  We consider a problem of optimal distribution of conductivities in a system
governed by a non-local diffusion law. The problem stems from applications in
optimal design and more specifically topology optimization. We propose a novel
parametrization of non-local material properties. With this parametrization the
non-local diffusion law in the limit of vanishing non-local interaction
horizons converges to the famous and ubiquitously used generalized Laplacian
with SIMP (Solid Isotropic Material with Penalization) material model. The
optimal control problem for the limiting local model is typically ill-posed and
does not attain its infimum without additional regularization. Surprisingly,
its non-local counterpart attains its global minima in many practical
situations, as we demonstrate in this work. In spite of this qualitatively
different behaviour, we are able to partially characterize the relationship
between the non-local and the local optimal control problems. We also
complement our theoretical findings with numerical examples, which illustrate
the viability of our approach to optimal design practitioners.
