Summary: ESAIM: Control, Optimisation and Calculus of Variations Will be set by the publisher
CONVERGENCE AND ASYMPTOTIC STABILIZATION FOR SOME DAMPED
HYPERBOLIC EQUATIONS WITH NONISOLATED EQUILIBRIA #
Felipe Alvarez 1, 2 and Hedy Attouch 3
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.
Mathematics Subject Classification. 34E10, 34G05, 35B40, 35L70, 58D25.
Received June 1, 2000. Revised April 3, 2001.
Classical methods to establish the asymptotic convergence of trajectories of dissipative dynamical systems
assume isolated equilibrium points (local uniqueness). However, in many interesting cases the set of all equilibria
is a continuum of stationary solutions so that local uniqueness does not hold. The aim of this article is to show
that in the case of some infinitedimensional secondorder in time evolution equations, global monotonicity
conditions allow one to overcome the lack of uniqueness of stationary solutions in order to establish asymptotic
convergence. This work is motivated by a recent result of the first author concerning the asymptotic convergence