Skip to main content
SpringerLink
Log in

A general theorem on error estimates with application to a quasilinear elliptic optimal control problem

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

A theorem on error estimates for smooth nonlinear programming problems in Banach spaces is proved that can be used to derive optimal error estimates for optimal control problems. This theorem is applied to a class of optimal control problems for quasilinear elliptic equations. The state equation is approximated by a finite element scheme, while different discretization methods are used for the control functions. The distance of locally optimal controls to their discrete approximations is estimated.

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. Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23, 201–229 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonnans, F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    MATH  Google Scholar 

  3. Bonnans, J.F., Zidani, H.: Optimal control problems with partially polyhedric constraints. SIAM J. Control Optim. 37(6), 1726–1741 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)

    MATH  Google Scholar 

  5. Casas, E.: Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM Control Optim. Calc. Var. 8, 345–374 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Casas, E.: Using piecewise linear functions in the approximation of semilinear elliptic control problems. Adv. Comput. Math. 26, 137–153 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Casas, E., Mateos, M., Tröltzsch, F.: Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 31, 193–220 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Casas, E., Raymond, J.P.: Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45, 1586–1611 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Casas, E., Tröltzsch, F.: First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations. SIAM J. Control Optim. 48, 688–718 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Casas, E., Tröltzsch, F.: Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations. ESAIM Control Optim. Calc. Var. (2009). doi:10.1051/cocv/2009010

    Google Scholar 

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

    MATH  Google Scholar 

  12. Deckelnick, K., Günther, A., Hinze, M.: Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains. SIAM J. Control Optim. 48(4), 2798–2819 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Deckelnick, K., Hinze, M.: Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal. 45, 1937–1953 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deckelnick, K., Hinze, M.: Numerical analysis of a control and state constrained elliptic control problem with piecewise constant control approximations. In: Proceedings of the ENUMATH (2007)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Geveci, T.: On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO Anal. Numér. 13, 313–328 (1979)

    MathSciNet  MATH  Google Scholar 

  17. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)

    MATH  Google Scholar 

  18. Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30, 45–63 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hinze, M., Tröltzsch, F.: Discrete concepts versus error analysis in PDE constrained optimization. In: GAMM-Mitteilungen, March (2011)

    Google Scholar 

  20. Meyer, C., Rösch, A.: Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43, 970–985 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Raviart, P.A., Thomas, J.M.: Introduction à L’analyse Numérique des Equations aux Dérivées Partielles. Masson, Paris (1983)

    MATH  Google Scholar 

  22. Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications, vol. 112. American Math. Society, Providence (2010)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to one of the referees for his detailed comments, in particular for pointing out a technical problem in the first version of the paper.

Author information

Authors and Affiliations

  1. Departmento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005, Santander, Spain

    Eduardo Casas

  2. Institut für Mathematik, Technische Universität Berlin, 10623, Berlin, Germany

    Fredi Tröltzsch

Authors
  1. Eduardo Casas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fredi Tröltzsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fredi Tröltzsch.

Additional information

This work was supported by Spanish Ministerio de Ciencia e Innovación under the project MTM2008-04206.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Casas, E., Tröltzsch, F. A general theorem on error estimates with application to a quasilinear elliptic optimal control problem. Comput Optim Appl 53, 173–206 (2012). https://doi.org/10.1007/s10589-011-9453-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-011-9453-8

Keywords

Search

Navigation