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

This paper is devoted to studying the null and approximate controllability of two linear coupled parabolic equations posed on a smooth domain Ω of R^N (N⩾ 1) with coupling terms of zero and first orders and one control localized in some arbitrary nonempty open subset ω of the domain Ω. We prove the null controllability under a new sufficient condition and we also provide the first example of a not approximately controllable system in the case where the support of one of the nontrivial coupling terms intersects the control domain ω.