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.
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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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Solution of the Allen-Cahn equation (1) with
Multiphase decomposition of a signal
Optimal control
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