Abstract

We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic PDE as the state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first- and necessary and sufficient second-order optimality conditions. Regularity estimates for optimal variables are also analyzed. We develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the convergence properties of discretizations and derive error estimates.