Null controllability of Grushin-type operators in dimension two

By Karine Beauchard, Piermarco Cannarsa, Roberto Guglielmi. 2011

89 citations recorded by the source corpus.

We study the null controllability of the parabolic equation associated with the Grushintype operator $A = \partial_x^2 + |x|^{2\gamma}\partial_y^2 (\gamma > 0)$ 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 fo $\gamma 1$. In the transition regime $\gamma=1$ and when $\omega$ is a strip $\omega = (a, b)\times (0, 1) (0 < a, b \leq 1)$, 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.

Subfields: Applied Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/c68c446cadbe78023a3a51535d75241052216d3f

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In Applied Mathematics

In Control and Systems Engineering

Citations in the corpus

Null-controllability of hypoelliptic quadratic differential equations (2016, unknown): “Furthermore, a positive minimal time may be required to get observability, see the works [4, 5] in the case of the Grushin equation, [3] for the Heisenberg heat equation, and [2] for the Kolmogorov equation.”
DEGENERATE PARABOLIC OPERATORS OF KOLMOGOROV TYPE WITH A GEOMETRIC CONTROL CONDITION (2015, unknown): “In [11], K.”
2d Grushin-type equations: Minimal time and null controllable data (2015, background): “These observability constants are sharper in the present article than the one proved in [6] by Carleman estimates.”