In this paper the authors study control problems constrained by nonlocal steady diffusion equations of the form
(1) \(-Lu=f\) on \(\Omega\), \(\nu u =g\) on \(\Omega_\mathcal{I},\)
where \(\nu\) denotes a linear operator of volume constraints acting on an interaction volume \(\Omega_\mathcal{I}\) that is disjoint from \(\Omega \subset \mathbb R^n\) (a bounded, open domain).
For the first time, the authors rigorously analyze optimal control problems for a class of nonlocal equations that significantly generalize many nonlocal models which are already in fairly common use and which are of increasing interest to the scientific community. These include fractional derivatives and fractional Laplacian models, e.g. for anomalous diffusion problems arising in many application areas. They show how to apply standard approaches of the classical control theory for differential equations to nonlocal problems.