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Pré-Publication, Document De Travail Année : 2018

Non-null-controllability of the fractional heat equation and of the Kolmogorov equation

Armand Koenig

Résumé

We prove in this article that the Kolmogorov-type equation $(\partial_t -\partial_v^2 + v^2\partial_x)f(t,x,v) = \mathbf 1_\omega u(t,x,v)$ for $(t,x)\in \mathbb T\times \Omega_v$ with $\Omega_v = \mathbb R$ or $(-1,1)$ is not null-controllable in any time if $\omega$ is a vertical band $\omega_x\times \Omega_v$. The idea is to remark that, for some families of solutions, the Kolmogorov equation behaves like what we'll call the rotated fractional heat equation $(\partial_t + \sqrt i(-\Delta)^{1/4})g(t,x) = \mathbf 1_\omega u(t,x)$, $x\in \mathbb T$ and to disprove the observability inequality for rotated fractional equation by looking at how coherent states evolve.
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Dates et versions

hal-01829289 , version 1 (04-07-2018)
hal-01829289 , version 2 (01-12-2020)

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Armand Koenig. Non-null-controllability of the fractional heat equation and of the Kolmogorov equation. 2018. ⟨hal-01829289v1⟩
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