Error estimates for fractional semilinear optimal control on Lipschitz polytopes

We adopt the integral definition of the fractional Laplace operator and analyze discretization techniques for a fractional, semilinear, and elliptic optimal control problem posed on a Lipschitz polytope. We consider two strategies of discretization: a semidiscrete scheme where control variables are not discretized -- the so-called variational discretization approach -- and a fully discrete scheme where control variables are discretized with piecewise constant functions. We discretize the corresponding state and adjoint equations with a finite element scheme based on continuous piecewise linear functions and derive error estimates. With these estimates at hand, we derive error bounds for the semidiscrete scheme on quasi-uniform and suitable graded meshes, and improve the ones that are available in the literature for the fully discrete scheme.