arXiv subjects

Gen Qi Xu

Publications and source records attributed to Gen Qi Xu.

5 recordsLinked to original sources

arXiv2024

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

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.

math.AP↗

arXiv2024

The analysis of vertex feedback stabilisability of a star-shaped network of fluid-conveying pipes

It is an outstanding problem whether a pipe-flow system on a star-shaped network is stabilisable by a feedback control on the common vertex. In the present paper we deal with this problem. In particular, we study the equation governing the small vibrations of a stretched elastic pipe conveying fluid in a star-shaped network and examine the question of vertex feedback stabilisability of such a system via control moments. Finding an answer to the question is not straightforward, for the system operator associated with the corresponding closed-loop system is unbounded and nonselfadjoint. An approach to the study of the stabilisation problem for the closed-loop system is presented based on the spectral approach previously introduced by the authors for star graphs of stretched elastic beams. When the tension in the pipes is greater than the square of the fluid-flow velocity, we establish a positive result that in fact gives the strong property of uniform exponential stability of the closed-loop system.

math.AP↗

arXiv2024

Spectral analysis of a viscoelastic tube conveying fluid with generalised boundary conditions

We study the spectral problem associated with the equation governing the small transverse motions of a viscoelastic tube of finite length conveying an ideal fluid. The boundary conditions considered are of general form, accounting for a combination of elasticity and viscous damping acting on both the slopes and the displacements of the ends of the tube. These include many standard boundary conditions as special cases such as the clamped, free, hinged, and guided conditions. We derive explicit asymptotic formulae for the eigenvalues for the case of generalised boundary conditions and specialise these results to the clamped case and the case in which damping acts on the slopes but not on the displacements. In particular, the dependence of the eigenvalues on the parameters of the problem is investigated and it is found that all eigenvalues are located in certain sectorial sets in the complex plane.

math.AP↗

arXiv2023

On the exponential stability of Beck's Problem on a star-shaped graph

We deal with the as yet unresolved exponential stability problem for Beck's Problem on a metric star graph with three identical edges. The edges are stretched Euler--Bernoulli beams which are simply supported with respect to the outer vertices. At the inner vertex we have viscoelastic damping acting on the slopes of the edges. We carry out a complete spectral analysis of the system operator associated with the abstract spectral problem in Hilbert space. Within this framework it is shown that the eigenvectors have the property of forming a Riesz (i.e.\ an unconditional) basis, which makes it possible to directly deduce the exponential stability of the corresponding $C_0$-semigroup using spectral information for the system operator alone. A physically interesting conclusion is that the particular choice of vertex conditions ensures the exponential stability even when the elasticity acting on the slopes of the edges is absent.

math.AP↗

arXiv2023

Decay Rates and Eigenvalue Asymptotics for Abstract Strongly Coupled Hyperbolic Equations with Infinite Memory

In this paper, the asymptotic behavior of abstract strongly coupled hyperbolic equations with one infinite memory term is investigated, one specific case of which is the model for describing the dynamical behaviour of magnetic effected piezoelectric beams. A fractional operator is involved in the memory term depending on the parameter $a\in [0,1)$. By means of frequency domain analysis, it is proved that the system can be indirectly stabilized polynomially by only one infinite memory term located on one of these two strongly coupled PDEs, and the explicit decay rates given as $t^{-\frac{1}{2-2a}}$ are only dependent on the parameter $a$. When considering the exponentially decreasing kernel, a detailed spectral analysis for the system operator is further provided. Specifically, the asymptotic expressions of the eigenvalues of the system operator are derived. Based on the expressions of the eigenvalues, the optimality of the obtained decay rates is further verified for this system.

math.AP↗