Null-controllability of some reaction–diffusion systems with one control force

Abstract

This work is concerned with the null-controllability of semilinear parabolic systems by a single control force acting on a subdomain.

Inverse problem for a parabolic system with two components by measurements of one component

“…For controllability for systems, see [2] - [4], [11] - [13]. Next we will prove the approximate controllability with control χ ω h to only one component.…”

“…Imanuvilov and Yamamoto [17] discuss the global exact zero controllability for a semilinear parabolic equation. Also see Ammar-Khodja, Benabdallah and Dupaix [2], and Ammar-Khodja, Benabdallah, Dupaix and Kostine [3], [4], González-Burgos and Pérez-García [12] for semilinear parabolic systems.…”

“…In [2] and [8], it is assumed that A 11 = A 12 = 0. In that case, the proof can be completed by directly substituting v by means of u in ω T .…”

Indirect controllability of locally coupled systems under geometric conditions

“…Ici, a, b, c, p sont des fonctions réelles régulières de x ∈ Ω, avec b ≥ 0 et p ≥ 0, δ > 0 est un paramètre, −∆ c est un opérateur autoadjoint uniformément elliptique sur Ω, et f est le contrôle. Le résultat général concernant ces systèmes, prouvé par différentes méthodes dans [16,4,7,9] est un théorème de contrôlabilitéà zéro dès qu'on suppose {p > 0} ∩ {b > 0} ∅. Qu'en est-il du cas {p > 0} ∩ {b > 0} = ∅?…”

“…The null-controllability problem under view is the following: given a time T > 0 and initial data, is it possible to find a control function f so that the state has been driven to zero in time T ? It has been proved in [16,4,7,9] with different methods that System (2) is null-controllable as soon as {p > 0} ∩ {b > 0} ∅. In these works, the case {p > 0} ∩ {b > 0} = ∅ has been left as an open problem.…”