Optimal control of the fidelity coefficient in a Cahn–Hilliard image inpainting model

We consider an inpainting model proposed by A. Bertozzi et al., which is based on a Cahn–Hilliard-type equation. This equation describes the evolution of an order parameter that represents an approximation of the original image occupying a bounded two-dimensional domain. The given image is assumed to be damaged in a fixed subdomain, and the equation is characterised by a linear reaction term. This term is multiplied by the so-called fidelity coefficient, which is a strictly positive bounded function defined in the undamaged region. The idea is that, given an initial image, the order parameter evolves towards the given image and this process properly diffuses through the boundary of the damaged region, restoring the damaged image, provided that the fidelity coefficient is large enough. Here, we formulate an optimal control problem based on this fact, namely, our cost functional accounts for the magnitude of the fidelity coefficient. Assuming a singular potential to ensure that the order parameter takes its values in between 0 and 1, we first analyse the control-to-state operator and prove the existence of at least one optimal control, establishing the validity of first-order optimality conditions. Then, under suitable assumptions, we demonstrate second-order optimality conditions.