  We prove that the thickness property is a necessary and sufficient geometric
condition that ensures the (rapid) stabilization or the approximate
null-controllability with uniform cost of a large class of evolution equations
posed on the whole space $\mathbb R^n$. These equations are associated with
operators of the form $F(\vert D_x\vert)$, the function
$F:[0,+\infty)\rightarrow\mathbb R$ being continuous and bounded from below. We
also provide explicit feedbacks and constants associated with these
stabilization properties. The notion of thickness is known to be a necessary
and sufficient condition for the null-controllability of the fractional heat
equations associated with the functions $F(t) = t^{2s}$ in the case $s>1/2$.
Our results apply in particular for this class of equations, but also for the
half heat equation associated with the function $F(t) = t$, which is the most
diffusive fractional heat equation for which null-controllability is known to
fail from general thick control supports.
