Hermite函数的测不准原理与密度衰减传感器集的零能控性
Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density
摘要：
建立了Hermite函数有限线性组合的一类测不准原理。更准确地说，我们给出了一个几何准则的子集$S\subset \RR^d$，确保$L^2$ -规范与$S$是等价的充分$L^2$ -规范$\RR^d$时，限制到一个给定的程度的Hermite函数的空间。我们给出了精确的估计如何等价常数取决于这个程度和几何参数的$S$。从这些估计我们推出的抛物型方程的生成元是谐振子是零可控的$S$。在我们的所有结果中，集合$S$可能具有次指数衰减的密度，特别是有限体积。我们还表明，有界集在这种情况下是没有效率的。
英文摘要：
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.
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