| Abstract: |
| We study an optimal control problem for a Cahn--Hilliard-type inpainting model, as originally proposed by Bertozzi et al. The model employs a linear reaction term, weighted by a fidelity coefficient, to restore a damaged image within a fixed subdomain. To ensure that the order parameter remains within the physically relevant range $[0,1]$, we employ a singular potential. We formulate a cost functional that accounts for the magnitude of the fidelity coefficient and analyze the resulting control-to-state operator. After proving the existence of optimal controls, we derive first-order optimality conditions and, under suitable assumptions, establish second-order optimality conditions. |
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