Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

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 Double-struck capital R-n. These equations are associated with operators of the form F(|D-x|), the function F : [0, + infinity) -> Double-struck capital 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 exact 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 exact null-controllability is known to fail from general thick control supports.