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 $\mathbb R^n$. 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 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 null-controllability is known to fail from general thick control supports.

Mots clés: Approximate null-controllability; Thick sets; Quasi-analytic sequences; Diffusive equations; Stabilization

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Paul Alphonse, Jérémy Martin. Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports. 2021. ⟨hal-03588500v2⟩