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

Abstract

We prove that the thickness 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 R n and associated with operators of the form F (|Dx|), the function F : [0, +∞) → R being bounded below and continuous. We also provide explicit feedbacks and constants associated with these stabilization properties. Our results apply in particular for the half heat equation associated with the function F (t) = t, for which null-controllability is known to fail from thick control supports. More generally, the notion of thickness was 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, and that this null-controllability property from thick control supports does not hold when 0 < s ≤ 1.

Approximate null-controllability with uniform cost for the hypoelliptic Ornstein-Uhlenbeck equations

“…In order to prove that this geometric condition is also sufficient, we adapt the strategy used in [4] (Subsection 5.2) where the authors proved that the thickness property (2.6) is a necessary and sufficient condition that ensures the cost-uniform approximate nullcontrollability for a large class of parabolic equations. In the present work, we begin by noticing that the geometric condition (2.5) implies that the set…”

“…Notice that we have to get strictly far from the final time T in order to avoid blow-up phenomena. This is the precise reason why we can use the same strategy as in the work [4]. First, we need to establish smoothing estimates in the time and space variables of the following form, by using the ellipticity condition (A T ),…”

“…These estimates are obtained in Section 5. Notice that when we work with the thickness condition (2.6), which does not depend on time, as in the work [4], we only have to consider the smoothing properties in space of the evolution equation at play. Then, by using the above estimates and elements of harmonic analysis, and more precisely the unique continuation property stated in Proposition 6.2, coming essentially from the second author's work [22], we obtain the following quantitative unique continuation property in Section 6:…”

“…The study of the (rapid) stabilization and the (approximate) null-controllability of parabolic equations [4,12,14,16,20,25,28] or degenerate parabolic equations of hypoelliptic type [3,7,8,9,11] posed on R n and taking the following form (E P ) ∂ t f (t, x) + P f (t, x) = h(t, x)½ ω (x), (t, x) ∈ (0, +∞) × R n ,…”

“…This notion of thickness has turned out to play a key role in the theories of stabilization and (approximate) null-controllability, since it was proven to be a necessary and sufficient geometric condition that ensures these notions for large classes of parabolic equations posed on R n , as the fractional heat equations for instance, see e.g. [3,4,12,14,21,27,28]. The thickness condition is also involved in the study of the exact null-controllability of the free Schrödinger equation, as highlighted in [23].…”