Computational Optimization and Applications

, Volume 42, Issue 1, pp 43–66 | Cite as

A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints

Article

Abstract

A Lavrentiev type regularization technique for solving elliptic boundary control problems with pointwise state constraints is considered. The main concept behind this regularization is to look for controls in the range of the adjoint control-to-state mapping. After investigating the analysis of the method, a semismooth Newton method based on the optimality conditions is presented. The theoretical results are confirmed by numerical tests. Moreover, they are validated by comparing the regularization technique with standard numerical codes based on the discretize-then-optimize concept.

Keywords

Boundary control State constraints Lavrentiev type regularization Semismooth Newton method Optimize-then-discretize Nested iteration 

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References

  1. 1.
    Alibert, J.-J., Raymond, J.-P.: Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 3&4, 235–250 (1997) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37, 1176–1194 (1999) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bergounioux, M., Kunisch, K.: Primal-dual active set strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22, 193–224 (2002) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 4, 1309–1322 (1986) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38(4), 1200–1216 (2000) (electronic) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hackbusch, W.: Multigrid Methods and Applications. Springer Series in Computational Mathematics, vol. 4. Springer, Berlin (1985) Google Scholar
  8. 8.
    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2003) MATHCrossRefGoogle Scholar
  9. 9.
    Hintermüller, M., Tröltzsch, F., Yousept, I.: Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems (2006) Google Scholar
  10. 10.
    Ito, K., Kunisch, K.: Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. TMA 41, 591–616 (2000) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control Lett. 50, 221–228 (2003) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lavrentiev, M.M.: Some Improperly Posed Problems of Mathematical Physics. Springer, New York (1967) MATHGoogle Scholar
  13. 13.
    Maurer, H., Mittelmann, H.D.: Optimization techniques for solving elliptic control problems with control and state constraints. I. Boundary control. J. Comput. Appl. Math. 16, 29–55 (2000) MATHMathSciNetGoogle Scholar
  14. 14.
    Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33, 209–228 (2006) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Meyer, C., Tröltzsch, F.: On an elliptic optimal control problem with pointwise mixed control-state constraints. In: Seeger, A. (ed.) Recent Advances in Optimization. Proceedings of the 12th French-German-Spanish Conference on Optimization held in Avignon, September 20–24, 2004. Lectures Notes in Economics and Mathematical Systems, vol. 563, pp. 187–204. Springer, Berlin (2006) CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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