We study the null controllability of the parabolic equation associated with the Grushin-type operator in the rectangle , under an additive control supported in an open subset of . We prove that the equation is null controllable in any positive time for and that there is no time for which it is null controllable for . In the transition regime and when is a strip , a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular geometric configuration of , null controllability is closely linked to the one-dimensional observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the Fourier frequency.
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Piermarco Cannarsa, Karine Beauchard, Roberto Guglielmi, Null controllability of Grushin-type operators in dimension two. J. Eur. Math. Soc. 16 (2014), no. 1, pp. 67–101DOI 10.4171/JEMS/428