IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v56y2013i1p153-185.html
   My bibliography  Save this article

On the optimal control of the Schlögl-model

Author

Listed:
  • Rico Buchholz
  • Harald Engel
  • Eileen Kammann
  • Fredi Tröltzsch

Abstract

Optimal control problems for a class of 1D semilinear parabolic equations with cubic nonlinearity are considered. This class is also known as the Schlögl model. Main emphasis is laid on the control of traveling wave fronts that appear as typical solutions to the state equation. The well-posedness of the optimal control problem and the regularity of its solution are proved. First-order necessary optimality conditions are established by standard adjoint calculus. The state equation is solved by the implicit Euler method in time and a finite element technique with respect to the spatial variable. Moreover, model reduction by Proper Orthogonal Decomposition is applied and compared with the numerical solution of the full problem. To solve the optimal control problems numerically, the performance of different versions of the nonlinear conjugate gradient method is studied. Various numerical examples demonstrate the capacities and limits of optimal control methods. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Rico Buchholz & Harald Engel & Eileen Kammann & Fredi Tröltzsch, 2013. "On the optimal control of the Schlögl-model," Computational Optimization and Applications, Springer, vol. 56(1), pages 153-185, September.
    • Handle: RePEc:spr:coopap:v:56:y:2013:i:1:p:153-185
    DOI: 10.1007/s10589-013-9550-y
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-013-9550-y
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10589-013-9550-y?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.

    Citations

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


    Cited by:

    1. Daraghmeh, Adnan & Hartmann, Carsten & Qatanani, Naji, 2019. "Balanced model reduction of linear systems with nonzero initial conditions: Singular perturbation approximation," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 295-307.
    2. Uzunca, M. & Karasözen, B. & Küçükseyhan, T., 2017. "Moving mesh discontinuous Galerkin methods for PDEs with traveling waves," Applied Mathematics and Computation, Elsevier, vol. 292(C), pages 9-18.
    3. Jens Lang & Bernhard A. Schmitt, 2024. "Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 985-1026, October.
    4. Peter Nestler & Eckehard Schöll & Fredi Tröltzsch, 2016. "Optimization of nonlocal time-delayed feedback controllers," Computational Optimization and Applications, Springer, vol. 64(1), pages 265-294, May.
    5. Eduardo Casas & Christopher Ryll & Fredi Tröltzsch, 2018. "Optimal control of a class of reaction–diffusion systems," Computational Optimization and Applications, Springer, vol. 70(3), pages 677-707, July.

    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:coopap:v:56:y:2013:i:1:p:153-185. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.