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Distributed optimal control of the Cahn-Hilliard system including the case of a double-obstacle homogeneous free energy density

Please always quote using this URN:urn:nbn:de:0296-matheon-7869
  • In this paper we study the distributed optimal control for the Cahn-Hilliard system. A general class of free energy potentials is allowed which, in particular, includes the double-obstacle potential. The latter potential yields an optimal control problem of a parabolic variational inequality which is of fourth-order in space. We show the existence of optimal controls to approximating problems where the potential is replaced by a mollified version of its Moreau-Yosida approximation. Corresponding first order optimality conditions for the mollified problems are given. For this purpose a new result on the continuous Fr\'echet differentiability of superposition operators with values in Sobolev spaces is established. Besides the convergence of optimal controls of the mollified problems to an optimal control of the original problem, we also derive first order optimality conditions for the original problem by a limit process. The newly derived stationarity system corresponds to a function space version of C-stationarity.

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Metadaten
Author:Michael Hintermüller, Donat Wegner
URN:urn:nbn:de:0296-matheon-7869
Referee:Ralf Kornhuber
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2011/02/23
Release Date:2011/02/23
Tag:
Institute:Humboldt-Universität zu Berlin
MSC-Classification:34-XX ORDINARY DIFFERENTIAL EQUATIONS / 34Gxx Differential equations in abstract spaces [See also 34Lxx, 37Kxx, 47Dxx, 47Hxx, 47Jxx, 58D25] / 34G25 Evolution inclusions
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K86 Nonlinear parabolic unilateral problems and nonlinear parabolic variational inequalities [See also 35R35, 49J40]
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Jxx Existence theories / 49J20 Optimal control problems involving partial differential equations
Preprint Number:770
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