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

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)

• 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.

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

 Citation: • • 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)

Figure 2.  Multiphase decomposition of a signal

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)

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)

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)

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)

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)

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)

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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