JournalsjemsVol. 16, No. 1pp. 67–101

Null controllability of Grushin-type operators in dimension two

  • Piermarco Cannarsa

    Università di Roma, Italy
  • Karine Beauchard

    École Polytechnique, Palaiseau, France
  • Roberto Guglielmi

    Università di Roma 'Tor Vergata', Italy
Null controllability of Grushin-type operators in dimension two cover
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Abstract

We study the null controllability of the parabolic equation associated with the Grushin-type operator A=x2+x2γy2,(γ>0),A=\partial_x^2+|x|^{2\gamma}\partial_y^2\,, (\gamma>0), in the rectangle Ω=(1,1)×(0,1)\Omega=(-1,1)\times(0,1), under an additive control supported in an open subset ω\omega of Ω\Omega. We prove that the equation is null controllable in any positive time for γ<1\gamma<1 and that there is no time for which it is null controllable for γ>1\gamma>1. In the transition regime γ=1\gamma=1 and when ω\omega is a strip ω=(a,b)×(0,1),(0<a,b1)\omega=(a,b)\times(0,1)\,, (0<a,b\le1), a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular geometric configuration of Ω\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.

Cite this article

Piermarco Cannarsa, Karine Beauchard, Roberto Guglielmi, Null controllability of Grushin-type operators in dimension two. J. Eur. Math. Soc. 16 (2014), no. 1, pp. 67–101

DOI 10.4171/JEMS/428