Skip to main content
SpringerLink
Log in

Conjugate-point conditions for variational problems with delayed argument

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

A variational problem with delayed argument is investigated. The existing necessary conditions are first reviewed. New results, two conjugate-point conditions, are then derived for this problem. The method of proof is similar to that used by Bliss for the classical problem. An example shows that the two conditions are not equivalent, and that the first-order necessary conditions, the strengthened Legendre conditions, and the conjugate-point conditions do not in general constitute a set of sufficient conditions for the delay problem. It is shown that a special case, referred to as the separated-integrand problem, leads to considerable simplification of the results for the general problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hughes, D. K.,Variational and Optimal Control Problems with Delayed Argument, Journal of Optimization Theory and Applications, Vol. 2, No. 1, 1968.

  2. Sabbagh, L. D.,Variational Problems with Lags, Journal of Optimization Theory and Applications, Vol. 3, No. 1, 1969.

  3. Bliss, G. A.,Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Illinois, 1946.

    Google Scholar 

  4. Reid, W. T.,Discontinuous Solutions in the Non-Parametric Problem of Mayer in the Calculus of Variations, American Journal of Mathematics, Vol. 57, No. 1, 1935.

  5. Palm, W. J.,Optimal Control Problems and Differential Games with Time Delays, Northwestern University, Ph.D. Thesis, 1971.

  6. Ewing, G. M.,Calculus of Variations, with Applications, W. W. Norton, New York, New York, 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Rhode Island, Kingston, Rhode Island

    W. J. Palm (Assistant Professor of Mechanical Engineering and Applied Mechanics)

  2. Northwestern University, Evanston, Illinois

    W. E. Schmitendorf (Associate Professor of Mechanical Engineering and Astronautical Sciences)

Authors
  1. W. J. Palm
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. E. Schmitendorf
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by L. D. Berkovitz

This research was supported in part by a National Defense Education Act Fellowship and the National Science Foundation under Grant No. GK-3341.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Palm, W.J., Schmitendorf, W.E. Conjugate-point conditions for variational problems with delayed argument. J Optim Theory Appl 14, 599–612 (1974). https://doi.org/10.1007/BF00932963

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00932963

Key Words

Search

Navigation