Moreau-Yosida Regularization for Optimal Control of Fractional Elliptic Problems with State and Control Constraints

Recently the authors have studied a state and control constrained optimal control problem with fractional elliptic PDE as constraints. The goal of this paper is to continue that program forward and introduce an algorithm to solve such optimal control problems. We shall employ the well-known Moreau-Yosida regularization to handle the state constraints. Similarly to the classical case, we establish the convergence (with rate) of the regularized control problem to the original one. We discretize the problem using a finite element method and establish convergence of our numerical scheme. We emphasize that due to the non-smooth nature of the fractional Laplacian, the proof for the classical case does not apply to the fractional case. The numerical experiments confirm all our theoretical findings.