Stabilization of a hybrid system of elasticity by feedback boundary damping
type: Item
instance of: scholarly article
title: Stabilization of a hybrid system of elasticity by feedback boundary damping
zbMATH Open document ID: 0664.73025
DOI: 10.1007/BF01766154
author: Walter Littman
author: L. Markus
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1988-01-01T00:00:00Z
review text: The authors consider the controlled vibration of a uniform beam to which a rigid body is attached at one end. The problem they discuss is to find linear feedback controls guaranteeing uniform asymptotic stability of strong solutions. The hybrid-type system can be reformulated as an Euler- Bernoulli beam equation with one end damped and the other under control. The boundary conditions at the controlled end, where the rigid body is attached to, are, however, not of the common form. First a semigroup generation result is given for feedback system. Then arguments based on the \(\omega\)-limit sets are given to ensure the trend to equilibrium. Furthermore, it is shown that the most natural absorbing boundary conditions do not lead to uniform exponential decay of the energy. As a matter of fact, the decay can be shown to be arbitrarily slow. [This is a continuation of the authors' earlier article in: Arch. Ration. Mech. Anal. 103, No.3, 193-236 (1988; Zbl 0656.73029).]
Mathematics Subject Classification ID: 74H45
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 70Q05
Mathematics Subject Classification ID: 74E30
zbMATH DE Number: 4087151
zbMATH Keywords: large-scale space-structure
zbMATH Keywords: long flexible mast
zbMATH Keywords: rigid antenna
zbMATH Keywords: closed- loop controllers
zbMATH Keywords: controlled vibration
zbMATH Keywords: uniform beam
zbMATH Keywords: rigid body is attached
zbMATH Keywords: linear feedback controls
zbMATH Keywords: uniform asymptotic stability
zbMATH Keywords: strong solutions
zbMATH Keywords: Euler-Bernoulli beam equation
zbMATH Keywords: one end damped
zbMATH Keywords: controlled end
zbMATH Keywords: semigroup generation
zbMATH Keywords: absorbing boundary conditions
zbMATH Keywords: decay
zbMATH Keywords: arbitrarily slow
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4087151