ equation with nonlinear boundary feedback: H 1 stability, well posedness, and simulation. Mathematical Problems in Engineering 6, 189–200 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bardos, C., Leroux, A.Y., Nedelec, J.C.: First order quasilinear equations with boundary conditions. Comm. Part. Diff. Eqns. 4(9), 1017–1034 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bayen, A.M., Raffard, R.L., Tomlin, C.J.: Network congestion alleviation using adjoint hybrid control: application to highways. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 95–110. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  10. Bressan, A.: Hyperbolic systems of conservation laws: the one dimensional Cauchy problem. Oxford Lecture Series in Mathematics and its Applications, vol. 20. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  11. Byrnes, C.I., Gilliam, D.S., Shubov, V.I.: Semiglobal stabilization of a boundary controlled viscous burgers’ equation. In: CDCnintynine, Phoenix, AZ, pp. 680–681 (December 1999)

    Google Scholar 

  12. Clawpack 4.2, http://www.amath.washington.edu/~claw/

  13. Crandall, M.G.: Viscosity solutions: a primer. In: Champarnaud, J.-M., Maurel, D., Ziadi, D. (eds.) WIA 1998. LNCS, vol. 1660, pp. 1–43. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  14. Crandall, M.G., Evans, L.C., Lions, P.-L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282, 487–502 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dafermos, C.: Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Math. J. 26, 1097–1119 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dafermos, C.: Hyberbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wissenchaften, vol. 325. Springer, Heidelberg (2000)

    Book  MATH  Google Scholar 

  17. Daganzo, C.: The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory. Transportation Research 28B(4), 269–287 (1994)

    Article  Google Scholar 

  18. Daganzo, C.: The cell transmission model, part II: network traffic. Transportation Research 29B(2), 79–93 (1995)

    Article  Google Scholar 

  19. Evans, L.C.: Partial differential equations. In: Graduate studies in Mathematics, vol. 19, A.M.S (2002)

    Google Scholar 

  20. Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. CRC Press, Boca Raton (1991)

    MATH  Google Scholar 

  21. Gazis, D.C.: Traffic theory. International series in Operations Research and Management Science. Kluwer Academic, Boston (2002)

    MATH  Google Scholar 

  22. Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Math. Sbornik 47, 271–306 (1959)

    MathSciNet  MATH  Google Scholar 

  23. Greenberg, H.: An analysis of traffic flow. Operation Res. 7(1), 79–85 (1959)

    Article  MathSciNet  Google Scholar 

  24. Greenshields, B.D.: A study of traffic capacity. Proc. Highway Res. Board 14, 448–477 (1935)

    Google Scholar 

  25. Jameson, A.: Analysis and design of numerical schemes for gas dynamics 1: Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence. International Journal of Computational Fluid Dynamics 4, 171–218 (1995)

    Article  Google Scholar 

  26. Jameson, A.: Analysis and design of numerical schemes for gas dynamics 2: Artificial diffusion and discrete shock structure. International Journal of Computational Fluid Dynamics 4, 1–38 (1995)

    Article  Google Scholar 

  27. Joseph, K.T., Veerappa Gowda, G.D.: Explicit formula for the solution of convex conservation laws with boundary condition. Duke Math. J. 62(2), 401–416 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Keyfitz, B.L.: Solutions with shocks. Comm. Pure and Appl. Math. 24, 125–132 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kobayashi, T.: Adaptive regulator design of a viscous Burgers’ system by boundary control. IMA Journal of Mathematical Control and Information 18(3), 427–437 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Krstić, M.: On global stabilization of Burgers’ equation by boundary control. Systems and Control Letters 37, 123–142 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kružkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR Sbornik 10, 217–243 (1970)

    Article  Google Scholar 

  32. Lax, P.D.: Functional analysis. Wiley-Interscience, Chichester (2002)

    MATH  Google Scholar 

  33. Le Floch, P.: Explicit formula for scalar non-linear conservation laws with boudary condition. Math. Meth. Appl. Sci. 10, 265–287 (1988)

    Article  MATH  Google Scholar 

  34. Leroux, A.Y.: Vanishing viscosity method for a quasilinear first order equation with boundary conditions. In: Conference on the Numerical Analysis of Singular Perturbation Problems, Nijnegen, Academic Press, London (1978)

    Google Scholar 

  35. Le Veque, R.J.: Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  36. Lighthill, M.J., Whitham, G.B.: On kinematic waves. II. A theory of traffic flow on long crowded roads. In: Proceedings of the Royal Society of London, vol. 229(1178), pp. 317–345 (1956)

    Google Scholar 

  37. Ly, H., Mease, K.D., Titi, E.S.: Distributed and boundary control of the viscous burgers’ equation. Numerical Functional Analysis and Optimization 18(1-2), 143–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  38. Nollet, R.: Technical report (July 2005)

    Google Scholar 

  39. Oleinik, O.A.: On discontinuous solutions of nonlinear differential equations. Uspekhi Mat. Nauk. 12, 3–73 (1957); English translation: American Mathematical Society, 2(26), pp. 95–172 (1963)

    MathSciNet  Google Scholar 

  40. Önder, M., Özbay, E.: Low dimensional modelling and dirichlet boundary controller design for burgers equation. International Journal of Control 77(10), 895–906 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. Freeway performance measurement system, http://pems.eecs.berkeley.edu

  42. Petit, N.: Systèmes à retard. Platitude en génie des procédés et contrôle de certaines équations des ondes. PhD thesis, Ecole Nationale Supérieure des Mines de Paris (2000)

    Google Scholar 

  43. Richards, P.I.: Shock waves on the highway. Operations Research 4(1), 42–51 (1956)

    Article  MathSciNet  Google Scholar 

  44. Schonbek, M.E.: Existence of solutions to singular conservation laws. J. Math. Anal. 15(6), 1125–1139 (1984)

    MathSciNet  MATH  Google Scholar 

  45. Serre, D.: Systems of conservation laws. Hyperbolicity, entropies, shock waves, vol. 1. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  46. Sun, X., Horowitz, R.: Localized switching ramp-metering controller with queue length regulator for congested freeways. In: ACCofive, Portland, pp. 2141–2146 (June 2005)

    Google Scholar 

  47. Terracina, A.: Comparison properties for scalar conservation laws with boundary conditions. Nonlinear Anal. Theory Meth. Appl. 28, 633–653 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  48. Vol’pert, A.I.: The spaces BV and quasilinear equations. Math. USSR Sbornik 2, 225–267 (1967)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Civil Systems, Department of Civil and Environmental Engineering, University of California, Berkeley, CA, 94720-1710, USA

    Issam S. Strub & Alexandre M. Bayen

Authors
  1. Issam S. Strub
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandre M. Bayen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Center for Control, Dynamical Systems and Computation, University of California, 93106, Santa Barbara, CA, USA

    João P. Hespanha

  2. SRI International, 333 Ravenswood Ave, Menlo Park, CA, USA

    Ashish Tiwari

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Strub, I.S., Bayen, A.M. (2006). Mixed Initial-Boundary Value Problems for Scalar Conservation Laws: Application to the Modeling of Transportation Networks. In: Hespanha, J.P., Tiwari, A. (eds) Hybrid Systems: Computation and Control. HSCC 2006. Lecture Notes in Computer Science, vol 3927. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11730637_41

Download citation

  • DOI: https://doi.org/10.1007/11730637_41

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-33170-4

  • Online ISBN: 978-3-540-33171-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation