环面$\u211D^d$上薛定谔方程的可控子空间与实部可观性
Controllable subspaces and real-part observability for Schrödinger equations on $\mathbb T^d$
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中文总结 AI 辅助
本文研究环面$\u211D^d$上实部控制的薛定谔方程内部可控性，刻画了最大零可控子空间，证明不可控方向为纯虚定常模式，通过实部紧性-唯一性等论证建立可观性不等式，为相关问题提供了显式描述。
中文摘要 AI 辅助
我们研究环面$\u211D^d$上薛定谔方程的内部可控性，其中控制仅通过方程的实部作用。该实部约束会导致一个实线性控制问题，可能无法实现完全零可控性。我们刻画了最大零可控子空间，并证明不可控方向恰好由纯虚定常模式给出。这些结果为具有实部控制的薛定谔方程提供了可控与不可控方向的显式描述，该刻画通过带有精确定常修正项的可观性不等式获得。对于柱形开控制区域，我们通过实部紧性-唯一性论证证明了该估计；对于移位自由情形下的容许可测乘积控制区域，我们针对粗糙时空控制集建立了相应估计。该证明结合了复值可观性输入与由实部观测中正向和反向薛定谔演化耦合产生的双谱分析。
英文摘要
We study internal controllability of Schrödinger equations on tori with controls acting only through their real parts. This real-part constraint leads to a real-linear control problem for which full null controllability may fail. We characterize the maximal null-controllable subspace and show that the uncontrollable directions are precisely given by the purely imaginary stationary modes. These results provide an explicit description of controllable and uncontrollable directions for Schrödinger equations with real-part controls. The characterization is obtained through observability inequalities with sharp stationary correction terms. For cylindrical open control regions, we prove such an estimate by a real-part compactness--uniqueness argument. For admissible measurable product control regions in the shifted free case, we establish the corresponding estimate for rough spacetime control sets. The proof combines complex-valued observability inputs with a double-spectrum analysis arising from the coupling of the forward and backward Schrödinger evolutions in the real-part observation.
