Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings

TL;DR

This paper analyzes the spectral properties and stabilization of a chain of alternating Euler-Bernoulli beams and strings, demonstrating energy decay under certain conditions using frequency domain methods.

Contribution

It provides new spectral analysis and stabilization results for a complex network of Euler-Bernoulli beams and strings, including decay rates independent of lengths.

Findings

Energy tends to zero under irrational length conditions.

Polynomial decay of energy for all regular initial data.

Spectral analysis of the characteristic equation.

Abstract

We consider $N$ Euler-Bernoulli beams and $N$ strings alternatively connected to one another and forming a particular network which is a chain beginning with a string. We study two stabilization problems on the same network and the spectrum of the corresponding conservative system: the characteristic equation as well as its asymptotic behavior are given. We prove that the energy of the solutions of the first dissipative system tends to zero when the time tends to infinity under some irrationality assumptions of the length of the strings and beams. On another hand we prove a polynomial decay result of the energy of the second system, independently of the length of the strings and beams, for all regular initial data. Our technique is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.