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

Alphonse, Paul; Martin, Jérémy

ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 16

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 ℝ$$. These equations are associated with operators of the form F(|D$$|), 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$$ 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.

Classification : 93D15, 93B05, 35R11, 26E10
Keywords: Stabilization, approximate null-controllability, thick sets, quasi-analytic sequences, diffusive equations

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