Stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions
type: Item
instance of: scholarly article
title: Stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions
zbMATH Open document ID: 0665.35011
DOI: 10.1016/0022-0396(88)90129-5
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: The paper studies the problem of local exponential stabilizability of abstract systems of the form \[ dy/dt+Ay+B_ 1Fy+B_ 2G(y)=0, \] where F, \(B_ i\) are linear, bounded or not, and G is a nonlinear perturbation. Conditions on the data are given in order to preserve local exponential stability for a large class of perturbations. The major novelty here is that unbounded perturbations can be taken into account. Then the choice of the feedback operator F becomes more critical. General results, both for bounded and unbounded feedback operators F, are given. In the case of bounded perturbation or analytic semigroups, roughly speaking, the principle of linearized stability works. Applications to the wave equation with interior or boundary stabilizing feedback, in the last case acting either in a Dirichlet or Neumann condition, are set up. A high degree of unboundedness is then needed. An application to parabolic problems with boundary stabilization on Dirichlet or Neumann conditions illustrates the ``analytic'' case.
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 4088116
zbMATH Keywords: local exponential stabilizability
zbMATH Keywords: abstract systems
zbMATH Keywords: nonlinear perturbation
zbMATH Keywords: unbounded perturbations
zbMATH Keywords: feedback operator
zbMATH Keywords: analytic semigroups
zbMATH Keywords: principle of linearized stability
zbMATH Keywords: wave equation
zbMATH Keywords: Dirichlet or Neumann condition
author: Irena Lasiecka
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90129-5
OpenAlex ID: W2107288043
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description: scientific article; zbMATH DE number 4088116