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- W3097029119 abstract "We consider the equation $(partial_t + rho(sqrt{-Delta}))f(t,x) = mathbf 1_omega u(t,x)$, $xin mathbb R$ 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 $(partial_t -partial_v^2 + v^2partial_x)f(t,x,v) = mathbf 1_omega u(t,x,v)$ for $(x,v)in Omega_xtimes Omega_v$ 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 $omega_xtimes Omega_v$. The idea is to remark that, for some families of solutions, the Kolmogorov equation behaves like the rotated fractional heat equation $(partial_t + sqrt i(-Delta)^{1/4})g(t,x) = mathbf 1_omega u(t,x)$, $xin mathbb T$." @default.
- W3097029119 created "2020-11-09" @default.
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- W3097029119 date "2020-01-01" @default.
- W3097029119 modified "2023-09-25" @default.
- W3097029119 title "Lack of Null-Controllability for the Fractional Heat Equation and Related Equations" @default.
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- W3097029119 doi "https://doi.org/10.1137/19m1256610" @default.
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