Paper detail

Lack of null-controllability for the fractional heat equation and related equations

We consider the equation $(\partial_t + ρ(\sqrt{-Δ}))f(t,x) = \mathbf 1_ωu(t,x)$, $x\in \mathbb R$ or $\mathbb T$. We prove it is not null-controllable if $ρ$ is analytic on a conic neighborhood of $\mathbb R_+$ and $ρ(ξ) = o(|ξ|)$. The proof relies essentially on geometric optics, i.e.\ estimates for the evolution of semiclassical coherent states. The method also applies to other equations. The most interesting example might be the Kolmogorov-type equation $(\partial_t -\partial_v^2 + v^2\partial_x)f(t,x,v) = \mathbf 1_ωu(t,x,v)$ for $(x,v)\in Ω_x\times Ω_v$ with $Ω_x = \mathbb R$ or $\mathbb T$ and $Ω_v = \mathbb R$ or $(-1,1)$. We prove it is not null-controllable in any time if $ω$ is a vertical band $ω_x\times Ω_v$. The idea is to remark that, for some families of solutions, the Kolmogorov equation behaves like the rotated fractional heat equation $(\partial_t + \sqrt i(-Δ)^{1/4})g(t,x) = \mathbf 1_ωu(t,x)$, $x\in \mathbb T$.