Volume 28, 2022
|Number of page(s)||30|
|Published online||24 February 2022|
Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports
Université de Lyon, ENSL, UMPA - UMR 5669,
2 Univ Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 29 January 2022
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.
Mathematics Subject Classification: 93D15 / 93B05 / 35R11 / 26E10
Key words: Stabilization / approximate null-controllability / thick sets / quasi-analytic sequences / diffusive equations
© The authors. Published by EDP Sciences, SMAI 2022
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