Non null controllability of the Grushin equation in 2D

Abstract

We are interested in the exact null controllability of the equation ${\partial }_{t}f-{\partial }_x^{2}f-x^2{\partial }_y^{2}f=1_{\omega }u$, 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.

References in corpus (1)

• Null-controllability of non-autonomous Ornstein-Uhlenbeck equations

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