  We consider an optimal control problem that entails the minimization of a
nondifferentiable cost functional, fractional diffusion as state equation and
constraints on the control variable. We provide existence, uniqueness and
regularity results together with first order optimality conditions. In order to
propose a solution technique, we realize fractional diffusion as the
Dirichlet-to-Neumann map for a nonuniformly elliptic operator and consider an
equivalent optimal control problem with a nonuniformly elliptic equation as
state equation. The rapid decay of the solution to this problem suggests a
truncation that is suitable for numerical approximation. We propose a fully
discrete scheme: piecewise constant functions for the control variable and
first--degree tensor product finite elements for the state variable. We derive
a priori error estimates for the control and state variables which are
quasi--optimal with respect to degrees of freedom.
