Skip to main content
SpringerLink
Log in

Method of penalization for the state equation for an elliptical optimal control problem

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We solve by finite difference method an optimal control problem of a system governed by a linear elliptic equation with pointwise control constraints and non-local state constraints. A discrete optimal control problem is approximated by a minimization problem with penalized state equation. We derive the error estimates for the distance between the exact and regularized solutions. We also prove the rate of convergence of block Gauss–Seidel iterative solution method for the penalized problem. We present and analyze the results of the numerical experiments.

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. Bergounioux, M., Haddou, M., Hintermu¨ ller, M., Kunisch, K. “A Comparison of a Moreau–Yosida-Based Active Set Strategy and Interior Point Methods for Constrained Optimal Control Problems,” SIAM J. Optim. 11, No. 2, 495–521 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  2. Bergounioux, M., Kunisch, K. “Primal-Dual Active Set Strategy for State-Constrained Optimal Control Problems,” Comput. Optim. Appl. 22, No. 2, 193–224 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  3. Tro¨ ltzsch, F., Yousept, I. “A Regularization Method for the Numerical Solution of Elliptic Boundary Control Problems with Pointwise State Constraints,” Comput. Optim. Appl. 42, No. 1, 43–66 (2009).

    Article  MathSciNet  Google Scholar 

  4. Hintermu¨ ller, M., Hinze, M. “Moreau–Yosida Regularization in State Constrained Elliptic Control Problems: Error Estimates and Parameter Adjustment,” SIAM J. Numer. Anal. 47, No. 3, 1666–1683 (2009).

    Article  MathSciNet  Google Scholar 

  5. Gra¨ ser, C., Kornhuber, R. “Nonsmooth Newton Methods for Set-Valued Saddle Point Problems,” SIAM J. Numer. Anal. 47, No. 2, 1251–1273 (2009).

    Article  MathSciNet  Google Scholar 

  6. Schiela, A., Gu¨ nther, A. “An Interior Point Algorithm with Inexact Step Computation in Function Space for State Constrained Optimal Control,” Numer.Math. 119, No. 2, 373–407 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  7. Hinze, M., Schiela, A. “Discretization of Interior Point Methods for State Constrained Elliptic Optimal Control Problems: Optimal Error Estimates and Parameter Adjustment,” Comput. Optim. Appl. 48, No. 3, 581–600 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  8. Laitinen, E., Lapin, A., Lapin, S. “On the Iterative Solution Methods of Finite-Dimensional Inclusions with Applications to Optimal Control Problems,”Comp. Methods Appl. Math. 10, No. 3, 283–301 (2010).

    MATH  MathSciNet  Google Scholar 

  9. Lapin, A. “Preconditioned Uzawa-Type Methods for Finite-Dimensional Constrained Saddle Point Problems,” Lobachevskii J.Math. 31, No. 4, 309–322 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  10. Lapin, A., Khasanov, M. “State-Constrained Optimal Control of an Elliptic Equation with Its Right-Hand Side Used as Control Function,” Lobachevskii J.Math. 32, No. 4, 453–462 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  11. Laitinen, E., Lapin, A. “Iterative Solution Methods for a Class of State Constrained Optimal Control Problems,” Appl.Math. 3, No. 12, 1862–1867 (2012).

    Article  Google Scholar 

  12. Zalyalov, D. G., Lapin, A. V. “Numerical Solution of an Optimal Control Problem of a System Governed by Linear Elliptic Equation in Presence of Nonlocal State Constraints,” Uchen. Zap. Kazan. Univ. 154 (3), 129–144 (2012).

    Google Scholar 

  13. Laitinen, E., Lapin, A. “Iterative SolutionMethods for the Large-Scale Constrained Saddle Point Problems,” in Numerical Methods for Differential Equations, Optimization, and Technological Problems, Comp. Meth. Appl. Sci. (Springer, Dordrecht, 2013), Vol. 27, pp. 19–39.

    MathSciNet  Google Scholar 

  14. Laitinen, E., Lapin, A., Lapin, S. “Iterative Solution Methods for Variational Inequalities with Nonlinear Main Operator and Constraints to Gradient of Solution,” Lobachevskii J.Math. 33, No. 4, 341–352 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  15. Barbu, V. Nonlinear Semigroups and Differential Equations in Banach Spaces (Editura Academiei Republicii Socialiste Romaˆnia, Bucharest; Noordhoff International Publishing, Leiden, 1976).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008, Russia

    A. V. Lapin & D. G. Zalyalov

Authors
  1. A. V. Lapin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. G. Zalyalov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Lapin.

Additional information

Original Russian Text © A.V. Lapin, D.G. Zalyalov, 2015, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2015, No. 7, pp. 10–24.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lapin, A.V., Zalyalov, D.G. Method of penalization for the state equation for an elliptical optimal control problem. Russ Math. 59, 31–43 (2015). https://doi.org/10.3103/S1066369X1507004X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X1507004X

Keywords

Search

Navigation