# Null controllability of Grushin-type operators in dimension two

Karine Beauchard; Piermarco Cannarsa; Roberto Guglielmi

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 1, page 67-101
- ISSN: 1435-9855

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topBeauchard, 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 -

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