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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37, 1176–1194 (1999)
Bergounioux, M., Kunisch, K.: Primal-dual active set strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22, 193–224 (2002)
Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 4, 1309–1322 (1986)
Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993)
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)
Hackbusch, W.: Multigrid Methods and Applications. Springer Series in Computational Mathematics, vol. 4. Springer, Berlin (1985)
Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2003)
Hintermüller, M., Tröltzsch, F., Yousept, I.: Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems (2006)
Ito, K., Kunisch, K.: Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. TMA 41, 591–616 (2000)
Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control Lett. 50, 221–228 (2003)
Lavrentiev, M.M.: Some Improperly Posed Problems of Mathematical Physics. Springer, New York (1967)
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)
Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33, 209–228 (2006)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors acknowledge support through DFG Research Center “Mathematics for Key Technologies” (FZT 86) in Berlin.
Rights and permissions
About this article
Cite this article
Tröltzsch, F., Yousept, I. A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints. Comput Optim Appl 42, 43–66 (2009). https://doi.org/10.1007/s10589-007-9114-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-007-9114-0