  The aim of this work is to give a broad panorama of the control properties of
fractional diffusive models from a numerical analysis and simulation
perspective. We do this by surveying several research results we obtained in
the last years, focusing in particular on the numerical computation of
controls. Our reference model will be a non-local diffusive dynamics driven by
the fractional Laplacian on a bounded domain $\Omega$. The starting point of
our analysis will be a Finite Element approximation for the associated elliptic
model in one and two space-dimensions, for which we also present error
estimates and convergence rates in the $L^2$ and energy norm. Secondly, we will
address two specific control scenarios: firstly, we consider the standard
interior control problem, in which the control is acting from a small subset
$\omega\subset\Omega$. Secondly, we move our attention to the exterior control
problem, in which the control region $\mathcal O\subset\Omega^c$ is located
outside $\Omega$. This exterior control notion extends boundary control to the
fractional framework, in which the non-local nature of the models does not
allow for controls supported on $\partial\Omega$. We will conclude by
discussing the interesting problem of simultaneous control, in which we
consider families of parameter-dependent fractional heat equations and we aim
at designing a unique control function capable of steering all the different
realizations of the model to the same target configuration. In this framework,
we will see how the employment of stochastic optimization techniques may help
in alleviating the computational burden for the approximation of simultaneous
controls. Our discussion is complemented by several open problems related with
fractional models which are currently unsolved and may be of interest for
future investigation.
