Skip to main content
SpringerLink
Log in

Optimal Control of PDEs with Regularized Pointwise State Constraints

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

Abstract

This paper addresses the regularization of pointwise state constraints in optimal control problems. By analyzing the associated dual problem, it is shown that the regularized problems admit Lagrange multipliers in L2-spaces. Under a certain boundedness assumption, the solution of the regularized problem converges to the one of the original state constrained problem. The results of our analysis are confirmed by numerical tests.

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. N. Arada and J.-P. Raymond, “Optimal control problems with mixed control-state constraints,” SIAM J. Control and Optimization, vol. 39, pp. 1391–1407, 2000.

    MathSciNet  Google Scholar 

  2. M. Bergounioux, K. Ito, and K. Kunisch, “Primal-dual strategy for constrained optimal control problem,” SIAM J. Control and Optimization, vol. 37, pp. 1176–1194, 1999.

    MathSciNet  Google Scholar 

  3. M. Bergounioux and K. Kunisch, “Primal-dual strategy for state-constrained optimal control problems,” Computational Optimization and Applications, vol. 22, pp. 193–224, 2002.

    Article  MathSciNet  Google Scholar 

  4. M. Bergounioux and F. Tröltzsch, “Optimal control of linear bottleneck problems,” ESAIM, Control, Optimisation and Calculus of Variations, vol. 3, pp. 235–250,1998.

  5. M. Bergounioux and F. Tröltzsch, “Optimal control of semilinear parabolic equations with state-constraints of bottleneck type,” ESAIM, Control, Optimisation and Calculus of Variations, vol. 4, pp. 595–608, 1999.

    MathSciNet  Google Scholar 

  6. E. Casas, “Boundary control of semilinear elliptic equations with pointwise state constraints,” SIAM J. Control and Optimization, vol. 31, pp. 993–1006, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  7. E. Casas, J.P. Raymond, and H. Zidani, “Pontryagin's principle for local solutions of control problems with mixed control-state contraints,” SIAM J. Control and Optimization, vol. 39, pp. 1182–1203, 2000.

    Article  MathSciNet  Google Scholar 

  8. K. Kunisch and A. Rösch, “Primal-dual active set strategy for a general class of constrained optimal control problems,” SIAM J. Optimization, vol. 13, pp. 321–334, 2002.

    Article  Google Scholar 

  9. J.P. Raymond, “Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints,” Discrete Contin. Dyn. Syst., vol. 3, pp. 341–370, 1997.

    MATH  MathSciNet  Google Scholar 

  10. F. Tröltzsch, “Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints,” SIAM J. Optimization, to appear.

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische, Universität Berlin, Str. des 17. Juni 136, D-10623, Berlin, Germany

    Christian Meyer, Arnd Rösch & Fredi Tröltzsch

Authors
  1. Christian Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arnd Rösch
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fredi Tröltzsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported by the DFG Research Center “Mathematics for key technologies” (FZT 86) in Berlin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Meyer, C., Rösch, A. & Tröltzsch, F. Optimal Control of PDEs with Regularized Pointwise State Constraints. Comput Optim Applic 33, 209–228 (2006). https://doi.org/10.1007/s10589-005-3056-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-005-3056-1

Keywords

Search

Navigation