Skip to main content
SpringerLink
Log in

Stability of semilinear elliptic optimal control problems with pointwise state constraints

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

Abstract

A semi-linear elliptic control problems with distributed control and pointwise inequality constraints on the control and the state is considered. The general optimization problem is perturbed by a certain class of perturbations, and we establish convergence of local solutions of the perturbed problems to a local solution of the unperturbed optimal control problem. This class of perturbations include finite element discretization as well as data perturbation such that the theory implies convergence of finite element approximation and stability w.r.t. noisy data.

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. Alibert, J.-J., Raymond, J.-P.: Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 18, 235–250 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Carstensen, C.: Quasi-interpolation and a posteriori error analysis in finite elements methods. Modél. Math. Anal. Numér. 33, 1187–1202 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Casas, E., Mateos, M.: Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21, 1726–1741 (2002)

    MathSciNet  Google Scholar 

  6. Casas, E., Tröltzsch, F.: Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control Cybern. 31, 695–712 (2002)

    MATH  Google Scholar 

  7. Casas, E., de los Reyes, J.-C., Tröltzsch, F.: Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19(2), 616–643 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. de los Reyes, J.C., Meyer, C., Vexler, B.: Finite element error analysis for state-constrained optimal control of the Stokes equations. Control Cybern. 37, 251–284 (2008)

    MATH  Google Scholar 

  9. 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 

  10. Deckelnick, K., Hinze, M.: A finite element approximation to elliptic control problems in the presence of control and state constraints. Preprint HBAM2007-01, Hamburger Beiträge zur Angewandten Mathematik, Universität Hamburg (2007)

  11. Deckelnick, K., Günther, A., Hinze, M.: Finite element approximation of elliptic control problems with constraints on the gradient. Numer. Math. 111, 435–455 (2009)

    Article  Google Scholar 

  12. Griesse, R.: Lipschitz stability of solutions to some state-constrained elliptic optimal control problems. J. Anal. Appl. 25, 435–455 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Griesse, R., Metla, N., Rösch, A.: Convergence analysis of the SQP method for nonlinear mixed-constrained elliptic optimal control problems. Z. Angew. Math. Mech. 88, 776–792 (2008)

    Article  MATH  Google Scholar 

  14. Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris (1992)

    MATH  Google Scholar 

  15. Gröger, K.: A W 1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283, 679–687 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Günther, A., Hinze, M.: Elliptic control problems with gradient constraints—variational discrete versus piecewise constant controls. Comput. Optim. Appl. (2009). doi:10.1007/s10589-009-9308-8

    Google Scholar 

  17. Hintermüller, M., Tröltzsch, F., Yousept, I.: Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems. Numer. Math. 108, 571–603 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Berlin (2009)

    MATH  Google Scholar 

  19. Krumbiegel, K., Meyer, C., Rösch, A.: A priori error analysis for linear quadratic elliptic Neumann boundary control problems with control and state constraints. SIAM J. Control Optim. 48, 5108–5142 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Luenberger, D.G.: Optimization by Vector Space Methods. Wiley, New York (1969)

    MATH  Google Scholar 

  21. Meyer, C.: Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern. 37, 51–85 (2008)

    MATH  Google Scholar 

  22. Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33, 187–208 (2006)

    Article  MathSciNet  Google Scholar 

  23. Meyer, C., Prüfert, U., Tröltzsch, F.: On two numerical methods for state-constrained elliptic control problems. Optim. Methods Softw. 22, 871–899 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nečas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques. Masson, Paris (1967)

    MATH  Google Scholar 

  25. Robinson, S.M.: Stability theory for systems of inequalities, part ii: differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rösch, A., Tröltzsch, F.: On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints. SIAM J. Control Optim. 46, 1098–1115 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schiela, A.: The control reduced interior point method. A function space oriented algorithmic approach. PhD thesis, Verlag Dr. Hut, Munich (2006)

  28. Sevilla, D., Wachsmuth, D.: Polynomial integration on regions defined by a triangle and a conic. In: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation ISSAC, vol. 2010, pp. 163–170 (2010)

    Chapter  Google Scholar 

  29. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second order à coéfficients discontinus. Ann. Inst. Fourier 15, 189–258 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zanger, D.Z.: The inhomogeneous Neumann problem in Lipschitz domains. Commun. Partial Differ. Equ. 25, 1771–1808 (2000)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Department, University Hamburg, Bundesstr. 55, 20146, Hamburg, Germany

    M. Hinze

  2. Graduate School of Computational Engineering, Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    C. Meyer

Authors
  1. M. Hinze
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Meyer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hinze, M., Meyer, C. Stability of semilinear elliptic optimal control problems with pointwise state constraints. Comput Optim Appl 52, 87–114 (2012). https://doi.org/10.1007/s10589-011-9410-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-011-9410-6

Keywords

Search

Navigation