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(|Dx|), 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) = t2s 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.
Highlights
∀x ∈ Rn, Leb(ω ∩ (x + [0, L]n)) ≥ γLn, where Leb denotes the Lebesgue measure in Rn. This notion of thickness has appeared to play a key role in the null-controllability theory since the works [7, 22], where the authors established that it is a necessary and sufficient geometric condition that ensures the null-controllability of the heat equation posed on Rn, which is the equation (EF ) associated with the function F (t) = t2
We prove that the thickness of the support ω ⊂ Rn is a necessary geometric condition that ensures the stabilization of the equation (EF ), and a sufficient one when assuming in addition that lim inf+∞ F > | inf F |
By taking advantage of this quasi-analytic regularity, we prove that the notion of thickness is a necessary and sufficient geometric condition that ensures the cost-uniform approximate null-controllability of the evolution equations (EF ) in any positive time
Summary
This paper is devoted to investigate the stabilization and approximate null-controllability for control systems of the following form:. ∀x ∈ Rn, Leb(ω ∩ (x + [0, L]n)) ≥ γLn, where Leb denotes the Lebesgue measure in Rn This notion of thickness has appeared to play a key role in the null-controllability theory since the works [7, 22], where the authors established that it is a necessary and sufficient geometric condition that ensures the null-controllability of the heat equation posed on Rn, which is the equation (EF ) associated with the function F (t) = t2. The study of the (rapid) stabilization of the control system (EF ), as for it, has been addressed very recently in the works [9, 17, 21] It has been proven in [9] (Thm. 1.1) that for all s > 0, the fractional heat equation (Et2s ) is exponentially stabilizable from the support ω if and only if ω is thick. Our results highlight the importance of the notion of thickness in the null-controllability theory, and for properties of stabilization and approximate null-controllability with uniform cost, as it turns out to be a necessary and sufficient geometric condition ensuring these two properties for a large class of diffusive equations (EF )
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