Exponential decay for semilinear distributed systems with damping
type: Item
instance of: scholarly article
title: Exponential decay for semilinear distributed systems with damping
zbMATH Open document ID: 1010.35010
DOI: 10.1007/BF02904267
author: Larbi Berrahmoune
publication date: 2003-03-18T00:00:00Z
review text: The author considers the abstract system \[ y''(t)+Ay(t)+Dy'(t)+F(y(t),y'(t))By(t)=0, \] \[ y(0)=y_0,\quad y'(0)=y_1, \] which is a model for vibrating processes where damping is accounted for by the term \(Dy'(t)\); \(B\) and \(D\) are linear bounded operators from \(H\) to \(H\), where \(H\) is real Hilbert space. The operator \(D\) is assumed to be selfadjoint and to satisfy: \((Dv,v)\geq 0\), \(|Dv|^2\leq \text{const} (Dv,v)\) for every \(v\in H\). The operator \(A\) is the unique linear operator from \(V\subset H\) to \(V'\) which is characterized by \((Av,w)_{V'V}=(v,w)_{V}\) for every \(v,w\in V\). Two different nonlinearities are considered: \(F(y(t),y'(t))=f((By(t),y'(t)))\) and \(F(y(t),y'(t))=f((By(t),y(t)))\). The stability of the given equation is studied. The main goal is to characterize the classes of damping operators \(D\) such that under additional assumptions the energy specified by \(E(t)=(|y(t)|^2+|y'(t)|^2)/2\) decays exponentially to zero when \(t\to\infty\). Local, nonuniform and uniform exponential stability is investigated. Applications to the wave and Euler-Bernoulli equations are given.
reviewed by: Josef Diblík
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 93D20
zbMATH DE Number: 1884298
zbMATH Keywords: vibrating processes
zbMATH Keywords: Euler-Bernoulli equations
MaRDI profile type: Publication
cites work: Shorter Notes: Strongly Continuous Semigroups, Weak Solutions, and the Variation of Constants Formula
cites work: On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations
cites work: Feedback stabilization of distributed semilinear control systems
cites work: Nonharmonic fourier series and the stabilization of distributed semi-linear control systems
cites work: Exponential decay for distributed bilinear control systems with damping
cites work: Q4729705
cites work: Exact Semi-Internal Control of an Euler–Bernoulli Equation
cites work: Q4040045
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Unique continuation for weak solutions of the wave equation plus a potential
cites work: Compact sets in the space \(L^ p(0,T;B)\)
cites work: Q4888162
cites work: Exponential Decay for The Semilinear Wave Equation with Locally Distributed Damping
cites work: Exponential decay for the semilinear wave equation with localized damping in unbounded domains
full work available at URL: https://doi.org/10.1007/bf02904267
OpenAlex ID: W2041032037
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1884298