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Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria

Published online by Cambridge University Press:  15 August 2002

Felipe Alvarez  and
Hedy Attouch
Felipe Alvarez
Affiliation:
Depto. Ingeniería Matemática, Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile; falvarez@dim.uchile.cl. Centro de Modelamiento Matemático (CNRS UMR 2071), Universidad de Chile, Blanco Encalada 2120, Santiago, Chile.
Hedy Attouch
Affiliation:
Laboratoire ACSIOM-CNRS FRE 2311, Département de Mathématiques, Université de Montpellier II, Place Eugène Bataillon, 34095 Montpellier Cedex 05, France; attouch@math.univ-montp2.fr.
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Abstract

It is established convergence to a particular equilibrium for weak solutions of abstract linear equations of the second order in time associated with monotone operators with nontrivial kernel. Concerning nonlinear hyperbolic equations with monotone and conservative potentials, it is proved a general asymptotic convergence result in terms of weak and strong topologies of appropriate Hilbert spaces. It is also considered the stabilization of a particular equilibrium via the introduction of an asymptotically vanishing restoring force into the evolution equation.

Keywords

Second-order in time equationlinear dampingdissipative hyperbolic equationweak solutionasymptotic behaviorstabilizationweak convergenceHilbert space.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 539 - 552
Copyright
© EDP Sciences, SMAI, 2001

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