  In this paper we consider the optimal control of semilinear fractional PDEs
with both spectral and integral fractional diffusion operators of order $2s$
with $s \in (0,1)$. We first prove the boundedness of solutions to both
semilinear fractional PDEs under minimal regularity assumptions on domain and
data. We next introduce an optimal growth condition on the nonlinearity to show
the Lipschitz continuity of the solution map for the semilinear elliptic
equations with respect to the data. We further apply our ideas to show
existence of solutions to optimal control problems with semilinear fractional
equations as constraints. Under the standard assumptions on the nonlinearity
(twice continuously differentiable) we derive the first and second order
optimality conditions.
