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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; [email protected]. 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; [email protected].
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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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References

Alvarez, F., On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control Optim. 38 (2000) 1102-1119. CrossRef
Attouch, H. and Cominetti, R., A dynamical approach to convex minimization coupling approximation with the steepest descent method. J. Differential Equations 128 (1996) 519-540. CrossRef
H. Attouch and M.O. Czarnecki, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Differential Equations (to appear).
Attouch, H., Goudou, X. and Redont, P., A dynamical method for the global exploration of stationary points of a real-valued mapping: The heavy ball method. Communications in Contemporary Math. 2 (2000) 1-34. CrossRef
Aulbach, B., Approach to hyperbolic manifolds of stationary solutions. Springer-Verlag, Lecture Notes in Math. 1017 (1983) 56-66. CrossRef
H. Brezis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam, Math. Studies 5 (1973).
Bruck, R.E., Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J. Funct. Anal. 18 (1975) 15-26. CrossRef
Brunovsky, P. and Polacik, P., The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension. J. Differential Equations 135 (1997) 129-181.
C.V. Coffman, R.J. Duffin and D.H. Shaffer, The fundamental mode of vibration of a clamped annular plate is not of one sign, Constructive Approaches to Math. Models. Academic Press, New York-London-Toronto, Ont. (1979) 267-277.
Dafermos, C.M. and Slemrod, M., Asymptotic behavior of nonlinear contraction semigroups. J. Funct. Anal. 13 (1973) 97-106. CrossRef
R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique, Vol. 8, Évolution : semi-groupe, variationnel. Masson, Paris (1988).
Furuya, H., Miyashiba, K. and Kenmochi, N., Asymptotic behavior of solutions to a class of nonlinear evolution equations. J. Differential Equations 62 (1986) 73-94. CrossRef
Ghidaglia, J.M. and Temam, R., Attractors for damped nonlinear hyperbolic equations. J. Math. Pures Appl. 66 (1987) 273-319.
Hale, J. and Raugel, G., Convergence in gradient-like systems with applications to PDE. Z. Angew. Math. Phys. 43 (1992) 63-124. CrossRef
Haraux, A., Asymptotics for some nonlinear hyperbolic equations with a one-dimensional set of rest points. Bol. Soc. Brasil. Mat. 17 (1986) 51-65. CrossRef
A. Haraux, Semilinear Hyperbolic Problems in Bounded Domains, Mathematical Reports 3(1). Harwood Academic Publishers, Gordon and Breach, London (1987).
Haraux, A. and Jendoubi, M.A., Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity. Calc. Var. Partial Differential Equations 9 (1999) 95-124. CrossRef
Jendoubi, M.A., Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity. J. Differential Equations 144 (1998) 302-312. CrossRef
Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967) 591-597. CrossRef
Pazy, A., On the asymptotic behavior of semigroups of nonlinear contractions in Hilbert space. J. Funct. Anal. 27 (1978) 292-307. CrossRef
Simon, L., Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. Math. 118 (1983) 525-571. CrossRef
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York, Appl. Math. Sci. 68 (1988).
Zuazua, E., Stability and decay for a class of nonlinear hyperbolic problems. Asymptot. Anal. 1 (1988) 161-185.