Geodesic
Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 539-552

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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.

Classification : 34E10, 34G05, 35B40, 35L70, 58D25
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://geodesic.mathdoc.fr/item/COCV_2001__6__539_0/}
}
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%A Attouch, Hedy
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%D 2001
%P 539-552
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Alvarez, Felipe; Attouch, Hedy. Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 539-552. http://geodesic.mathdoc.fr/item/COCV_2001__6__539_0/