An a posteriori error analysis for an optimal control problem involving the fractional Laplacian

In a previous work, we introduced a discretization scheme for a control-constrained optimal control problem involving the fractional Laplacian. For such a problem, we derived a near optimal a priori error estimate, for the approximation of the optimal control variable, that demands the convexity of the domain and some compatibility conditions on the data. To relax such restrictions, in this article, we introduce and analyse 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 a posteriori error indicator. The latter is defined as the sum of three contributions that 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. We present an analysis that 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.