Skip to main content
SpringerLink
Log in

Finite element methods for optimal control problems governed by integral equations and integro-differential equations

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we analyze finite-element Galerkin discretizations for a class of constrained optimal control problems that are governed by Fredholm integral or integro-differential equations. The analysis focuses on the derivation of a priori error estimates and a posteriori error estimators for the approximation schemes.

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. Ainsworth, M., Oden, J.T.: A posteriori error estimators in finite element analysis. Comput. Meth. Appl. Mech. Engrg., 142, 1–88 (1997)

    Google Scholar 

  2. Alt, W.: On the approximation of infinite optimisation problems with an application to optimal control problems. Appl. Math. Optim., 12, 15–27 (1984)

    Google Scholar 

  3. Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge; 1997

  4. Babuška, I., Aziz, A.K.: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Academic Press, New York, 1972

  5. Babuška, I., Miller, A. D.: A feedback finite element method with a posteriori error estimation Part 1. Comput. Methods Appl. Mech. Engrg., 61, 1–40 (1987)

    Google Scholar 

  6. Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optimization, 39, 113–132 (2000)

    Google Scholar 

  7. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978

  8. Dreyer, T., Maar, B., Schulz, V.: Multigrid optimization in application. J. Comput. Appl. Math., 120, 67–84 (2000)

    Google Scholar 

  9. Falk, F.S.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl., 44, 28–47 (1973)

    Google Scholar 

  10. French, D.A., King, J.T.: Approximation of an elliptic control problem by the finite element method. Numer. Funct. Anal. Appl., 12, 299–315 (1991)

    Google Scholar 

  11. Hackbusch, W., Will, Th.: A numerical method for parabolic bang-bang problem. In: Hoffmann, K.H., Krabs, W. (eds.) Optimal control of partial differential equations, Vol. 68 of ISNM, 20–45, Birkhäuser, Basel, 1982

  12. Hackbusch, W., Will, Th.: A numerical method for parabolic bang-bang problem. Control Cybernet, 12, 99–116 (1983)

    Google Scholar 

  13. Huang, Y.C., Li, R., Liu, W.B., Yan, N.: Efficient discretization for finite element approximation of constrained optimal control problem. to appear

  14. Krasnosel’skii, M.A., Zabreiko, P.P., et al.: Integral Operators in Spaces of Summable Functions. Noordhoff International Publishers, Leyden, 1976

  15. Kress, R.: Linear Integral Equations (2nd Edition). Springer-Verlag, New York, 1999

  16. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin, 1971

  17. Lions, J.L.: Some Methods in the Mathematical Analysis of Systems and Their Control. Science Press, Beijing, 1981

  18. Liu, W., Yan, N.: A posteriori error estimates for optimal boundary control. SIAM J. Numer. Anal., 39, 73–99 (2001)

    Google Scholar 

  19. Liu, W.B., Yan, N.: A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math., 15, 285–309 (2002)

    Google Scholar 

  20. Neittaanmaki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. Marcel Dekker, New York, 1994

  21. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp., 54, 483–493 (1990)

    Google Scholar 

  22. Tiba, D.: Lectures on the optimal control of elliptic equations. University of Jyvaskyla Press, Finland, 1995

  23. Verfurth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement. Wiley-Teubner, 1996

  24. Zabreiko, P.P., et al.: Integral Equations – A Reference Text. Noordhoff International Publishers, Leyden, 1975

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada, A1C 5S7

    Hermann Brunner

  2. Institute of System Sciences, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, China, 100080

    Ningning Yan

Authors
  1. Hermann Brunner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ningning Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hermann Brunner.

Additional information

Grants, communicated-by lines, or other notes about the article will be placed here between rules. Such notes are optional.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brunner, H., Yan, N. Finite element methods for optimal control problems governed by integral equations and integro-differential equations. Numer. Math. 101, 1–27 (2005). https://doi.org/10.1007/s00211-005-0608-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-005-0608-3

Mathematics Subject Classification (2000)

Search

Navigation