Optimal distributed control of a Allen-Cahn/Cahn-Hilliard system with temperature
type: Item
instance of: scholarly article
title: Optimal distributed control of a Allen-Cahn/Cahn-Hilliard system with temperature
zbMATH Open document ID: 1486.49004
author: Bosheng Chen
author: Chein-Shan Liu
published in: Applied Mathematics and Optimization
publication date: 2022-03-25T00:00:00Z
review text: Different kinds of optimal control problems for Allen-Cahn/Cahn-Hilliard equations and systems have been extensively studied, and numerous publications in form of research articles, surveys, preprints or monographs are available. The optimal control problem considered in the present article refers to the Allen-Cahn/Cahn-Hilliard system in the form described in a series of papers by Miranville \&Co, actually the system in: [\textit{A. Miranville} et al., Commun. Pure Appl. Anal. 19, No. 4, 2257--2288 (2020; Zbl 1437.35084)] \[ \frac {\partial u}{\partial t} -\Delta ^2u = \Delta( f(u+v) + f(u-v)),\text{ for }(x,t)\in \Omega_T, \] \[ \frac{\partial v}{\partial t} -\Delta v = \theta-f(u+v) + f(u-v),\text{ for }(x,t) \in \Omega_T, \] \[ \frac{\partial {\theta}v}{\partial t}- \Delta \theta = -\frac{\partial v}{\partial t} +\phi,\text{ for } (x,t) \in\Omega_T, \] \[u= v = \theta =0,\text{ on }\partial \Omega\times (0,T) , \] \[u(x,0) =u_0(x), v(x,0) = v_0(x),\theta(x,0)=\theta_0(x), \text{ for } x \in \Omega, \] where \(\Omega_T=\Omega\times (0,T)\), \(\Omega\subset\mathbb{R}\). Variable \(\phi\) denotes the control and one assumes it is admissible, \(\phi\in\Phi_{ad}\). If \((u,v,\theta)\) is a solution of the initial boundary value problem to the Allen-Chan/Cahn-Hilliard system, one defines a quadratic performance functional \(\mathcal J(\phi;u,v,\theta)\), and the goal is to minimize it for \(\phi\) in a suitable space of admissible controls. Some basics are reviewed in the second section of the paper. The third section is devoted to the well-posedness of the system. One transforms the system in an equivalent one by introducing a new variable \(\mu= -\Delta u+f(u+v) + f(u-v)\), and under some assumptions on initial data, one proves the existence of a weak solution for the transformed system. A next result states the existence of a strong solution for \(t\in[ 0, T]\) and two estimates are proved. In the fourth section one considers the system in a linearized form and one…
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 49J50
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K52
zbMATH DE Number: 7498417
zbMATH Keywords: optimal distributed control
zbMATH Keywords: Allen-Cahn/Cahn-Hilliard system
zbMATH Keywords: first-order necessary optimality conditions
MaRDI profile type: Publication
full work available at URL: s00245-021-09807-2
OpenAlex ID: W3185374770
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DOI: 10.1007/S00245-021-09807-2
description: scientific article; zbMATH DE number 7498417