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Recently the authors have studied a state and control constrained optimalcontrol problem with fractional elliptic PDE as constraints. The goal of thispaper is to continue that program forward and introduce an algorithm to solvesuch optimal control problems. We shall employ the well-known Moreau-Yosidaregularization to handle the state constraints. Similarly to the classicalcase, we establish the convergence (with rate) of the regularized controlproblem to the original one. We discretize the problem using a finite elementmethod and establish convergence of our numerical scheme. We emphasize that dueto the non-smooth nature of the fractional Laplacian, the proof for theclassical case does not apply to the fractional case. The numerical experimentsconfirm all our theoretical findings.

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