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Optimal Control for the Thin Film Equation: Convergence of a Multi-Parameter Approach to Track State Constraints Avoiding Degeneracies

  • Markus Klein and Andreas Prohl EMAIL logo
Published/Copyright: September 22, 2016

Abstract

We consider an optimal control problem subject to the thin-film equation. The PDE constraint lacks well-posedness for general right-hand sides due to possible degeneracies; state constraints are used to circumvent this problematic issue and to ensure well-posedness. Necessary optimality conditions for the optimal control problem are then derived. A convergent multi-parameter regularization is considered which addresses both, the possibly degenerate term in the equation and the state constraint. Some computational studies are then reported which evidence the relevant role of the state constraint, and motivate proper scalings of involved regularization and numerical parameters.

Award Identifier / Grant number: SPP 1253

Funding statement: The work supported by a DFG grant within the Priority Program SPP 1253 (Optimization with Partial Differential Equations). This work was performed on the computational resource bwUniCluster funded by the Ministry of Science, Research and Arts and the Universities of the State of Baden-Württemberg, Germany, within the framework program bwHPC; cf. [bwGRiD, Member of the German D-Grid initiative, funded by the Ministry of Education and Research (Bundesministerium für Bildung und Forschung) and the Ministry for Science, Research and Arts Baden-Württemberg (Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg), technical report, Universities of Baden-Württemberg, 2007–2010, www.bw-grid.de].

Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments, which helped to improve the quality of this article.

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Received: 2016-5-29
Revised: 2016-6-23
Accepted: 2016-8-3
Published Online: 2016-9-22
Published in Print: 2016-10-1

© 2016 by De Gruyter

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