Very recently Warma (2019 SIAM J. Control Optim . to appear) has shown that for nonlocal PDEs associated with the fractional Laplacian, the classical notion of controllability from the boundary does not make sense and therefore it must be replaced by a control that is localized outside the open set where the PDE is solved. Having learned from the above mentioned result, in this paper we introduce a new class of source identification and optimal control problems where the source/control is located outside the observation domain where the PDE is satisfied. The classical diffusion models lack this flexibility as they assume that the source/control is located either inside or on the boundary. This is essentially due to the locality property of the underlying operators. We use the nonlocality of the fractional operator to create a framework that now allows placing a source/control outside the observation domain. We consider the Dirichlet, Robin and Neumann source identification or optimal control problems. These problems require dealing with the nonlocal normal derivative (that we shall call interaction operator). We create a functional analytic framework and show well-posedness and derive the first order optimality conditions for these problems. We introduce a new approach to approximate, with convergence rate, the Dirichlet problem with nonzero exterior condition. The numerical examples confirm our theoretical findings and illustrate the practicality of our approach.

In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order s ∈ (0,1). We identify minimal conditions on the nonlinear term and the source which lead to existence of weak solutions and uniform L ∞-bound on the solutions. Next we realize the fractional Laplacian as a Dirichlet-to-Neumann map via the Caffarelli−Silvestre extension. We introduce a first-degree tensor product finite elements space to approximate the truncated problem. We derive a priori error estimates and conclude with an illustrative numerical example.