Exponential stability of semigroups associated with some operator models in mechanics.
type: Item
instance of: scholarly article
title: Exponential stability of semigroups associated with some operator models in mechanics.
zbMATH Open document ID: 1061.34042
DOI: 10.1023/A:1024052419431
author: R. O. Griniv
author: A. A. Shkalikov
published in: Mathematical Notes
publication date: 2004-12-02T00:00:00Z
review text: The authors consider equations in a Hilbert space \(H\) of the form \[ \ddot x + B\dot x + Ax = 0\quad t\geq 0, \] where \(B\) is an unbounded symmetric operator in \(H\) (it corresponds to damping), and \(A\) is a uniformly positive and selfadjoint operator in \(H\) (it corresponds to the potential energy of the system), and \(x(t)\) is a function with values in \(H\). Such equations arise as consequences of linearizations in various problems of elasticity theory and hydromechanics. It is proved that the linear solution operator associated with this equation is a generator of a \(C_0\)-semigroup in the energy space. In addition, the mentioned semigroup is exponentially stable if the operator \(B\) is uniformly positive and the operator \(A\) dominates \(B\) in the sense of quadratic forms.
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 2120972
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: semigroup
zbMATH Keywords: exponential stability
zbMATH Keywords: energy space
zbMATH Keywords: dissipative operator
zbMATH Keywords: Hilbert space
zbMATH Keywords: generalized spectrum
reviewed by: Michael I. Gil'
MaRDI profile type: Publication
full work available at URL: a:1024052419431
OpenAlex ID: W135558868
description: scientific article; zbMATH DE number 2120972