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

¹ Université de Lyon, ENSL, UMPA - UMR 5669, 69364 Lyon, France.
² Univ Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France.

^* Corresponding author: paul.alphonse@ens-lyon.fr

Received: 5 July 2021
Accepted: 29 January 2022

Abstract

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 ℝⁿ. These equations are associated with operators of the form F(|D_x|), the function F : [0, + ∞) → 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.