  We consider the integral definition of the fractional Laplacian and analyze a
linear-quadratic optimal control problem for the so-called fractional heat
equation; control constraints are also considered. We derive existence and
uniqueness results, first order optimality conditions, and regularity estimates
for the optimal variables. To discretize the state equation equation we propose
a fully discrete scheme that relies on an implicit finite difference
discretization in time combined with a piecewise linear finite element
discretization in space. We derive stability results and a novel
$L^2(0,T;L^2(\Omega))$ a priori error estimate. On the basis of the
aforementioned solution technique, we propose a fully discrete scheme for our
optimal control problem that discretizes the control variable with piecewise
constant functions and derive a priori error estimates for it. We illustrate
the theory with one- and two-dimensional numerical experiments.
