IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v154y2012i3d10.1007_s10957-012-0050-5.html
   My bibliography  Save this article

Optimal Control and Applications to Aerospace: Some Results and Challenges

Author

Listed:
  • E. Trélat

    (Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France)

Abstract

This article surveys the usual techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem, in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the usual tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 1980s and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric reentry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem toward a simpler one and then of solving a series of parameterized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low-cost interplanetary space missions. The article ends with open problems and perspectives.

Suggested Citation

  • E. Trélat, 2012. "Optimal Control and Applications to Aerospace: Some Results and Challenges," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 713-758, September.
  • Handle: RePEc:spr:joptap:v:154:y:2012:i:3:d:10.1007_s10957-012-0050-5
    DOI: 10.1007/s10957-012-0050-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-012-0050-5
    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/s10957-012-0050-5?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. M. Gerdts, 2003. "Direct Shooting Method for the Numerical Solution of Higher-Index DAE Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 117(2), pages 267-294, May.
    2. W. Grimm & A. Markl, 1997. "Adjoint Estimation from a Direct Multiple Shooting Method," Journal of Optimization Theory and Applications, Springer, vol. 92(2), pages 263-283, February.
    3. J.B. Caillau & J. Gergaud & J. Noailles, 2003. "3D Geosynchronous Transfer of a Satellite: Continuation on the Thrust," Journal of Optimization Theory and Applications, Springer, vol. 118(3), pages 541-565, September.
    4. F. Bonnans & P. Martinon & E. Trélat, 2008. "Singular Arcs in the Generalized Goddard’s Problem," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 439-461, November.
    Full references (including those not matched with items on IDEAS)

    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. M. Soledad Aronna & J. Frédéric Bonnans & Pierre Martinon, 2013. "A Shooting Algorithm for Optimal Control Problems with Singular Arcs," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 419-459, August.
    2. M. Gerdts, 2006. "Local Minimum Principle for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two," Journal of Optimization Theory and Applications, Springer, vol. 130(3), pages 443-462, September.
    3. Gerdts, Matthias, 2008. "A nonsmooth Newton’s method for control-state constrained optimal control problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(4), pages 925-936.
    4. Matthias Gerdts & Martin Kunkel, 2011. "A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems," Computational Optimization and Applications, Springer, vol. 48(3), pages 601-633, April.
    5. M. Gerdts, 2006. "Representation of the Lagrange Multipliers for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two," Journal of Optimization Theory and Applications, Springer, vol. 130(2), pages 231-251, August.
    6. E. Cristiani & P. Martinon, 2010. "Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 321-346, August.
    7. Jinhai Chen & Herschel Rabitz, 2019. "On Lifting Operators and Regularity of Nonsmooth Newton Methods for Optimal Control Problems of Differential Algebraic Equations," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 518-535, February.

    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:154:y:2012:i:3:d:10.1007_s10957-012-0050-5. 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.