Skip to main content
SpringerLink
Log in

Modeling and Analysis of Modal Switching in Networked Transport Systems

  • Published:
Applied Mathematics and Optimization Aims and scope Submit manuscript

Abstract

We consider networked transport systems defined on directed graphs: the dynamics on the edges correspond to solutions of transport equations with space dimension one. In addition to the graph setting, a major consideration is the introduction and propagation of discontinuities in the solutions when the system may discontinuously switch modes, naturally or as a hybrid control. This kind of switching has been extensively studied for ordinary differential equations, but not much so far for systems governed by partial differential equations. In particular, we give well-posedness results for switching as a control, both in finite horizon open loop operation and as feedback based on sensor measurements in the system.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amin, S., Hante, F.M., Bayen, A.M.: On stability of switched linear hyperbolic conservation laws with reflecting boundaries. In: Egerstedt, M., Mishra, B. (eds.) Hybrid Systems: Computation and Control. LNCS, vol. 4981, pp. 602–605 (2008)

  2. Antsaklis, P.J., Koutsoukos, X.D., Zaytoon, J.: On hybrid control of complex systems: a survey. J. Eur. Syst. Autom. 32, 1023–1045 (1998)

    Google Scholar 

  3. Bayen, A., Raffard, R., Tomlin, C.: Network congestion alleviation using adjoint hybrid control: application to highways. In: Alur, R., Pappas, G. (eds.) Hybrid Systems: Computation and Control. LNCS, vol. 2993, pp. 95–110 (2004)

  4. Capuzzo-Dolcetta, I., Evans, L.C.: Optimal switching for ordinary differential equations. SIAM J. Control. Optim. 22, 143–161 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  5. Courant, R., Hilbert, D.: Methods of Mathematical Physics, Part II: Partial Differential Equations. Interscience, New York (1962)

    Google Scholar 

  6. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Series of Comprehensive Studies in Mathematics, vol. 325. Springer, Berlin (2000)

    MATH  Google Scholar 

  7. de Halleux, J., Prieur, C., Coron, J.M., d’Andrea-Novel, B., Bastin, G.: Boundary feedback control in networks of open channels. Automatica 39, 1365–1376 (2003)

    Article  MATH  Google Scholar 

  8. Gugat, M., Leugering, G., Schmidt, E.J.P.G.: Global controllability between steady supercritical flows in channel networks. Math. Methods Appl. Sci. 27, 781–802 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gugat, M., Herty, M., Klar, A., Leugering, G.: Optimal control for traffic flow networks. J. Optim. Theory Appl. 162, 589–616 (2005)

    Article  MathSciNet  Google Scholar 

  10. Kramar, M., Sikolya, E.: Spectral properties and asymptotic periodicity of flows in networks. Math. Z. 249, 139–162 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Krasnosel’skiĭ, M.A., Pokrovskiĭ, A.V.: Systems with Hysteresis. Nauka, Moscow (1983). [English transl.: Springer, Berlin, 1989]

    Google Scholar 

  12. Leugering, G., Schmidt, E.J.P.G.: On the modelling and stabilisation of flows in networks of open canals. SIAM J. Control Optim. 41, 164–180 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Li, T.-T.: Exact boundary controllability of unsteady flows in a network of open canals. Math. Nachr. 278, 278–289 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mátrai, T., Sikolya, E.: Asymptotic behaviour of flows in networks. Forum Math. 19, 429–461 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Rauch, J., Reed, M.: Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: Creation and propagation. Commun. Math. Phys. 81, 203–227 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Seidman, T.I.: Some problems with thermostat nonlinearities. In: Proc. 27th IEEE Conf. Decision and Control, pp. 1255–1259 (1988)

  17. Seidman, T.I.: Switching systems I. Control Cybern. 19, 63–92 (1990). Note also: Switching systems: thermostats and periodicity, UMBC Math. Res. Report 83-07 (1983)

    MATH  MathSciNet  Google Scholar 

  18. Seidman, T.I.: The residue of model reduction. In: Alur, R., Henzinger, T.A., Sontag, E.D. (eds.) Hybrid Systems III: Verification and Control. LNCS, vol. 1066, pp. 201–207 (1996)

  19. Seidman, T.I.: Some aspects of modeling with discontinuities. Int. J. Evol. Equ. 3(4), 129–143 (2008).

    Google Scholar 

  20. van der Schaft, A., Schumacher, H.: An Introduction to Hybrid Dynamical Systems. LNCIS, vol. 251. Springer, Berlin (2000)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department Mathematik, Universität Erlangen-Nürnberg, 91058, Erlangen, Germany

    Falk M. Hante & Günter Leugering

  2. Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD, 21250, USA

    Thomas I. Seidman

Authors
  1. Falk M. Hante
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Günter Leugering
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas I. Seidman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Falk M. Hante.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hante, F.M., Leugering, G. & Seidman, T.I. Modeling and Analysis of Modal Switching in Networked Transport Systems. Appl Math Optim 59, 275–292 (2009). https://doi.org/10.1007/s00245-008-9057-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-008-9057-6

Keywords

Search

Navigation