  We adopt the integral definition of the fractional Laplace operator and
analyze solution techniques for fractional, semilinear, and elliptic optimal
control problems posed on Lipschitz polytopes. We consider two strategies of
discretization: a semidiscrete scheme where the admissible control set is not
discretized and a fully discrete scheme where such a set is discretized with
piecewise constant functions. As an instrumental step, we derive error
estimates for finite element discretizations of fractional semilinear elliptic
partial differential equations (PDEs) on quasi-uniform and graded meshes. With
these estimates at hand, we derive error bounds for the semidiscrete scheme and
improve the ones that are available in the literature for the fully discrete
scheme.
