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In a previous work, we introduced a discretization scheme for a constrainedoptimal control problem involving the fractional Laplacian. For such a controlproblem, we derived optimal a priori error estimates that demand the convexityof the domain and some compatibility conditions on the data. To relax suchrestrictions, in this paper, we introduce and analyze an efficient and, undercertain assumptions, reliable a posteriori error estimator. We realize thefractional Laplacian as the Dirichlet-to-Neumann map for a nonuniformlyelliptic problem posed on a semi--infinite cylinder in one more spatialdimension. This extra dimension further motivates the design of an posteriorierror indicator. The latter is defined as the sum of three contributions, whichcome from the discretization of the state and adjoint equations and the controlvariable. The indicator for the state and adjoint equations relies on ananisotropic error estimator in Muckenhoupt weighted Sobolev spaces. Theanalysis is valid in any dimension. On the basis of the devised a posteriorierror estimator, we design a simple adaptive strategy that exhibits optimalexperimental rates of convergence.

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