  We study the numerical approximation of linear-quadratic optimal control
problems subject to the fractional Laplace equation with its spectral
definition. We compute an approximation of the state equation using a
discretization of the Balakrishnan formula that is based on a finite element
discretization in space and a sinc quadrature approximation of the additionally
involved integral. A tailored approach for the numerical solution of the
resulting linear systems is proposed. Concerning the discretization of the
optimal control problem we consider two schemes. The first one is the
variational approach, where the control set is not discretized, and the second
one is the fully discrete scheme where the control is discretized by piecewise
constant functions. We derive finite element error estimates for both methods
and illustrate our results by numerical experiments.
