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.
Funding source: Deutsche Forschungsgemeinschaft
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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© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Numerical Approximation of Multi-Phase Penrose–Fife Systems
- Tailored Finite Point Method for Parabolic Problems
- A Weakly Penalized Discontinuous Galerkin Method for Radiation in Dense, Scattering Media
- Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast
- Chinese–German Computational and Applied Mathematics
- Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms
- Super-Exponentially Convergent Parallel Algorithm for Eigenvalue Problems with Fractional Derivatives
- A Nonconforming Finite Element Approximation for Optimal Control of an Obstacle Problem
- Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes
- Optimal Control for the Thin Film Equation: Convergence of a Multi-Parameter Approach to Track State Constraints Avoiding Degeneracies
Articles in the same Issue
- Frontmatter
- Numerical Approximation of Multi-Phase Penrose–Fife Systems
- Tailored Finite Point Method for Parabolic Problems
- A Weakly Penalized Discontinuous Galerkin Method for Radiation in Dense, Scattering Media
- Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast
- Chinese–German Computational and Applied Mathematics
- Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms
- Super-Exponentially Convergent Parallel Algorithm for Eigenvalue Problems with Fractional Derivatives
- A Nonconforming Finite Element Approximation for Optimal Control of an Obstacle Problem
- Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes
- Optimal Control for the Thin Film Equation: Convergence of a Multi-Parameter Approach to Track State Constraints Avoiding Degeneracies