Skip to main content
SpringerLink
Log in

The derivation of cubic splines with obstacles by methods of optimization and optimal control

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

Interpolating splines which are restricted in their movement by the presence of obstacles are investigated. For simplicity we mainly treat cubic splines which are required to be non-negative. The extension to splines of higher order and to certain other forms of obstacles is straightforward. Methods of optimization and of optimal control are used to obtain necessary optimality criteria. These criteria are applied to derive an algorithm to compute splines which are restricted to constant lower or upper bounds. There is a numerical example which illustrates the method presented.

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. De Boor, C.: A practical guide to splines. Berlin, Heidelberg, New York: Springer 1978

    Google Scholar 

  2. Bryson, Jr., A.E., Denham, W.F., Dreyfus, S.E.: Optimal programming problems with inequality constraints. I. Necessary conditions for extremal solutions. AIAA J.1, 2544–2550 (1963)

    Google Scholar 

  3. Bulirsch, R., Rutishauser, H.: Interpolation und genäherte Quadratur. R. Sauer and I. Szabó (eds.), Mathematische Hilfsmittel des Ingenieurs, III, pp. 232–319. Berlin, Heidelberg, New York: Springer 1968

    Google Scholar 

  4. Gelfand, F.M., Fomin, S.V.: Calculus of variations. Englewood Cliffs, N.J., Prentice-Hall 1963

    Google Scholar 

  5. Holladay, J.C.: A smoothest curve approxination, Math. Tables Aids Comput.11, 233–243 (1957)

    Google Scholar 

  6. Jacobson, D.H., Lele, M.M., Speyer, J.L.: New necessary conditions of optimality for control problems with state-variable inequality constraints. J. Math. Anal. Appl.35, 255–284 (1971)

    Google Scholar 

  7. Link, H.: Über den geraden Knickstab mit begrenzter Durchbiegung. Ing. Arch.22, 237–250 (1954)

    Google Scholar 

  8. Luenberger, D.G.: Optimization by vector space methods. New York: Wiley 1969

    Google Scholar 

  9. Mangasarian, O.L., Schumaker, L.L.: Splines via optimal control. I.J. Schoenberg (ed.). Approximations with special emphasis on spline functions (Proc. Symp. Univ. of Wisconsin, Madison, Wis.), pp. 119–156. New York, London: Academic Press 1969

    Google Scholar 

  10. Maurer, H.: On the minimum principle for optimal control problems with state constraints. Schriftenreihe des Rechenzentrums der Universität Münster, ISSN 0344-0842, Nr. 41, 1979

  11. Maurer, H.: Differential stability in optimal control problems. Appl. Math. Optim.5, 283–295 (1979)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Insitute of Applied Mathematics, University of Hamburg, Bundesstrasse 55, D-2000, Hamburg 13, Federal Republic of Germany

    Gerhard Opfer & Hans Joachim Oberle

Authors
  1. Gerhard Opfer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans Joachim Oberle
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Günter Meinardus on the occasion of his 60th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Opfer, G., Joachim Oberle, H. The derivation of cubic splines with obstacles by methods of optimization and optimal control. Numer. Math. 52, 17–31 (1987). https://doi.org/10.1007/BF01401019

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01401019

Subject classifications

Search

Navigation