Nonlinear boundary stabilization of nonuniform Timoshenko beam
type: Item
instance of: scholarly article
title: Nonlinear boundary stabilization of nonuniform Timoshenko beam
zbMATH Open document ID: 1062.74031
author: Huichao Zou
author: Q. Yan
author: Dexing Feng
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 2004-09-22T00:00:00Z
review text: The aim is to study the stabilization problem for nonuniform Timoshenko beam by using nonlinear boundary feedback controls. The boundary stabilization problem is characterized by the distributed system of the form \[ \begin{cases} \rho\ddot w+ (K(\varphi- w'))'= 0\quad &0< x< \ell,\;t> 0,\\ I_\rho\ddot\varphi- (EI\varphi')'+ K(\varphi- w')= 0\quad & 0< x< \ell,\;t> 0,\\ w(0, t)= \varphi(0,t)= 0\quad & t> 0,\\ K(\ell)(\varphi(\ell, t)- w'(\ell,t))\in f_1(\dot w(\ell, t))\quad & t> 0,\\ -EI(\ell)\varphi'(\ell, t)\in f_2(\dot\varphi(\ell, t))\quad &t> 0,\end{cases}\tag{1} \] where \(\rho\) is the mass per unit length, \(w(x,t)\) is the deflection of the beam, \(\varphi(x, t)\) is the total rotatory angle of the beam at \(x\), \(I_\rho\) \(EI\) are respectively the mass moment of inertia and rigidity coefficient of the cross-section, and \(K\) is the shear modulus of elasticity. Two functions \(f_1(\dot w(\ell,t))\) and \(f_2(\dot\varphi(\ell, t))\) are the control force and moment applied to the free end of the beam. The main result: Using the nonlinear semigroup theory and exponential multiplier method, it is proved that the energy associated with the solution to system (1) decays as a negative power of time \(t\) and exponentially in some special case as \(t\to\infty\). The precise proofs of all lemmas are presented. Finally, two examples show that a great variety of controls (including saturated ones) can act as boundary controls of the system (1).
Mathematics Subject Classification ID: 74M05
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 74H40
Mathematics Subject Classification ID: 93D15
zbMATH DE Number: 2101811
zbMATH Keywords: feedback controls
zbMATH Keywords: nonlinear semigroups theory
zbMATH Keywords: exponential multiplier method
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
full work available at URL: s10255-003-0099-x
OpenAlex ID: W1974701011
cites work: Q4094029
cites work: Nonlinear feedback control of Timoshenko beam
cites work: Exponential Decay of Energy of the Euler--Bernoulli Beam with Locally Distributed Kelvin--Voigt Damping
cites work: Boundary stabilization of Timoshenko beam
DOI: 10.1007/S10255-003-0099-X
description: scientific article; zbMATH DE number 2101811