The Riesz basisness of the eigenfunctions and eigenvectors connected to the stability problem of a fluid-conveying tube with boundary control

Abstract

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(\lambda\right)=\lambda^2G+\lambda D+C\quad\text{and}\quad\mathcal{P}\left(\lambda\right)=\lambda I-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.