IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-1-4612-5651-9_2.html
   My bibliography  Save this book chapter

Control Theory and Singular Riemannian Geometry

In: New Directions in Applied Mathematics

Author

Listed:
  • R. W. Brockett

    (Harvard University, Division of Applied Sciences)

Abstract

This paper discusses the qualitative and quantitative aspects of the solution of a class of optimal control problems, together with related questions concerning a corresponding stochastic differential equation. The class has been chosen to reveal what one may expect for the structure of the set of conjugate points for smooth problems in which existence of optimal trajectories is not an issue but for which Lie bracketing is necessary to reveal the reachable set. It is, perhaps, not too surprising that in thinking about this problem various geometrical analogies are useful and, in the final analysis, provide a convenient language to express the results. Indeed, the geodesic problem of Riemannian geometry is commonly taken to be the paradigm in the calculus of variations; a point of view which is supported by a variety of variational principles such as the theorem of Euler which identifies the path of a freely moving particle on a manifold with a geodesic and the whole theory of general relativity. Nonetheless, the class of variational problems considered here can only be thought of as geodesic problems in some limiting sense in which the metric tends to infinity. For this reason the geodesic analogy has to be developed rather carefully.

Suggested Citation

  • R. W. Brockett, 1982. "Control Theory and Singular Riemannian Geometry," Springer Books, in: Peter J. Hilton & Gail S. Young (ed.), New Directions in Applied Mathematics, pages 11-27, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-5651-9_2
    DOI: 10.1007/978-1-4612-5651-9_2
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:sprchp:978-1-4612-5651-9_2. 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.