Blank, Luise and Meisinger, Johannes (2022) Optimal Control of a Quasilinear Parabolic Equation and its Time Discretization. APPLIED MATHEMATICS AND OPTIMIZATION, 86 (3): 34. ISSN 0095-4616, 1432-0606
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In this paper we discuss the optimal control of a quasilinear parabolic state equation. Its form is leaned on the kind of problems arising for example when controlling the anisotropic Allen-Cahn equation as a model for crystal growth. Motivated by this application we consider the state equation as a result of a gradient flow of an energy functional. The quasilinear term is strongly monotone and obeys a certain growth condition and the lower order term is non-monotone. The state equation is discretized implicitly in time with piecewise constant functions. The existence of the control-to-state operator and its Lipschitz-continuity is shown for the time discretized as well as for the time continuous problem. Latter is based on the convergence proof of the discretized solutions. Finally we present for both the existence of global minimizers. Also convergence of a subsequence of time discrete optimal controls to a global minimizer of the time continuous problem can be shown. Our results hold in arbitrary space dimensions.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | CAHN-HILLIARD SYSTEM; 3 SPATIAL DIMENSIONS; BOUNDARY CONTROL; EXISTENCE; GROWTH; MODEL; Quasilinear parabolic equation; Allen-Cahn equation; Anisotropy; Optimal control; Implicit discretization; Convergence analysis |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics Mathematics > Prof. Dr. Luise Blank |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 29 Jan 2024 10:51 |
| Last Modified: | 29 Jan 2024 12:56 |
| URI: | https://pred.uni-regensburg.de/id/eprint/56608 |
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