  We consider an optimal control problem of diffusion equation with missing
data governed by the fractional Laplacian with homogeneous Dirichlet boundary
conditions on an arbitrary interaction domain disjoint from the domain of the
state equation. We assume that the unknown initial condition belongs to an
appropriate space of infinite dimension, the so-called space of uncertainties.
The key tools we used in order to characterize the optimal control is the
no-regret and low-regret control developed by J.L Lions.
