论文标题
HERMITE功能的不确定性原理和与传感器腐烂密度的无效控制性
Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density
论文作者
论文摘要
我们建立了一个不确定性原则的家族，以实现Hermite功能的有限线性组合。更准确地说，我们在子集$ s \ subset \ rr^d $上给出了几何标准，以确保与$ s $相关的$ l^2 $ -Seminorm等于$ \ rr^d $的完整$ l^2 $ - norm，当$ \ rr^d $上限制在Hermite的空间中，最高为给定程度。我们给出准确的估计，等价常数如何取决于该度和$ s $的几何参数。根据这些估计，我们得出的是，发电机是谐波振荡器的抛物线方程可从$ s $ null控制。 在我们的所有结果中，集合$ s $可能具有亚指数的腐烂密度，尤其是有限的量。我们还表明，在这种情况下，有限集并非有效。
We establish a family of uncertainty principles for finite linear combinations of Hermite functions. More precisely, we give a geometric criterion on a subset $S\subset \RR^d$ ensuring that the $L^2$-seminorm associated to $S$ is equivalent to the full $L^2$-norm on $\RR^d$ when restricted to the space of Hermite functions up to a given degree. We give precise estimates how the equivalence constant depends on this degree and on geometric parameters of $S$. From these estimates we deduce that the parabolic equation whose generator is the harmonic oscillator is null-controllable from $S$. In all our results, the set $S$ may have sub-exponentially decaying density and, in particular, finite volume. We also show that bounded sets are not efficient in this context.
