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

Paul Alphonse (1) , Jérémy Martin (2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Stabilization; approximate null-controllability; thick sets; quasi-analytic sequences; diffusive equations

Domaines: Mathématiques [math]

Paul Alphonse, Jérémy Martin. Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports. ESAIM: Control, Optimisation and Calculus of Variations, 2022, 28, pp.16. ⟨10.1051/cocv/2022009⟩. ⟨hal-03588500v4⟩