IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v171y2016i1d10.1007_s10957-016-0984-0.html
   My bibliography  Save this article

Hierarchical Control for the Wave Equation with a Moving Boundary

Author

Listed:
  • Isaías Pereira Jesus

    (Universidade Federal do Piauí)

Abstract

This paper addresses the study of the hierarchical control for the one-dimensional wave equation in intervals with a moving boundary. This equation models the motion of a string where an endpoint is fixed and the other one is moving. When the speed of the moving endpoint is less than the characteristic speed, the controllability of this equation is established. We assume that we can act on the dynamic of the system by a hierarchy of controls. According to the formulation given by Stackelberg (Marktform und Gleichgewicht. Springer, Berlin, 1934), there are local controls called followers and global controls called leaders. In fact, one considers situations where there are two cost (objective) functions. One possible way is to cut the control into two parts, one being thought of as “the leader” and the other one as “the follower.” This situation is studied in the paper, with one of the cost functions being of the controllability type. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control.

Suggested Citation

  • Isaías Pereira Jesus, 2016. "Hierarchical Control for the Wave Equation with a Moving Boundary," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 336-350, October.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:1:d:10.1007_s10957-016-0984-0
    DOI: 10.1007/s10957-016-0984-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-016-0984-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-016-0984-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. A.M. Ramos & R. Glowinski & J. Periaux, 2002. "Pointwise Control of the Burgers Equation and Related Nash Equilibrium Problems: Computational Approach," Journal of Optimization Theory and Applications, Springer, vol. 112(3), pages 499-516, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Madureira, Rodrigo L.R. & Rincon, Mauro A. & Aouadi, Moncef, 2019. "Global existence and numerical simulations for a thermoelastic diffusion problem in moving boundary," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 410-431.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. A.M. Ramos & R. Glowinski & J. Periaux, 2002. "Nash Equilibria for the Multiobjective Control of Linear Partial Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 112(3), pages 457-498, March.
    2. B. Ivorra & A. M. Ramos & B. Mohammadi, 2007. "Semideterministic Global Optimization Method: Application to a Control Problem of the Burgers Equation," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 549-561, December.
    3. A. M. Croicu & M. Y. Hussaini, 2008. "Multiobjective Stochastic Control in Fluid Dynamics via Game Theory Approach: Application to the Periodic Burgers Equation," Journal of Optimization Theory and Applications, Springer, vol. 139(3), pages 501-514, December.
    4. T. Roubíček, 2007. "On Nash Equilibria for Noncooperative Games Governed by the Burgers Equation," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 41-50, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:171:y:2016:i:1:d:10.1007_s10957-016-0984-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.