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 equation, linear damping, dissipative hyperbolic equation, weak solution, asymptotic behavior, stabilization, weak convergence, Hilbert space

@article{COCV_2001__6__539_0, author = {Alvarez, Felipe and Attouch, Hedy}, title = {Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {539--552}, publisher = {EDP-Sciences}, volume = {6}, year = {2001}, mrnumber = {1849415}, zbl = {1004.34045}, language = {en}, url = {http://www.numdam.org/item/COCV_2001__6__539_0/} }

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 SP - 539 EP - 552 VL - 6 PB - EDP-Sciences UR - http://www.numdam.org/item/COCV_2001__6__539_0/ LA - en ID - COCV_2001__6__539_0 ER -

%0 Journal Article %A Alvarez, Felipe %A Attouch, Hedy %T Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria %J ESAIM: Control, Optimisation and Calculus of Variations %D 2001 %P 539-552 %V 6 %I EDP-Sciences %U http://www.numdam.org/item/COCV_2001__6__539_0/ %G en %F COCV_2001__6__539_0

Alvarez, Felipe; Attouch, Hedy. Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria. ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), pp. 539-552. http://www.numdam.org/item/COCV_2001__6__539_0/

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