发表机构
School of Mathematics (Zhuhai), Sun Yat-sen University; Graduate School of System Informatics, Kobe University; Hetao Institute of Mathematics and Interdisciplinary Studies (Shenzhen)（中山大学数学学院（珠海）; 神户大学系统信息学研究科; 河套数学与交叉科学研究院（深圳））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究了自由薛定谔方程最优稳定化前置因子的渐近增长，证明了其以 $e^{c\sqrt{\delta}}$ 和 $e^{C\sqrt{\delta}}$ 为界，并应用于多种受控设置。
AI 中文摘要
我们研究了自由薛定谔方程的最优稳定化前置因子在所有有界平稳线性反馈下的渐近行为。对于给定的衰减率 $\delta>0$，令 $\widehat{C}_S(\delta)$ 表示该最优前置因子。假设非受控区域包含一个非空开集，且可观测性代价满足 $C(T)\le C_0e^{C_0/T}$ 对于小的 $T>0$，我们证明 $$ e^{c\sqrt{\delta}}\le \widehat{C}_S(\delta)\le e^{C\sqrt{\delta}}, \qquad \delta\gg1. $$ 下界基于非受控开孔和适用于拉普拉斯算子的定量截断构造，而上界结合了指数加权的 Gramian 矩阵和小时间可观测性代价 $C(T)\lesssim e^{C/T}$。这些结果适用于三种设置下的受控薛定谔方程：平坦环面，包括一个超出几何控制条件的带状控制示例；全空间 $\mathbb{R}^n$ 具有外部球控制；以及满足广义测地线几何控制条件的有界光滑区域。
英文摘要
We study the asymptotic behavior of the optimal stabilization prefactor for the free Schrödinger equation, optimized over all bounded stationary linear feedbacks. For a prescribed decay rate $δ>0$, let $\widehat{C}_S(δ)$ denote this optimal prefactor. Assuming that the uncontrolled region contains a nonempty open set and that the observability cost satisfies $C(T)\le C_0e^{C_0/T}$ for small $T>0$, we prove $$ e^{c\sqrtδ}\le \widehat{C}_S(δ)\le e^{C\sqrtδ}, \qquad δ\gg1. $$ The lower bound is based on the uncontrolled open hole and a quantitative cutoff construction adapted to the Laplacian, while the upper bound combines an exponentially weighted Gramian with the small-time observability cost $C(T)\lesssim e^{C/T}$. The results apply to controlled Schrödinger equations in three settings: flat tori, including a strip-control example beyond the geometric control condition; the whole space $\mathbb{R}^n$ with exterior-ball control; and bounded smooth domains satisfying the geometric control condition for generalized geodesics.
Comments19 pages
