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arXiv · 1701.06467

Non null controllability of the Grushin equation in 2D

Armand Koenig
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Abstract

We are interested in the exact null controllability of the equation $\partial_t f - \partial_x^2 f - x^2 \partial_y^2f = \mathbf 1_\omega u$, with control $u$ supported on $\omega$. We show that, when $\omega$ does not intersect a horizontal band, the considered equation is never null-controllable. The main idea is to interpret the associated observability inequality as an $L^2$ estimate on entire functions, which Runge's theorem disproves. To that end, we study in particular the first eigenvalue of the operator $-\partial_x^2 + (nx)^2$ with Dirichlet conditions on $(-1,1)$ and we show a quite precise estimation it satisfies, even when $n$ is in some complex domain.

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Armand Koenig. 2017-01-23. Non null controllability of the Grushin equation in 2D. https://doi.org/10.1016/j.crma.2017.10.021

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