Approximation of Sparse Controls in Semilinear Elliptic Equations

  • Eduardo Casas
  • Roland Herzog
  • Gerd Wachsmuth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7116)


Semilinear elliptic optimal control problems involving the L 1 norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for three different discretizations for the control problem are given. These discretizations differ in the use of piecewise constant, piecewise linear and continuous or non-discretized controls, respectively. Numerical results and implementation details are provided.


optimal control of partial differential equations sparse controls non-differentiable objective finite element discretization a priori error estimates semilinear equations 


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  1. 1.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  2. 2.
    Carstensen, C.: Quasi-interpolation and a posteriori error analysis in finite element methods. M2AN Math. Model. Numer. Anal. 33(6), 1187–1202 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Casas, E., Herzog, R., Wachsmuth, G.: Analysis of an elliptic control problem with non-differentiable cost functional. Technical report, TU Chemnitz (2010)Google Scholar
  4. 4.
    Casas, E., Herzog, R., Wachsmuth, G.: Approximation of sparse controls in semilinear equations by piecewise linear functions. Technical report, TU Chemnitz (2011)Google Scholar
  5. 5.
    CGAL. Computational Geometry Algorithms Library,
  6. 6.
    Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM Journal on Numerical Analysis 38, 1200–1216 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Clason, C., Kunisch, K.: A duality-based approach to elliptic control problems in non-reflexive Banach spaces. In: ESAIM: COCV (2010), doi:10.1051/cocv/2010003Google Scholar
  8. 8.
    Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics 57(11), 1413–1457 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    FEniCS. FEniCS project (2007),
  10. 10.
    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM Journal on Optimization 13(3), 865–888 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hinze, M.: A variational discretization concept in control constrained optimization: The linear-quadratic case. Comp. Optim. Appls. 30, 45–61 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Logg, A., Wells, G.N.: Dolfin: Automated finite element computing. ACM Trans. Math. Softw. 37, 20:1–20:28 (2010)Google Scholar
  13. 13.
    Stadler, G.: Elliptic optimal control problems with L 1-control cost and applications for the placement of control devices. Comp. Optim. Appls. 44(2), 159–181 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Wachsmuth, G., Wachsmuth, D.: Convergence and regularization results for optimal control problems with sparsity functional. In: ESAIM: COCV (2010), doi:10.1051/cocv/2010027Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eduardo Casas
    • 1
  • Roland Herzog
    • 2
  • Gerd Wachsmuth
    • 2
  1. 1.Departmento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de TelecomunicaciónUniversidad de CantabriaSantanderSpain
  2. 2.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany

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