AI 中文总结
研究由斜自伴算子生成的二阶无穷维演化系统的弱可观测性，通过引入谱强制性概念建立谱刻画，揭示椭圆算子预解式估计与演化生成元弱可观测性的联系，并在矩形域波动方程上应用得到弱可观测性估计。
AI 中文摘要
本文研究由形如\(iA_0\)的斜自伴算子生成的二阶无穷维演化系统的弱可观测性，其中\(A_0\)是自伴椭圆算子。首先通过引入观测算子的谱强制性概念并证明其与合适的预解式估计等价，建立了弱可观测性的谱刻画。主要结果揭示了椭圆算子\(A_0\)的预解式估计与相关演化生成元\(A\)的弱可观测性之间的直接联系。作为应用，在几种观测区域的几何配置下，为矩形域上的波动方程建立了明确的弱可观测性估计。分析结合了频域方法、预解式估计和傅里叶分析，对无穷维系统中预解式不等式、谱强制性和弱可观测性之间的相互作用有了新见解。
英文摘要
In this paper, we investigate the weak observability of second-order infinite-dimensional evolution systems generated by skew-adjoint operators of the form $iA_0$, where $A_0$ is a self-adjoint elliptic operator. We first establish a spectral characterization of weak observability by introducing the notion of spectral coercivity for the observation operator and proving its equivalence to a suitable resolvent estimate. Our main result reveals a direct link between resolvent estimates for the elliptic operator $A_0$ and the weak observability of the associated evolution generator $A$. More precisely, we prove that a resolvent inequality for $A_0$ implies a Hautus-type spectral observability estimate for $A$, which guarantees the weak observability of the system. This provides a unified spectral framework for weak observability based on the coercivity properties of the observation operator. As an application, we establish explicit weak observability estimates for the wave equation on a rectangular domain under several geometric configurations of the observation region. The analysis combines frequency-domain methods, resolvent estimates, and Fourier analysis, yielding new insights into the interplay between resolvent inequalities, spectral coercivity, and weak observability in infinite-dimensional systems.
