与具有边界控制的输液管道稳定性问题相关的特征函数和特征向量的 Riesz 基性质
The Riesz basisness of the eigenfunctions and eigenvectors connected to the stability problem of a fluid-conveying tube with boundary control
AI总结：
本文研究了具有边界控制的拉伸输液管道的稳定性问题，通过分析相关算子束的特征值与 Riesz 基性质，建立了闭环系统的适定性与指数稳定性，并给出了比现有文献更大的参数稳定区域。
摘要：
本文研究具有边界控制的拉伸输液管道的稳定性问题。抽象谱问题涉及在不同 Hilbert 乘积空间中取值的如下形式的算子束：$\mathcal{M}(\lambda)=\lambda^2G+\lambda D+C$ 以及 $\mathcal{P}(\lambda)=\lambda I-T$。深入分析了复平面中特征值的位置与渐近性，以及相应特征函数和特征向量的 Riesz 基性质。在半群框架下，建立了由抽象方程 $\dot{x}(t)=Tx(t)$ 的初值问题所表示的闭环系统的适定性，以及解按特征向量的展开和闭环系统算子 $T$ 的稳定性。针对问题参数，给出了比文献中更大的区域，在这些区域内，一端简支、另一端施加边界控制器的拉伸输液管道可以实现指数稳定。
英文摘要：
In the present paper we study the stability problem for a stretched tube conveying fluid with boundary control. The abstract spectral problem concerns operator pencils of the forms \begin{equation*} \mathcal{M}\left(λ\right)=λ^2G+λD+C\quad\text{and}\quad\mathcal{P}\left(λ\right)=λI-T \end{equation*} taking values in different Hilbert product spaces. Thorough analysis is made of the location and asymptotics of eigenvalues in the complex plane and Riesz basisness of the corresponding eigenfunctions and eigenvectors. Well-posedness of the closed-loop system represented by the initial-value problem for the abstract equation \begin{equation*} \dot{x}\left(t\right)=Tx\left(t\right) \end{equation*} is established in the framework of semigroups as well as expansions of the solutions in terms of eigenvectors and stability of the closed-loop system operator $T$. For the parameters of the problem we give regions, larger than those in the literature, in which a stretched tube with flow, simply supported at one end, with a boundary controller applied at the other end, can be made exponentially stable.
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