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Title: Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential
Authors: Colli, PierluigiGilardi, GianniSprekels, Jürgen
Issue Date: 2016
Published in: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 2277, ISSN 2198-5855
Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Abstract: This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of CahnHilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. The local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors the nonlocal model studied here consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called deep quench approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential.
Keywords: Parameter identification; semilinear wave equation; nonlinear inverse problem; Tikhonov regularization; approximate source condition; conditional stability
DDC: 510
License: This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
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