Null controllability of Grushin-type operators in dimension two

Karine Beauchard (1) , Piermarco Cannarsa (2) , Roberto Guglielmi (2)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 DIPMAT - Dipartimento di Matematica [Roma II]

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 the rectangle. We prove that the equation is null controllable in any positive time when the degeneracy is not too strong, and that there is no time for which it is null controllable when the degeneracy is too strong. In the transition regime and when the control support 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 con guration, 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.

Domaines: Equations aux dérivées partielles [math.AP]

Karine Beauchard, Piermarco Cannarsa, Roberto Guglielmi. Null controllability of Grushin-type operators in dimension two. 2013. ⟨hal-00826116⟩