IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v186y2020i1d10.1007_s10957-020-01684-z.html
   My bibliography  Save this article

Optimal Control of Fractional Elliptic PDEs with State Constraints and Characterization of the Dual of Fractional-Order Sobolev Spaces

Author

Listed:
  • Harbir Antil

    (George Mason University)

  • Deepanshu Verma

    (George Mason University)

  • Mahamadi Warma

    (George Mason University)

Abstract

This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic partial differential equations. Several mathematical tools are developed during the process to study these problems, for instance, the characterization of the dual of fractional-order Sobolev spaces and the well-posedness of fractional elliptic equations with measure-valued data. These tools are widely applicable. We show well-posedness of the optimal control problem and derive the first- order optimality conditions. Notice that the adjoint equation is a fractional partial differential equation with a measure as the right-hand-side datum. We use the characterization of the fractional- order dual spaces to study the regularity of solutions of the state and adjoint equations. We emphasize that the classical case was considered by E. Casas, but almost none of the existing results are applicable to our fractional case. As an application of the regularity result of the adjoint equation, we establish the Sobolev regularity of the optimal control. In addition, under this setup, even weaker controls can be used.

Suggested Citation

  • Harbir Antil & Deepanshu Verma & Mahamadi Warma, 2020. "Optimal Control of Fractional Elliptic PDEs with State Constraints and Characterization of the Dual of Fractional-Order Sobolev Spaces," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 1-23, July.
    • Handle: RePEc:spr:joptap:v:186:y:2020:i:1:d:10.1007_s10957-020-01684-z
    DOI: 10.1007/s10957-020-01684-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-020-01684-z
    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/s10957-020-01684-z?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. Wang, Fangyuan & Chen, Chuanjun & Zhou, Zhaojie, 2024. "Integral constraint regularization method for fractional optimal control problem with pointwise state constraint," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).

    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:186:y:2020:i:1:d:10.1007_s10957-020-01684-z. 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.