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.

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
Mots clés : second-order in time equation, linear damping, dissipative hyperbolic equation, weak solution, asymptotic behavior, stabilization, weak convergence, Hilbert space
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author = {Alvarez, Felipe and Attouch, H\'edy},
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},
zbl = {1004.34045},
mrnumber = {1849415},
language = {en},
url = {http://www.numdam.org/item/COCV_2001__6__539_0/}
}
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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://www.numdam.org/item/COCV_2001__6__539_0/

[1] F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control Optim. 38 (2000) 1102-1119. | MR 1760062 | Zbl 0954.34053

[2] H. Attouch and R. Cominetti, A dynamical approach to convex minimization coupling approximation with the steepest descent method. J. Differential Equations 128 (1996) 519-540. | MR 1398330 | Zbl 0886.49024

[3] H. Attouch and M.O. Czarnecki, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Differential Equations (to appear). | MR 1883745 | Zbl 1007.34049

[4] H. Attouch, X. Goudou and P. Redont, 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. | MR 1753136 | Zbl 0983.37016

[5] B. Aulbach, Approach to hyperbolic manifolds of stationary solutions. Springer-Verlag, Lecture Notes in Math. 1017 (1983) 56-66. | MR 726568 | Zbl 0525.34034

[6] H. Brezis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam, Math. Studies 5 (1973). | Zbl 0252.47055

[7] R.E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J. Funct. Anal. 18 (1975) 15-26. | MR 377609 | Zbl 0319.47041

[8] P. Brunovsky and P. Polacik, The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension. J. Differential Equations 135 (1997) 129-181. | Zbl 0868.35062

[9] 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. | MR 559499 | Zbl 0475.73049

[10] C.M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups. J. Funct. Anal. 13 (1973) 97-106. | MR 346611 | Zbl 0267.34062

[11] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique, Vol. 8, Évolution : semi-groupe, variationnel. Masson, Paris (1988).

[12] H. Furuya, K. Miyashiba and N. Kenmochi, Asymptotic behavior of solutions to a class of nonlinear evolution equations. J. Differential Equations 62 (1986) 73-94. | MR 830048 | Zbl 0563.47041

[13] J.M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations. J. Math. Pures Appl. 66 (1987) 273-319. | MR 913856 | Zbl 0572.35071

[14] J. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE. Z. Angew. Math. Phys. 43 (1992) 63-124. | MR 1149371 | Zbl 0751.58033

[15] A. Haraux, Asymptotics for some nonlinear hyperbolic equations with a one-dimensional set of rest points. Bol. Soc. Brasil. Mat. 17 (1986) 51-65. | MR 901595 | Zbl 0637.35010

[16] A. Haraux, Semilinear Hyperbolic Problems in Bounded Domains, Mathematical Reports 3(1). Harwood Academic Publishers, Gordon and Breach, London (1987). | Zbl 0875.35054

[17] A. Haraux and M.A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity. Calc. Var. Partial Differential Equations 9 (1999) 95-124. | MR 1714129 | Zbl 0939.35122

[18] M.A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity. J. Differential Equations 144 (1998) 302-312. | MR 1616964 | Zbl 0912.35028

[19] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967) 591-597. | MR 211301 | Zbl 0179.19902

[20] A. Pazy, On the asymptotic behavior of semigroups of nonlinear contractions in Hilbert space. J. Funct. Anal. 27 (1978) 292-307. | MR 477932 | Zbl 0377.47045

[21] L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. Math. 118 (1983) 525-571. | MR 727703 | Zbl 0549.35071

[22] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York, Appl. Math. Sci. 68 (1988). | MR 953967 | Zbl 0662.35001

[23] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems. Asymptot. Anal. 1 (1988) 161-185. | MR 950012 | Zbl 0677.35069