Optimal Control of Allen-Cahn Systems

Blank, Luise; Farshbaf-Shaker, M. Hassan; Hecht, Claudia; Michl, Josef; Rupprecht, Christoph

doi:10.1007/978-3-319-05083-6_3

Cited by 1 publication

(6 citation statements)

References 37 publications

(68 reference statements)

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“…With Lemma 2 at hand we can show the existence of a unique weak solution to the time discretized state equation (9). Note that the following bound (20) (and likewise (33) in the time continuous case) will be crucial for showing the existence of an optimal control later.…”

Section: Solution To the Time Discretized And To The Time Continuous ...mentioning

confidence: 95%

“…To the best of our knowledge there does not exist any mathematical treatment on the optimal control of anisotropic phase-field models so far. Optimal control of isotropic Allen-Cahn variational equations are studied, e.g., in [8][9][10][11][12] and of Cahn-Hilliard variational (in-)equalities in [13][14][15] and references therein. Let us mention results given in the context of anisotropic Allen-Cahn equations.…”

Section: E(y) :=mentioning

confidence: 99%

“…The same holds respectively for integration over . Together with the monotonicity of A this enables the usual steps of the proof that solving the variational inequality (10) is equivalent to solving the variational equality (9) with f instead of ψ.…”

Section: Solution To the Time Discretized And To The Time Continuous ...mentioning

confidence: 99%

“…Proof We test (9) with the difference y j − y j−1 and obtain 1 τ j y j − y j−1 2 + A (∇ y j ), ∇ y j − ∇ y j−1 + ψ (y j ), y j − y j−1 = 0. (…”

Section: Theoremunclassified

“…Theorem 11 Let Assumptions 1 be fulfilled and y ∈ L 2 ( ) hold. Consider a sequence of global optimal controls (u τ , y τ ) τ of (8) subject to (9) belonging to a sequence of discretizations with τ → 0. Then there exists a subsequence with u τ → u in L 2 (0, T ; L 2 ( )) and with y τ converging to y = S(u) in the sense of (31) where (u, y) solves (3) subject to (4).…”

Section: Theorem 10mentioning

confidence: 99%

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Optimal Control of a Quasilinear Parabolic Equation and its Time Discretization

Blank

Meisinger

2022

Appl Math Optim

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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.

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Section: Solution To the Time Discretized And To The Time Continuous ...mentioning

confidence: 95%