  Very recently M. Warma 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.
