Exponential stability of laminated Timoshenko beams with boundary/internal controls
type: Item
instance of: scholarly article
title: Exponential stability of laminated Timoshenko beams with boundary/internal controls
zbMATH Open document ID: 1435.35047
author: Margareth Alves
author: Rodrigo Nunes Monteiro
published in: Journal of Mathematical Analysis and Applications
publication date: 2019-11-28T00:00:00Z
review text: The authors deal with the initial-boundary value problem describing the behavior of the two-layered cantilever Timoshenko beam configured over the interval \(I=(0,\ell)\), composed of the two identical beams of uniform thickness bonded by an adhesive layer of negligible thickness. After introducing the variable \(\xi=3s-\psi\) denoting the effective rotation angle they obtain the system \begin{align*} &mw_{tt}+G(3s-\xi-w_x)_x=0,\\ &I_m\xi_{tt}-G(3s-\xi-w_x)-D\xi_{xx}=0,\\ &I_m s_{tt}+G(3s-\xi-w_x)+\frac {4\gamma}3s+\frac {4\beta}3s_t=0 \ \text{in}\ (0,\ell) \times (0,\infty) \end{align*} with the dissipative boundary conditions \begin{align*} &w(0,t)=\xi(0,t)=s(0,t)=0,\\ &(3s-\xi-w_x)(\ell,t)=\kappa_1w_t(\ell,t),\ \xi_x(\ell,t)=-\kappa_2\xi_t(0,t),\ s_x(\ell,t)=0,\ t\ge 0 \end{align*} and the initial condition \[ (w,w_t,\xi,\xi_t,s,s_t)=(w_0(x),w_1(x),\xi_0(x),\xi_1(x),s_0(x),s_1(x))\ \text{in}\ (0,\ell). \] Using the semigroup framework the authors prove the existence, uniqueness and continuous dependence of a solution with respect to initial data. The main result is the exponential stability of the solution semigroup.
reviewed by: Igor Bock
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 35L53
zbMATH DE Number: 7138374
zbMATH Keywords: laminated Timoshenko systems
zbMATH Keywords: dissipative systems
zbMATH Keywords: exponential stability
MaRDI profile type: Publication
full work available at URL: j.jmaa.2019.123516
OpenAlex ID: W2977056041
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Wikidata QID: Q115570262
DOI: 10.1016/J.JMAA.2019.123516
description: scientific article; zbMATH DE number 7138374