  We adopt the integral definition of the fractional Laplace operator and study
an optimal control problem on Lipschitz domains that involves a fractional
elliptic partial differential equation (PDE) as state equation and a control
variable that enters the state equation as a coefficient; pointwise constraints
on the control variable are considered as well. We establish the existence of
optimal solutions and analyze first and, necessary and sufficient, second order
optimality conditions. Regularity estimates for optimal variables are also
analyzed. We develop two finite element discretization strategies: a
semidiscrete scheme in which the control variable is not discretized, and a
fully discrete scheme in which the control variable is discretized with
piecewise constant functions. For both schemes, we analyze the convergence
properties of discretizations and derive error estimates.
