Numerische Mathematik

, Volume 120, Issue 4, pp 733–762 | Cite as

A-posteriori error estimates for optimal control problems with state and control constraints

  • Arnd Rösch
  • Daniel Wachsmuth


We discuss the full discretization of an elliptic optimal control problem with pointwise control and state constraints. We provide the first reliable a-posteriori error estimator that contains only computable quantities for this class of problems. Moreover, we show, that the error estimator converges to zero if one has convergence of the discrete solutions to the solution of the original problem. The theory is illustrated by numerical tests.

Mathematics Subject Classification (2000)

Primary 65N15 Secondary 49M25 65N30 65K10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Apel T., Benedix O., Sirch D., Vexler B.: A priori mesh grading for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal. 49(3), 992–1005 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Araya R., Behrens E., Rodríguez R.: A posteriori error estimates for elliptic problems with Dirac delta source terms. Numer. Math. 105(2), 193– (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Benedix O., Vexler B.: A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints. Comput. Optim. Appl. 44(1), 3–25 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Braess D.: Finite Elemente—Theorie, Schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer, Berlin (1997)Google Scholar
  5. 5.
    Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)MATHGoogle Scholar
  6. 6.
    Casas E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24(6), 1309–1318 (1986)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Cherednichenko, S., Krumbiegel, K., Rösch, A.: Error estimates for the Lavrentiev regularization of elliptic optimal control problems. Inverse Probl. 24(5), 055003, 21 (2008)Google Scholar
  8. 8.
    Ciarlet P.G., Raviart P.-A.: Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2, 17–31 (1973)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Deckelnick K., Hinze M.: Convergence of a finite element approximation to a state-constrained elliptic control problem. SIAM J. Numer. Anal. 45(5), 1937–1953 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Grisvard P.: Elliptic Problems in Nonsmooth Domains. Pitman, London (1985)MATHGoogle Scholar
  11. 11.
    Günther A., Hinze M.: A posteriori error control of a state constrained elliptic control problem. J. Numer. Math. 16(4), 307–322 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hintermüller M., Hoppe R.H.W.: Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations. SIAM J. Control Optim. 48, 5468–5487 (2010)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hintermüller M., Hoppe R.H.W., Iliash Y., Kieweg M.: An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM Control Optim. Calc. Var. 14(3), 540–560 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hinze M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. J. Comput. Optim. Appl. 30, 45–63 (2005)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hinze M., Pinnau R., Ulbrich M., Ulbrich S.: Optimization with PDE Constraints, Volume 23 of Mathematical Modelling: Theory and Applications. Springer, New York (2009)Google Scholar
  16. 16.
    Hoppe R.H.W., Kieweg M.: A posteriori error estimation of finite element approximations of pointwise state constrained distributed control problems. J. Numer. Math. 17(3), 219–244 (2009)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Krumbiegel K., Rösch A.: A new stopping criterion for iterative solvers for control constrained optimal control problems. Arch. Control Sci. 18(1), 17–42 (2008)MathSciNetMATHGoogle Scholar
  18. 18.
    Li R., Liu W., Yan N.: A posteriori error estimates of recovery type for distributed convex optimal control problems. J. Sci. Comput. 33(2), 155–182 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Meyer C.: Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern. 37(1), 51–83 (2008)MATHGoogle Scholar
  20. 20.
    Meyer C., de los Reyes J.C., Vexler B.: Finite element error analysis for state-constrained optimal control of the Stokes equations. Control Cybern. 37, 251–284 (2008)MATHGoogle Scholar
  21. 21.
    Nochetto R.H., Schmidt A., Siebert K.G., Veeser A.: Pointwise a posteriori error estimates for monotone semi-linear equations. Numer. Math. 104(4), 515–538 (2006)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Plum M.: Computer-assisted enclosure methods for elliptic differential equations. Linear Algebra Appl. 324(1–3), 147–187 (2001)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Tröltzsch F.: Optimal Control of Partial Differential Equations, Volume 112 of Graduate Studies in Mathematics. AMS, Providence (2010)Google Scholar
  24. 24.
    Wollner W.: A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints. Comput. Optim. Appl. 47(1), 133–159 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität Duisburg-EssenDuisburgGermany
  2. 2.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria

Personalised recommendations