Computational Optimization and Applications

, Volume 63, Issue 3, pp 825–853 | Cite as

Dirichlet control of elliptic state constrained problems

  • Mariano Mateos
  • Ira Neitzel


We study a state constrained Dirichlet optimal control problem and derive a priori error estimates for its finite element discretization. Additional control constraints may or may not be included in the formulation. The pointwise state constraints are prescribed in the interior of a convex polygonal domain. We obtain a priori error estimates for the \(L^2(\varGamma )\)-norm of order \(h^{1-1/p}\) for pure state constraints and \(h^{3/4-1/(2p)}\) when additional control constraints are present. Here, p is a real number that depends on the largest interior angle of the domain. Unlike in e.g. distributed or Neumann control problems, the state functions associated with \(L^2\)-Dirichlet control have very low regularity, i.e. they are elements of \(H^{1/2}(\varOmega )\). By considering the state constraints in the interior we make use of higher interior regularity and separate the regularity limiting influences of the boundary on the one-hand, and the measure in the right-hand-side of the adjoint equation associated with the state constraints on the other hand. We note in passing that in case of control constraints, these may be interpreted as state constraints on the boundary.


State constraints Dirichlet control Optimality conditions Finite elements A priori error estimates 

Mathematics Subject Classification

49M25 49M05 49K20 65N15 



The first author was partially supported by the Spanish Ministerio of Economía y Competitividad under project MTM2011-22711.


  1. 1.
    Apel, T., Mateos, M., Pfefferer, J., Rösch, A.: On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains (2014). SubmittedGoogle Scholar
  2. 2.
    Berggren, M.: Approximations of very weak solutions to boundary-value problems. SIAM J. Numer. Anal. 42(2), 860–877 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37(4), 1176–1194 (1999). doi: 10.1137/S0363012997328609. (electronic)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Bourdaud, G., Sickel, W.: Composition operators on function spaces with fractional order of smoothness. In: Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku Bessatsu, B26, pp. 93–132. Res. Inst. Math. Sci. (RIMS), Kyoto (2011)Google Scholar
  5. 5.
    Bramble, J.H., King, J.T.: A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries. Math. Comput. 63(207), 1–17 (1994). doi: 10.2307/2153559 CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24(6), 1309–1318 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Casas, E., Günther, A., Mateos, M.: A paradox in the approximation of Dirichlet control problems in curved domains. SIAM J. Control Optim. 49(5), 1998–2007 (2011). doi: 10.1137/100794882 CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Casas, E., Mateos, M.: Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21(1), 67–100 (2002). Special issue in memory of Jacques-Louis LionsMathSciNetzbMATHGoogle Scholar
  10. 10.
    Casas, E., Mateos, M.: Numerical approximation of elliptic control problems with finitely many pointwise constraints. Comput. Optim. Appl. 51, 1319–1343 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Casas, E., Mateos, M., Raymond, J.P.: Penalization of Dirichlet optimal control problems. ESAIM Control Optim. Calc. Var. 15(4), 782–809 (2009). doi: 10.1051/cocv:2008049 CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Casas, E., Mateos, M., Vexler, B.: New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM Control Optim. Calc. Var. 20(3), 803–822 (2014). doi: 10.1051/cocv/2013084 CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Casas, E., Raymond, J.P.: Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45(5), 1586–1611 (2006). doi: 10.1137/050626600. (electronic)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Casas, E., Raymond, J.P.: The stability in \(W^{s, p}(\varGamma )\) spaces of \(L^2\)-projections on some convex sets. Numer. Funct. Anal. Optim. 27(2), 117–137 (2006). doi: 10.1080/01630560600569940 CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, Vol. II, Handbook of Numerical Analysis, II, pp. 17–351. North-Holland (1991)Google Scholar
  16. 16.
    Deckelnick, K., Günther, A., Hinze, M.: Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains. SIAM J. Control Optim. 48(4), 2798–2819 (2009). doi: 10.1137/080735369 CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    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)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)zbMATHGoogle Scholar
  19. 19.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, vol. 23. Springer, New York (2009)Google Scholar
  20. 20.
    Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control Lett. 50(3), 221–228 (2003). doi: 10.1016/S0167-6911(03)00156-7 CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    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(8), 5108–5142 (2010). doi: 10.1137/090746148 CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Leykekhman, D., Meidner, D., Vexler, B.: Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints. Comput. Optim. Appl. 55(3), 769–802 (2013). doi: 10.1007/s10589-013-9537-8 CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    May, S., Rannacher, R., Vexler, B.: Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems. SIAM J. Control Optim. 51(3), 2585–2611 (2013). doi: 10.1137/080735734 CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Merino, P., Neitzel, I., Tröltzsch, F.: On linear-quadratic elliptic control problems of semi-infinite type. Appl. Anal. 90(6), 1047–1074 (2011). doi: 10.1080/00036811.2010.489187 CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Merino, P., Tröltzsch, F., Vexler, B.: Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space. M2AN. Math. Model. Numer. Anal. 44(1), 167–188 (2010). doi: 10.1051/m2an/2009045 CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    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)zbMATHGoogle Scholar
  27. 27.
    Rösch, A., Steinig, S.: A priori error estimates for a state-constrained elliptic optimal control problem. ESAIM Math. Model. Numer. Anal. 46(5), 1107–1120 (2012). doi: 10.1051/m2an/2011076 CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comput. 31(138), 414–442 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15, 189–258 (1965)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Departmento de Matemáticas, E.P.I. GijónUniversidad de OviedoGijónSpain
  2. 2.Centre for the Mathematical Sciences, M17Technische Universität MünchenGarching b. MünchenGermany

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