AI 中文总结
研究紧致黎曼流形上带大参数缩放势的波动方程控制成本扰动依赖性，引入几何控制条件\eqref{GCC+}，证明其对一致可观性成本存在的必要性与充分性，给出示例并估计不成立时可观性成本爆破率，证明依赖半经典和二阶微局部缺陷测度。
AI 中文摘要
本文研究了紧致黎曼流形上，由大参数$\lambda$缩放的与时间无关的势$\lambda V$对波动方程控制成本的扰动依赖性。引入几何控制条件\eqref{GCC+}，它是Bardos--Lebeau--Rauch--Taylor几何控制条件的变体，以适应势$V$的影响。证明了\eqref{GCC+}对于关于大参数$\lambda$的一致可观性成本的存在是必要且充分的。给出满足\eqref{GCC+}的几何示例，并估计其不成立时可观性成本的爆破率。证明依赖于半经典和二阶微局部缺陷测度。
英文摘要
This paper investigates the dependence of the control cost for a wave equation with respect to perturbation by a time-independent potential $\lmbd V$ scaled by a large parameter $\lmbd$ on a compact Riemannian manifold. We introduce the geometric control condition~\eqref{GCC+}, a variant of the geometric control condition of Bardos--Lebeau--Rauch--Taylor, tailored to accommodate the influence of the potential $V$. We show that~\eqref{GCC+} is necessary and sufficient for the existence of a uniform \emph{observability cost} with respect to the large parameter $\lmbd$. We provide geometric examples satisfying~\eqref{GCC+} and estimate the blow-up rate of the \emph{observability cost} in situations where it fails. The proofs rely on semiclassical and second microlocal defect measures.
