Adaptive finite element approximation of bilinear optimal control with fractional Laplacian

We investigate the application of a posteriori error estimates to a fractional optimal control problem with pointwise control constraints. Specifically, we address a problem in which the state equation is formulated as an integral form of the fractional Laplacian equation, with the control variable embedded within the state equation as a coefficient. We propose two distinct finite element discretization approaches for an optimal control problem. The first approach employs a fully discrete scheme where the control variable is discretized using piecewise constant functions. The second approach, a semi-discrete scheme, does not discretize the control variable. Using the first-order optimality condition, the second-order optimality condition, and a solution regularity analysis for the optimal control problem, we devise a posteriori error estimates. We subsequently demonstrate the reliability and efficiency of the proposed error estimators. Based on the established error estimates framework, an adaptive refinement strategy is developed to help achieve the optimal convergence rate. The effectiveness of the refinement strategy is verified by numerical experiments.