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

Paul Alphonse (1) , Jérémy Martin

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We prove that the thickness 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 $\mathbb R^n$ and associated with operators of the form $F(\vert D_x\vert)$, the function $F:[0,+\infty)\rightarrow\mathbb R$ being bounded below and continuous. We also provide explicit feedbacks and constants associated with these stabilization properties. Our results apply in particular for the half heat equation associated with the function $F(t) = t$, for which null-controllability is known to fail from thick control supports. More generally, the notion of thickness was 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 $s>1$, and that this null-controllability property from thick control supports does not hold when $0

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 of a large class of diffusive equations from thick control supports. 2021. ⟨hal-03588500v1⟩