Fractional semilinear optimal control: optimality conditions, convergence, and error analysis

Abstract

We analyze an optimal control problem for a fractional semilinear PDE; control constraints are also considered. We adopt the integral definition of the fractional Laplacian and establish the well-posedness of a fractional semilinear PDE; we also analyze suitable finite element discretizations. We thus derive the existence of optimal solutions and first and second order optimality conditions for our optimal control problem; regularity properties are also studied. We devise a fully discrete scheme that approximates the control variable with piecewise constant functions; the state and adjoint equations are discretized via piecewise linear finite elements. We analyze convergence properties of discretizations and obtain a priori error estimates.