  In a previous work, we introduced a discretization scheme for a constrained
optimal control problem involving the fractional Laplacian. For such a control
problem, we derived optimal a priori error estimates that demand the convexity
of the domain and some compatibility conditions on the data. To relax such
restrictions, in this paper, we introduce and analyze an efficient and, under
certain assumptions, reliable a posteriori error estimator. We realize the
fractional Laplacian as the Dirichlet-to-Neumann map for a nonuniformly
elliptic problem posed on a semi--infinite cylinder in one more spatial
dimension. This extra dimension further motivates the design of an posteriori
error indicator. The latter is defined as the sum of three contributions, which
come from the discretization of the state and adjoint equations and the control
variable. The indicator for the state and adjoint equations relies on an
anisotropic error estimator in Muckenhoupt weighted Sobolev spaces. The
analysis is valid in any dimension. On the basis of the devised a posteriori
error estimator, we design a simple adaptive strategy that exhibits optimal
experimental rates of convergence.
