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

TL;DR

This paper investigates the stabilisability of a star-shaped pipe network conveying fluid using vertex feedback control, establishing conditions for exponential stability when tension exceeds the fluid flow velocity squared.

Contribution

It introduces a spectral approach to analyze vertex feedback stabilisability of fluid-conveying pipe networks, providing new stability results for certain tension conditions.

Findings

System is exponentially stable when tension > fluid velocity squared

Spectral approach effectively analyzes unbounded, nonselfadjoint operators

Positive stabilisation result under specific tension conditions

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.