Optimal control of the coefficient for fractional and regional fractional {$p$}-{L}aplace equations: Approximation and convergence
This work addresses theoretical foundations for optimal control of fractional PDEs, which is relevant for applications in nonlocal diffusion and image processing, but the results are incremental as they extend existing regularization techniques to a specific class of operators.
The paper studies optimal control problems constrained by fractional and regional fractional p-Laplace equations, introducing a regularization to handle degeneracy. It proves existence and uniqueness of solutions to regularized state equations and optimal control problems, and establishes convergence of regularized solutions.
In this paper we study optimal control problems with either fractional or regional fractional $p$-Laplace equation, of order $s$ and $p\in [2,\infty)$, as constraints over a bounded open set with Lipschitz continuous boundary. The control, which fulfills the pointwise box constraints, is given by the coefficient of the involved operator. To overcome the degeneracy of both fractional $p$-Laplacians, we introduce a regularization for both operators. We show existence and uniqueness of solution to the regularized state equations and existence of solution to the regularized optimal control problems. We also prove several auxiliary results for the regularized problems which are of independent interest. We conclude with the convergence of the regularized solutions.