  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.
