论文标题
光谱不平等，用于组合Hermite功能和享受Gelfand-Shilov平滑效果的进化方程的无效控制性
Spectral inequalities for combinations of Hermite functions and null-controllability for evolution equations enjoying Gelfand-Shilov smoothing effects
论文作者
论文摘要
这项工作致力于研究Hermite功能有限组合的不确定性原理。我们为对照亚集建立了一些光谱不平等，这些子集相对于一些在无穷大的无限密度生长而厚，并就这些光谱估计中出现的常数提供了HERMITE函数的能量水平，提供了定量估计值。这些光谱不平等允许在任何积极的时间内的进化方程中得出无效的可控制性，并具有特定的正则化效果。更准确地说，对于给定的索引$ \ frac {1} {2} \leqμ<1 $，我们在控制子集上推断出足够的几何条件，以确保进化方程式的无效连接性在对称的gelfand-shilov space $ s^μ___（\ s^μ__mathbb {\ rathbb {r）$ s^n）中享受正规化效果。这些结果尤其适用于在与某些类别的低纤维化非偏爱二次二次操作员或分数谐波振荡器相关的任何积极时间中得出零可控性。
This work is devoted to the study of uncertainty principles for finite combinations of Hermite functions. We establish some spectral inequalities for control subsets that are thick with respect to some unbounded densities growing almost linearly at infinity, and provide quantitative estimates, with respect to the energy level of the Hermite functions seen as eigenfunctions of the harmonic oscillator, for the constants appearing in these spectral estimates. These spectral inequalities allow to derive the null-controllability in any positive time for evolution equations enjoying specific regularizing effects. More precisely, for a given index $\frac{1}{2} \leq μ<1$, we deduce sufficient geometric conditions on control subsets to ensure the null-controllability of evolution equations enjoying regularizing effects in the symmetric Gelfand-Shilov space $S^μ_μ(\mathbb{R}^n)$. These results apply in particular to derive the null-controllability in any positive time for evolution equations associated to certain classes of hypoelliptic non-selfadjoint quadratic operators, or to fractional harmonic oscillators.
