Summary
The paper analyzes the stability of a fluid-conveying tube with boundary control, using a spectral approach to study the eigenvalues and eigenvectors of the system. The results show that the system is exponentially stable when the tension in the tube is greater than the square of the fluid-flow velocity.
Highlights
- The system is exponentially stable when the tension in the tube is greater than the square of the fluid-flow velocity.
- The eigenvalues of the system approach a vertical asymptote in the left half-plane, a finite distance away from the imaginary axis.
- The eigenvectors of the system form a Riesz basis for the state space X.
- The system's energy decay rate is determined by the spectrum of the CLS operator.
- The Riesz basisness of the eigenfunctions of the boundary-eigenvalue problem is proven.
- The system's exponential stability is proven using the spectral approach.
- The results have implications for the study of fluid-structure interactions and the stability of pipes conveying fluid.
Key Insights
- The paper's use of a spectral approach to study the stability of the fluid-conveying tube system provides a rigorous and systematic way to analyze the system's behavior.
- The result that the system is exponentially stable when the tension in the tube is greater than the square of the fluid-flow velocity has important implications for the design and operation of fluid-conveying systems.
- The proof of the Riesz basisness of the eigenfunctions of the boundary-eigenvalue problem is a key technical contribution of the paper, and provides a foundation for further analysis of the system's behavior.
- The paper's use of the CLS operator to study the system's stability provides a powerful tool for analyzing the behavior of fluid-structure interactions.
- The results of the paper have implications for the study of fluid-structure interactions and the stability of pipes conveying fluid, and provide a foundation for further research in this area.
- The paper's focus on the exponential stability of the system provides a key insight into the system's long-term behavior, and has important implications for the design and operation of fluid-conveying systems.
- The use of Bari's theorem to prove the Riesz basisness of the eigenfunctions of the boundary-eigenvalue problem is a key technical contribution of the paper, and provides a powerful tool for analyzing the behavior of fluid-structure interactions.
Mindmap
Citation
Mahinzaeim, M., Xu, G. Q., & Feng, X. X. (2022). The Riesz basisness of the eigenfunctions and eigenvectors connected to the stability problem of a fluid-conveying tube with boundary control. arXiv. https://doi.org/10.48550/ARXIV.2204.01432