Null controllability of Grushin-type operators in dimension two

Beauchard, Karine, Cannarsa, Piermarco, and Guglielmi, Roberto. "Null controllability of Grushin-type operators in dimension two." Journal of the European Mathematical Society 016.1 (2014): 67-101. <http://eudml.org/doc/277760>.

@article{Beauchard2014,
abstract = {We study the null controllability of the parabolic equation associated with the Grushin-type operator $A=\partial ^2_x + \left|x\right|^\{2\gamma \}\partial ^2_\gamma , (\gamma >0)$, in the rectangle $\Omega =(-1,1)\times (0,1)$, 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 $\omega =(a,b)\times (0,1)(0<a,b\le 1)$), 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.},
author = {Beauchard, Karine, Cannarsa, Piermarco, Guglielmi, Roberto},
journal = {Journal of the European Mathematical Society},
keywords = {null controllability; degenerate parabolic equations; Carleman estimates; Carleman estimates},
language = {eng},
number = {1},
pages = {67-101},
publisher = {European Mathematical Society Publishing House},
title = {Null controllability of Grushin-type operators in dimension two},
url = {http://eudml.org/doc/277760},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Beauchard, Karine
AU - Cannarsa, Piermarco
AU - Guglielmi, Roberto
TI - Null controllability of Grushin-type operators in dimension two
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 1
SP - 67
EP - 101
AB - We study the null controllability of the parabolic equation associated with the Grushin-type operator $A=\partial ^2_x + \left|x\right|^{2\gamma }\partial ^2_\gamma , (\gamma >0)$, in the rectangle $\Omega =(-1,1)\times (0,1)$, 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 $\omega =(a,b)\times (0,1)(0<a,b\le 1)$), 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.
LA - eng
KW - null controllability; degenerate parabolic equations; Carleman estimates; Carleman estimates
UR - http://eudml.org/doc/277760
ER -