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

Felipe Alvarez; Hedy Attouch

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • Volume: 6, page 539-552
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Alvarez, Felipe, and Attouch, Hedy. "Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 539-552. <http://eudml.org/doc/90607>.

@article{Alvarez2001,
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.},
author = {Alvarez, Felipe, Attouch, Hedy},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {second-order in time equation; linear damping; dissipative hyperbolic equation; weak solution; asymptotic behavior; stabilization; weak convergence; Hilbert space; second-order in time; linear abstract equation},
language = {eng},
pages = {539-552},
publisher = {EDP-Sciences},
title = {Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria},
url = {http://eudml.org/doc/90607},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Alvarez, Felipe
AU - Attouch, Hedy
TI - Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 539
EP - 552
AB - 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.
LA - eng
KW - second-order in time equation; linear damping; dissipative hyperbolic equation; weak solution; asymptotic behavior; stabilization; weak convergence; Hilbert space; second-order in time; linear abstract equation
UR - http://eudml.org/doc/90607
ER -

References

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