Null controllability of Grushin-type operators in dimension two

作者：Beauchard K^*; Cannarsa P; Guglielmi R

来源：Journal of the European Mathematical Society, 2014, 16(1): 67-101.

DOI：10.4171/JEMS/428

• Null controllability
• degenerate parabolic equations
• Carleman estimates

We study the null controllability of the parabolic equation associated with the Grushin-type operator A = partial derivative(2)(x) + vertical bar x vertical bar(2 gamma) partial derivative(2)(y) (gamma > 0) in the rectangle Omega = (-1, 1) x (0, 1), under an additive control supported in an open subset omega of Omega. We prove that the equation is null controllable in any positive time for gamma < 1 and that there is no time for which it is null controllable for gamma > 1. In the transition regime gamma = 1 and when omega is a strip omega = (a, b) x (0, 1) (0 < a, b <= 1), a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular geometric configuration of Omega, 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.

• 出版日期2014