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

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 $\mathbb R^n$. 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 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.

Analyticity and observability for fractional order parabolic equations in the whole space

“…quasi-analytic). This, together with unique continuation of analytic functions, implies the approximate null-controllability result, see e.g [2]…”

“…For the exponential stabilization of the fractional heat equation, this restriction can be removed, see[2] or[17], Lemma 2.2. We also note that if s = 1 (resp.…”

Null-controllability of evolution equations associated with fractional Shubin operators through quantitative Agmon estimates

“…(5) We use the notation H q (R n ) for the Sobolev spaces, with q ⩾ 0 non-negative real numbers, and we denote by Ḣq (R n ) their homogeneous counterparts. (6) The space C ∞ b (R n ) stands for the set of smooth functions g ∈ C ∞ (R n ) with bounded derivatives. (7) For all measurable subset ω ⊂ R n , 1 ω stands for the characteristic function of ω.…”

“…It is also known from [24,25] ANNALES DE L'INSTITUT FOURIER that in the cases 0 < s ⩽ 1/2, the fractional heat equations are not nullcontrollable from thick control supports anymore. In the recent work [6], the notion of thickness has appeared to be a necessary and sufficient condition to ensure the stabilization or the approximate null-controllability with uniform cost (which are notions weaker than the null-controllability) of a very large class of diffusive equations posed on R n , including in particular the half heat equation associated with the operator (−∆) 1/2 . Finally, let us mention that other classes of degenerate parabolic equations of hypoelliptic type, as evolution equations associated with accretive quadratic operators or (non-autonomous) Ornstein-Uhlenbeck operators, were proven to be null-controllable from thick control supports, see [3,7,8].…”

Null-controllability of evolution equations associated with fractional Shubin operators through quantitative Agmon estimates

“…It is also known from [24,25] that in the cases 0 < s ≤ 1/2, the fractional heat equations are not null-controllable from thick control supports anymore. In the recent work [6], the notion of thickness has appeared to be a necessary and sufficient condition to ensure the stabilization or the approximate nullcontrollability with uniform cost (which are notions weaker than the null-controllability) of a very large class of diffusive equations posed on R n , including in particular the half heat equation associated with the operator (−∆) 1/2 . Finally, let us mention that other classes of degenerate parabolic equations of hypoelliptic type, as evolution equations associated with accretive quadratic operators or (non-autonomous) Ornstein-Uhlenbeck operators, were proven to be null-controllable from thick control supports, see [3,7,8].…”