Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential

Pierluigi Colli; Gianni Gilardi; Jürgen Sprekels

doi:10.3934/eect.2017003

Article Contents

2017, Volume 6, Issue 1: 35-58. Doi: 10.3934/eect.2017003

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Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential

1.
Dipartimento di Matematica "F. Casorati", Universitá di Pavia, Via Ferrata 1,27100 Pavia, Italy
2.
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin
3.
Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

* Corresponding author: Pierluigi Colli
* Corresponding author: Pierluigi Colli

Received: June 30, 2016

Revised: August 31, 2016

Published: March 2018

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Abstract

This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors nonlocal model here studied 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:

Distributed optimal control,
phase field systems,
double obstacle potentials,
nonlocal operators,
first-order necessary optimality conditions.

Mathematics Subject Classification: Primary: 35K55, 49K20; Secondary: 74A15.

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