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We design and analyze solution techniques for a linear-quadratic optimalcontrol problem involving the integral fractional Laplacian. We deriveexistence and uniqueness results, first order optimality conditions, andregularity estimates for the optimal variables. We propose two strategies todiscretize the fractional optimal control problem: a semidiscrete approachwhere the control is not discretized - the so-called variational discretizationapproach - and a fully discrete approach where the control variable isdiscretized with piecewise constant functions. Both schemes rely on thediscretization of the state equation with the finite element space ofcontinuous piecewise polynomials of degree one. We derive a priori errorestimates for both solution techniques. We illustrate the theory withtwo-dimensional numerical tests.

A priori error estimates for the optimal control of the integral fractional Laplacian