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In this paper we study optimal control problems with either fractional orregional fractional $p$-Laplace equation, of order $s$ and $p\in [2,\infty)$,as constraints over a bounded open set with Lipschitz continuous boundary. Thecontrol, which fulfills the pointwise box constraints, is given by thecoefficient of the involved operator. To overcome the degeneracy of bothfractional $p$-Laplacians, we introduce a regularization for both operators. Weshow existence and uniqueness of solution to the regularized state equationsand existence of solution to the regularized optimal control problems. We alsoprove several auxiliary results for the regularized problems which are ofindependent interest. We conclude with the convergence of the regularizedsolutions.

Optimal control of the coefficient for fractional and regional fractional {$p$}-{L}aplace equations: Approximation and convergence