发表机构
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明凸多面体上具有H^s(s>d/2)正则性初值的薛定谔方程，在边界奇异集任意非空开邻域为控制区域时可观测且可控，方法基于预解估计与HUM论证。
AI 中文摘要
我们考虑多面体上薛定谔方程的可观测性与可控性。我们证明，在$\mathbb{R}^d\\,(d\geq2)$中的凸多面体上，当初始数据具有$H^s\\,(s>d/2)$正则性时，可观测性与可控性成立，其中控制区域为边界奇异集的任意非空开邻域，这与特征函数的控制区域相同。我们的可观测性证明是对Burq--Zworski黑箱控制工作中方法的具体改编，其中可观测性不等式通过预解估计来证明。因此，在证明可观测性之前，我们给出了多面体上的可观测性预解估计。该预解估计的证明思路来自Cekić--Georgiev--Mukherjee考虑的多面体上特征函数集中情形，该情形使用半经典测度，并紧密依赖于多面体中台球流的动力学性质。最后，通过标准的HUM论证，我们证明可观测性蕴含可控性。
英文摘要
We consider the observability and controllability for the Schrödinger equation on polyhedra. We show that for the Schrödinger equation on convex polyhedra in $\mathbb{R}^d\,(d\geq2)$, when the initial data possess $H^s\,(s>d/2)$ regularity, observability and controllability hold, with the control region being an arbitrary nonempty open neighborhood of the singular set of the boundary, which is the same as the control region for eigenfunctions. Our proof of observability is a concrete adaptation of the method in the Burq--Zworski black box control work, where the observability inequality is proved via a resolvent estimate. Hence, before proving observability, we present an observability resolvent estimate on polyhedra. The idea of the proof of this resolvent estimate comes from the case of eigenfunction concentration on polyhedra considered by Cekić--Georgiev--Mukherjee, which uses semiclassical measures and relies closely on the dynamical properties of the billiard flow in polyhedra. Finally, by a standard HUM argument, we show that observability implies controllability.
Comments17 pages
