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

TL;DR

This paper proves the exponential stability of Beck's problem on a star-shaped graph with viscoelastic damping, using spectral analysis to show the system's eigenvectors form a Riesz basis, ensuring stability.

Contribution

It provides a complete spectral analysis demonstrating exponential stability for Beck's problem on a star graph with specific boundary conditions, even without elasticity.

Findings

Eigenvectors form a Riesz basis

System exhibits exponential stability

Stability holds without elasticity on slopes

Abstract

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.