A compact perturbation method for the boundary stabilization of the Rayleigh beam equation
type: Item
instance of: scholarly article
title: A compact perturbation method for the boundary stabilization of the Rayleigh beam equation
zbMATH Open document ID: 0847.93052
DOI: 10.1007/BF01204704
author: Yanyan Li
published in: Applied Mathematics and Optimization
publication date: 1996-10-08T00:00:00Z
review text: The purpose of this important work is to investigate the stabilization of a clamped Rayleigh beam equation by only one boundary feedback control moment, or control force: \[ y_{tt}- \gamma y_{xxtt}+ y_{xxxx} =0, \quad 0< x< 1, \qquad y(0,t)= y_x (0, t)= 0, \] \[ y_{xx} (1, t)+ \alpha y_{xt} (1, t) =0, \qquad y_{xxx} (1, t)- \gamma y_{xtt} (1, t)- \beta y_t (1, t) =0, \] where \(\gamma>0\) is a physical constant, and where \(\alpha y_{xt} (1, t)\) and \(\beta y_t (1,t)\) designate respectively the boundary feedback control moment and control force. The main result is the following: The Rayleigh beam equation can be uniformly exponentially stabilized by only one control moment. Moreover the strong asymptotic stabilization and the lack of uniform exponential stabilization in the case of only one control force are proved.
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 58J37
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 35Q72
zbMATH DE Number: 883410
zbMATH Keywords: compact perturbation
zbMATH Keywords: uniform decay rate of energy
zbMATH Keywords: stabilization
zbMATH Keywords: clamped Rayleigh beam
zbMATH Keywords: boundary feedback control moment
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
cites work: A Note on Weak Stabilizability of Contraction Semigroups
cites work: Modeling, Stabilization and Control of Serially Connected Beams
cites work: Powers of a contraction in Hilbert space
cites work: A Note on Stabilization of Infinite Dimensional Linear Oscillators by Compact Linear Feedback
cites work: Q4838262
cites work: Q3995611
cites work: Stabilizability of coupled wave equations in parallel under various boundary conditions
cites work: Q5286577
cites work: Q4313480
cites work: Uniform Stabilization of a Hybrid System of Elasticity
cites work: Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods
cites work: Controllability and Stabilizability of the Third-Order Linear Dispersion Equation on a Periodic Domain
cites work: Lack of Uniform Stabilization for Noncontractive Semigroups Under Compact Perturbation
full work available at URL: https://doi.org/10.1007/bf01204704
OpenAlex ID: W2043165465
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description: scientific article; zbMATH DE number 883410