Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation

Jean-Paul Chehab; Alejandro A. Franco; Youcef Mammeri

doi:10.3934/dcdss.2017005

Article Contents

2017, Volume 10, Issue 1: 87-100. Doi: 10.3934/dcdss.2017005

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Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation

1.
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée (LAMFA), CNRS UMR 7352, Université de Picardie Jules Verne, 33, rue Saint Leu, 80039 Amiens, France
2.
Laboratoire de Réactivité et de Chimie des Solides (LRCS), CNRS UMR 7314, Université de Picardie Jules Verne, 33, rue Saint Leu, 80039 Amiens, France
3.
Réseau sur le Stockage Electrochimique de l'Energie (RS2E), FR CNRS 3459, France, ALISTORE European Research Institute, FR CNRS 3104,80039 Amiens, France

Author Bio: E-mail address: Jean-Paul.Chehab@u-picardie.fr; E-mail address: Alejandro.Franco@u-picardie.fr; E-mail address: Youcef.Mammeri@u-picardie.fr

Received: March 2015

Revised: May 2016

Published: February 2018

Prof. Franco deeply acknowledges the Conseil Regional de Picardie and the European Regional Development Fund for the support through the project MASTERS, as well as the National Research Agency ANR for the support through the Project ALIBABA (reference ANR-11-PRGE-0002)

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Abstract

The identification of optimal structures in reaction-diffusion models is of great importance in numerous physicochemical systems. We propose here a simple method to monitor the number of interphases formed after long simulated times by using a boundary flux condition as a control parameter. We consider as an illustration a 1-D Allen-Cahn equation with Neumann boundary conditions. Numerical examples are provided and perspectives for the application of this approach to electrochemical systems are discussed.

Keywords:

Mathematics Subject Classification: Primary: 35Q93, 65M06; Secondary: 65M12.

Citation:

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Figure 1. Solution of the Allen-Cahn equation (1) with $f(u)=u(u^2-1)$ for $\epsilon=0.004$ (A) and for $\epsilon=0.001$ (B)

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Figure 2. Multiphase decomposition of a signal

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Figure 3. Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, \ N=127$ starting from the smooth initial datum $u_0=\cos(20\pi x)$. Solution obtained without control (C) and with the optimal control (D)

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Figure 4. Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.005, \ T=0.5, \ N=127$ starting from the smooth initial datum $u_0=\cos(20\pi x)$. Solution obtained without control (C) and with the optimal control (D)

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Figure 5. Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, \ N=255$ starting from the smooth initial datum $u_0=\cos(20\pi x)$. Solution obtained without control (C) and with the optimal control (D)

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Figure 6. Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, \ N=511$ starting from the smooth initial datum $u_0=\cos(20\pi x)$. Solution obtained without control (C) and with the optimal control (D)

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Figure 7. Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.00001, \ T=0.001, \ N=127$ starting from a random initial datum. Solution obtained without control (C) and with the optimal control (D)

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Figure 8. Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.00001, \ T=0.001, \ N=255$ starting from a random initial datum. Solution obtained without control (C) and with the optimal control (D)

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Figure 9. Optimal control $\alpha(t)$ (A) and merit function $J(u)=10 \parallel u-1\parallel +\parallel u+1\parallel$ (B) computed with $\epsilon=0.005, \ \Delta t = 0.00001, \ T=0.001, \ N=255$ starting from a random initial datum. Solution obtained without control (C) and with the optimal control (D)

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