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

Abstract

We consider the equation $({\partial }_t+\rho (\sqrt{-\Delta }))f(t,x)=1_{\omega }u(t,x)$, $x\in \mathbb{R}$ or $\mathbb{T}$. We prove it is not null-controllable if $\rho$ is analytic on a conic neighborhood of ${\mathbb{R}}_{+}$ and $\rho (\xi )=o(|\xi |)$. 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 $({\partial }_t-{\partial }_v^2+v^2{\partial }_x)f(t,x,v)=1_{\omega }u(t,x,v)$ for $(x,v)\in {\Omega }_x\times {\Omega }_v$ with ${\Omega }_x=\mathbb{R}$ or $\mathbb{T}$ and ${\Omega }_v=\mathbb{R}$ or $(-1,1)$. We prove it 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 the rotated fractional heat equation $({\partial }_t+\sqrt{i}(-\Delta {)}^{1/4})g(t,x)=1_{\omega }u(t,x)$, $x\in \mathbb{T}$.