SIAM Journal on Control and Optimization, Volume 58, Issue 6, Page 3130-3160, January 2020.
We consider the equation $(\partial_t + \rho(\sqrt{-\Delta}))f(t,x) = \mathbbm{1}_\omega u(t,x)$, $x\in \mathbb R$ 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 $(\partial_t -\partial_v^2 + v^2\partial_x)f(t,x,v) = \mathbbm{1}_\omega u(t,x,v)$ 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 note that, for some families of solutions, the Kolmogorov equation behaves like the rotated fractional heat equation $(\partial_t + \sqrt i(-\Delta)^{1/4})g(t,x) = \mathbbm{1}_\omega u(t,x)$, $x\in \mathbb T$.

中文翻译：

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分数热方程和相关方程的零可控性缺乏

SIAM控制与优化杂志，第58卷，第6期，第3130-3160页，2020年1月。
我们考虑方程$（\ partial_t + \ rho（\ sqrt {-\ Delta}））f（t，x）= \ mathbbm {1} _ \ omega u（t，x）$，$ x \ in \ mathbb R $或$ \ mathbb T $。我们证明，如果$ \ rho $是在$ \ mathbb R _ + $和$ \ rho（\ xi）= o（| \ xi |）$的圆锥邻域上分析的，则它不是null可控的。该证明基本上依赖于几何光学，即对半经典相干态演化的估计。该方法也适用于其他方程式。最有趣的示例可能是Kolmogorov型方程$（\ partial_t-\ partial_v ^ 2 + v ^ 2 \ partial_x）f（t，x，v）= \ mathbbm {1} _ \ omega u（t，x， v）$（x，v）\ in \ Omega_x \ times \ Omega_v $中的$ \ Omega_x = \ mathbb R $或$ \ mathbb T $和$ \ Omega_v = \ mathbb R $或$（-1,1 ）$。如果$ \ omega $是垂直带$ \ omega_x \ times \ Omega_v $，我们证明它在任何时候都不是可空控制的。这个想法是要注意，对于某些解决方案系列，