Controllability of a 2 × 2 parabolic system by one force with space-dependent coupling term of order one

Abstract

Abstract. This paper is devoted to the controllability of linear systems of two coupled parabolic equations when the coupling involves a space dependent first order term. This system is set on an bounded interval I ⊂⊂ R, and the first equation is controlled by a force supported in a subinterval of I or on the boundary. In the case where the intersection of the coupling and control domains is nonempty, we prove null controllability at any time. Otherwise, we provide a minimal time for null controllability. Finally we give a necessary and sufficient condition for the approximate controllability. The main technical tool for obtaining these results is the moment method.Mathematics Subject Classification. 93B05, 93B07.

A block moment method to handle spectral condensation phenomenon in parabolic control problems

“…Since then, the minimal null-control time property was investigated on various examples, still in the setting of coupled one dimensional parabolic systems [AKBGBdT16, Dup17,Oua20,Sam19] or in the setting of degenerate parabolic scalar equations [BC17,BDE20,BHHR15,BMM15,DK20]. For coupled parabolic systems a geometric control condition may also be needed for approximate controllability to hold [BO14,Oli14], proving once again that hyperbolic-like behavior can be observed in the parabolic setting.…”

Controllability of a 4 × 4 quadratic reaction–diffusion system

“…e 2sα (sφ) 24 (|ϕ 1 | 2 + |ϕ 2 | 2 + |∆ϕ 3 | 2 ) + (sφ) 22 (|∇ϕ 1 | 2 + |∇ϕ 2 | 2 + |∇∆ϕ 3 | 2 ) . (1, 43)), for any ε ′ > 0, we have…”

Positive and negative results on the internal controllability of parabolic equations coupled by zero- and first-order terms

“…In this section, we will prove that it is possible to replace locally the coupling operator g 21 · ∇ + a 21 by ∂ x1 , where x 1 is the first direction in space. This kind of simplification has already been used in [12, Lemma 2.6] for example, and we refer to this article for a more detailed proof (see also [18]). Let us remark that the regularities stated in Lemma 2.1 are higher than the one stated in Theorem 1 due to technical reasons appearing in the proofs of Lemmas 2.1 and 2.2.…”