AI 中文总结
该研究针对一类一维奇异/退化波动方程，借助奇异Sturm-Liouville理论，结合贝塞尔函数谱分解、Ingham型不等式等，证明其精确边界可控性，同时处理次临界等多种情形。
AI 中文摘要
我们证明了一类一维Sturm-Liouville型奇异/退化波动方程的精确边界可控性，该方程为：$u_{tt}-(x^\alpha u_x)_x-\beta x^{\alpha-1}u_x-\mu x^{\alpha-2}u=0$，其中$x\in(0,1)$，在正则端点处为Dirichlet条件，在奇异端点$x=0$处施加边界控制。分析在与相应奇异Sturm-Liouville算子关联的分数阶能量空间$X=\mathcal H^{\nu+1/2}\times \mathcal H^{\nu-1/2}$中进行。利用贝塞尔函数诱导的谱分解，我们建立了边界观测算子的容许性，并推导了观测系数的精确下界估计。通过Ingham型不等式结合酉群的抽象可观性结果，得到了精确可观性。该证明借助奇异Sturm-Liouville理论及通过奇异Sturm-Liouville表达式的拉格朗日括号确定的边界迹，同时处理了次临界、临界对数和极限点三种情形。
英文摘要
We prove exact boundary controllability for a class of one-dimensional singular/degenerate wave equations of Sturm--Liouville type, \[ u_{tt}-(x^αu_x)_x-βx^{α-1}u_x-μx^{α-2}u=0, \qquad x\in(0,1), \] with a Dirichlet condition at the regular endpoint and a boundary control acting at the singular endpoint \(x=0\). The analysis is carried out in the fractional energy space % \[ X=\mathcal H^{ν+1/2}\times \mathcal H^{ν-1/2}, \] associated with the corresponding singular Sturm--Liouville operator. Using the spectral decomposition induced by Bessel functions, we establish admissibility of the boundary observation operator and derive precise lower estimates for the observation coefficients. Exact observability is obtained through Ingham-type inequalities together with an abstract observability result for unitary groups. The proof treats simultaneously the subcritical, critical logarithmic, and limit-point regimes by means of singular Sturm--Liouville theory and boundary traces identified through the Lagrange bracket of the singular Sturm--Liouville expression.
Comments17 pages
