On Optimal sparse control formulation for reconstruction of noise-affected images

We discuss the optimal control formulation for enhancement and denoising of satellite multiband images and propose to take it in the form of an $L^1$ -control problem for quasi-linear parabolic equation with non-local $p[u]$-Laplacian and with a cost functional of a tracking type. The main characteristic features of the considered parabolic problem is that the variable exponent$p(t, x)$ and the diffusion anisotropic tensor $D(t,x)$ are not well predefined a priori, but instead thesecharacteristics non-locally depend on the form of the solution of this problem, i.e., $p_u = p(t, x, u)$ and$D_u = D (t, x, u)$. We prove the existence of optimal pairs with sparse $L^1$ -controls using for that the indirect approach and a special family of approximation problems.