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

TL;DR

This paper demonstrates the lack of null-controllability for certain fractional heat and related equations, using geometric optics and semiclassical analysis, with implications for equations like the Kolmogorov-type equation.

Contribution

It establishes non-null controllability results for fractional heat equations and related models, extending the understanding of control limitations in these systems.

Findings

Fractional heat equations are not null-controllable under specified conditions.

The method applies geometric optics and semiclassical estimates.

Results include non-controllability of Kolmogorov-type equations in certain domains.

Abstract

We consider the equation $(\partial_t + \rho(\sqrt{-\Delta}))f(t,x) = \mathbf 1_\omega u(t,x)$, $x\in \mathbb R$ 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 $(\partial_t -\partial_v^2 + v^2\partial_x)f(t,x,v) = \mathbf 1_\omega u(t,x,v)$ 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 $(\partial_t + \sqrt i(-\Delta)^{1/4})g(t,x) = \mathbf 1_\omega u(t,x)$, $x\in \mathbb T$.