Skip to main content

Advertisement

Springer Nature Link
Log in

Elliptic control problems with gradient constraints—variational discrete versus piecewise constant controls

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

Abstract

We consider an elliptic optimal control problem with pointwise bounds on the gradient of the state. To guarantee the required regularity of the state we include the L r-norm of the control in our cost functional with r>d (d=2,3). We investigate variational discretization of the control problem (Hinze in Comput. Optim. Appl. 30:45–63, 2005) as well as piecewise constant approximations of the control. In both cases we use standard piecewise linear and continuous finite elements for the discretization of the state. Pointwise bounds on the gradient of the discrete state are enforced element-wise. Error bounds for control and state are obtained in two and three space dimensions depending on the value of r.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Brenner, S.C., Scott, R.L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Texts in Applied Mathematics, vol. 15. Springer, New York (2008)

    Book  MATH  Google Scholar 

  2. Casas, E., Fernández, L.: Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state. Appl. Math. Optim. 27, 35–56 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dauge, M.: Neumann and mixed problems on curvilinear polyhedra. Integr. Equ. Oper. Theory 15(2), 227–261 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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  Google Scholar 

  5. Deckelnick, K., Günther, A., Hinze, M.: Finite element approximation of elliptic control problems with constraints on the gradient. Numer. Math. 111, 335–350 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Appl. Math. Sci., vol. 159. Springer, Berlin (2004)

    MATH  Google Scholar 

  7. Friedman, A.: Partial Differential Equations. Holt, Rinehart & Winston, New York (1969)

    MATH  Google Scholar 

  8. Grisvard, P.: Singularities in Boundary Value Problems. Recherches en Mathématiques Appliquées (Research in Applied Mathematics), vol. 22. Masson, Paris (1992)

    MATH  Google Scholar 

  9. 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 

  10. Guzmán, J., Leykekhman, D., Roßmann, J., Schatz, A.H.: Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods. Numer. Math. 112, 221–243 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  13. Jerison, D., Kenig, C.E.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ortner, C., Wollner, W.: A priori error estimates for optimal control problems with pointwise constraints on the gradient of the state. DFG Schwerpunktprogramm 1253, Preprint No. SPP1253-071 (2009)

Download references

Author information

Authors and Affiliations

  1. Bereich Optimierung und Approximation, Universität Hamburg, Bundesstr. 55, 20146, Hamburg, Germany

    Andreas Günther & Michael Hinze

Authors
  1. Andreas Günther
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Hinze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Hinze.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Günther, A., Hinze, M. Elliptic control problems with gradient constraints—variational discrete versus piecewise constant controls. Comput Optim Appl 49, 549–566 (2011). https://doi.org/10.1007/s10589-009-9308-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-009-9308-8