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arXiv · 2106.00129

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

Mahyar Mahinzaeim·Gen Qi Xu·Hai E Zhang
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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.

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Mahyar Mahinzaeim, Gen Qi Xu, Hai E Zhang. 2021-05-31. On the exponential stability of Beck's Problem on a star-shaped graph. https://doi.org/10.1016/j.jde.2023.07.040

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