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The analysis of vertex feedback stabilisability of a star-shaped network of fluid-conveying pipes
Analysis of PDEs · Submitted 18 Sep 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | gap 0.80 | 0.04 | 0.03 | asserted 0.68 |
| s1 | In the present paper we deal with this problem. | contribution 0.90 | 0.59 | 0.02 | asserted 0.62 |
| s2 | 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. | contribution 0.99 | 0.89 | 0.05 | asserted 0.79 |
| s3 | Finding an answer to the question is not straightforward, for the system operator associated with the corresponding closed-loop system is unbounded and nonselfadjoint. | background 0.63 | 0.17 | 0.02 | asserted 1.00 |
| s4 | 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. | method 0.97 | 0.55 | 0.03 | asserted 0.98 |
| s5 | 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. | result 1.00 | 0.94 | 0.90 | asserted 1.00 |
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The paper
Original abstract
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.
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What it cites · 8 sentences without a confident label
Uses the method or tool of 1 paper
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So the characteristic equation has an asymptotic representation of the form [equation] or [equation] (The idea of the proof would now be to establish, by applying Rouche's theorem, that, choosing $f(ρ)= cosρ$ and $g(ρ)= -𝒪 (ρ^-1)$, $|g|≤1< |f|$ on a closed contour in the complex plane around $ρ=(n+1/2)π$ for large $n$, and $f$ must therefore have only one zero within the contour; see [ref] .) Let the sequence ${λ_n}$ represent the roots of [ref] and set $ρ_n=(n+1/2)π+z_n$.
The Riesz basisness of the eigenfunctions and eigenvectors connected to the stability problem of a fluid-conveying tube with boundary control · Asymptotics of eigenvalues · 0.93
Cites as background 1 paper
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A resurgence of interest in studying stabilisation problems for pipe-flow systems (or systems related to them) can be seen in the recent papers [ref] .
The Riesz basisness of the eigenfunctions and eigenvectors connected to the stability problem of a fluid-conveying tube with boundary control · Introduction · 0.89
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