AI 中文总结
本文针对控制区域与退化集相交的退化双曲方程，引入基于形状设计的近似方法，通过一致双曲方程近似并结合谱近似等，最终得到原退化方程的能观性不等式。
AI 中文摘要
本文研究一类控制区域与退化集相交的退化双曲方程的能观性与能控性。现有结果主要处理控制区域与退化部分分离的情形，本文考虑控制区域延伸至退化部分的情况。为应对退化带来的困难，本文引入基于形状设计的近似方法，通过一族一致双曲方程近似退化方程。该证明依赖于相关算子的谱近似、弱解的精确估计以及乘子法。本文首先针对近似方程建立与近似参数无关常数的能观性不等式，随后通过取退化极限，得到原退化方程的能观性不等式。
英文摘要
In this paper, we study the observability and controllability of a class of degenerate hyperbolic equations with a control region intersecting the degenerate set. Unlike the existing results that mainly deal with control regions separated from the degeneracy, we consider the case where the control region reaches the degenerate part. To handle the difficulty caused by the degeneracy, we introduce a shape-design-based approximation method based on shape design by approximating the degenerate equation with a family of uniformly hyperbolic equations. The proof relies on the spectral approximation of the associated operators, precise estimates for weak solutions, and the multiplier method. We first establish observability inequalities for the approximating equations with constants independent of the approximation parameter. Then, by passing to the degenerate limit, we obtain the observability inequality for the original degenerate equation.
