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Error estimates for integral constraint regularization of state-constrained elliptic control problems

Author

Listed:
  • B. Jadamba

    () (Rochester Institute of Technology)

  • A. Khan

    () (Rochester Institute of Technology)

  • M. Sama

    () (E.T.S.I.I. Universidad Nacional de Educación a Distancia)

Abstract

Abstract In this paper, we study new aspects of the integral contraint regularization of state-constrained elliptic control problems (Jadamba et al. in Syst Control Lett 61(6):707–713, 2012). Besides giving new results on the regularity and the convergence of the regularized controls and associated Lagrange multipliers, the main objective of this paper is to give abstract error estimates for the regularization error. We also consider a discretization of the regularized problems and derive numerical estimates which are uniform with respect to the regularization parameter and the discretization parameter. As an application of these results, we prove that this discretization is indeed a full discretization of the original problem defined in terms of a problem with finitely many integral constraints. Detailed numerical results justifying the theoretical findings as well as a comparison of our work with the existing literature is also given.

Suggested Citation

  • B. Jadamba & A. Khan & M. Sama, 2017. "Error estimates for integral constraint regularization of state-constrained elliptic control problems," Computational Optimization and Applications, Springer, vol. 67(1), pages 39-71, May.
    • Handle: RePEc:spr:coopap:v:67:y:2017:i:1:d:10.1007_s10589-016-9885-2
    DOI: 10.1007/s10589-016-9885-2
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    References listed on IDEAS

    as
    1. Klaus Krumbiegel & Ira Neitzel & Arnd Rösch, 2012. "Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints," Computational Optimization and Applications, Springer, vol. 52(1), pages 181-207, May.
    2. M. Hinze & C. Meyer, 2010. "Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems," Computational Optimization and Applications, Springer, vol. 46(3), pages 487-510, July.
    3. Dmitriy Leykekhman & Dominik Meidner & Boris Vexler, 2013. "Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints," Computational Optimization and Applications, Springer, vol. 55(3), pages 769-802, July.
    4. Eduardo Casas & Mariano Mateos, 2012. "Numerical approximation of elliptic control problems with finitely many pointwise constraints," Computational Optimization and Applications, Springer, vol. 51(3), pages 1319-1343, April.
    5. Michael Hinze & Anton Schiela, 2011. "Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment," Computational Optimization and Applications, Springer, vol. 48(3), pages 581-600, April.
    6. K. Krumbiegel & A. Rösch, 2009. "A virtual control concept for state constrained optimal control problems," Computational Optimization and Applications, Springer, vol. 43(2), pages 213-233, June.
    7. Bienvenido Jiménez & Vicente Novo & Miguel Sama, 2013. "An extension of the Basic Constraint Qualification to nonconvex vector optimization problems," Journal of Global Optimization, Springer, vol. 56(4), pages 1755-1771, August.
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