A new method to obtain decay rate estimates for dissipative systems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 4 (1999), pp. 419-444.
@article{COCV_1999__4__419_0,
     author = {Martinez, Patrick},
     title = {A new method to obtain decay rate estimates for dissipative systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {419--444},
     publisher = {EDP-Sciences},
     volume = {4},
     year = {1999},
     mrnumber = {1693904},
     zbl = {0923.35027},
     language = {en},
     url = {http://archive.numdam.org/item/COCV_1999__4__419_0/}
}
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Martinez, Patrick. A new method to obtain decay rate estimates for dissipative systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 4 (1999), pp. 419-444. http://archive.numdam.org/item/COCV_1999__4__419_0/

[1] M. Aassila, On a quasilinear wave equation with a strong damping. Funkcial. Ekvac. 41 ( 199867-78. | MR | Zbl

[2] V. Barbu, Analysis and control of nonlinear infinite dimensional systems. Academic Press, New York ( 1993). | MR | Zbl

[3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 ( 1992) 1024-1065. | MR | Zbl

[4] A. Carpio, Sharp estimates of the energy for the solutions of some dissipative second order evolution equations. Potential Anal. 1 ( 1992) 265-289. | MR | Zbl

[5] G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain. J. Math. Pures Appl. 58 ( 1979) 249-274. | MR | Zbl

[6] G. Chen and H. Wang, Asymptotic behavior of solutions of the one dimensional wave equation with a nonlinear boundary stabilizer. SIAM J. Control Optim. 27 ( 1989) 758-775. | MR | Zbl

[7] F. Chentouh, Décroissance de l'énergie pour certaines équations hyperboliques semilinéaires dissipatives. Thèse de 3e cycle, Université Pierre et Marie Curie ( 1984).

[8] F. Conrad, J. Leblond and J. P. Marmorat, Stabilization of second order evolution equations by unbounded nonlinear feedback in. Proc. of the Fifth IFAC Symposium on Control of Distributed Parameter Systems, Perpignan ( 1989) 101-116. | Zbl

[9] F. Conrad and B. Rao, Decay of solutions of wave equations in a star-shaped domain with non-linear boundary feedback. Asymptotic Analysis 7 ( 1993) 159-177. | MR | Zbl

[10] C.M. Dafermos, Asymptotic behavior of solutions of evolutions equationsNonlinear evolution equations, M.G. Crandall, Ed., Academic Press, New-York ( 1978) 103-123. | MR | Zbl

[11] A. Haraux, Comportement à l'infini pour une équation des ondes non linéaire dissipative. C. R. Acad. Sci. Paris Sér. A 287 ( 1978507-509. | Zbl

[12] A. Haraux, Oscillations forcées pour certains systèmes dissipatifs non linéaires. Publication du Laboratoire d'Analyse Numérique No. 78010, Université Pierre et Marie Curie, Paris ( 1978).

[13] A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems. Arch. Rat. Mech. Anal. 100 ( 1988) 191-206. | MR | Zbl

[14] M.A. Horn and I. Lasiecka, Global stabilization of a dynamic Von Karman plate with nonlinear boundary feedback. Appl. Math. Optim. 31 ( 1995) 57-84. | MR | Zbl

[15] M.A. Horn and I. Lasiecka, Nonlinear boundary stabilization of parallelly connected Kirchhoff plates. Dynamics and Control 6 ( 1996) 263-292. | MR | Zbl

[16] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation. J. Maths Pures Appl. 69 ( 1990) 33-54. | MR | Zbl

[17] V. Komornik, On the nonlinear boundary stabilization of the wave equation. Chinese Ann. Math. Ser. B. 14 ( 1993153-164. | MR | Zbl

[18] V. Komornik, Exact Controllability and Stabilization RAM: Research in Applied Mathematics. Masson, Paris; John Wiley, Ltd., Chichester ( 1994). | MR | Zbl

[19] S. Kouémou Patcheu, On the decay of solutions of some semilinear hyperbolic problems. Panamer. Math. J. 6 ( 1996) 69-82. | MR | Zbl

[20] J.E. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differential Equations 50 ( 1983163-182. | Zbl

[21] J.E. Lagnese, Boundary stabilization of thin plates. SIAM Studies in Appl. Math., Philadelphia, 1989. | MR | Zbl

[22] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. J. Diff. Integr. Eq. 6 ( 1993) 507-533. | MR | Zbl

[23] I. Lasiecka, Uniform stabilizability of a full Von Karman System with nonlinear boundary feedback. SIAM J. Control Optim. 36 ( 1998) 1376-1422. | MR | Zbl

[24] I. Lasiecka, Boundary stabilization of a 3-dimensional structural acoustic model. J. Math. Pures Appl. 78 ( 1999203-232. | MR | Zbl

[25] J.L. Lions, Contrôlabilité exacte et stabilisation de systèmes distribués, Vol. 1, Masson, Paris ( 1988). | MR | Zbl

[26] W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equation, preprint. | MR

[27] P. Martinez, Decay of solutions of the wave equation with a local highly degenerate dissipationAsymptotic Analysis 19 ( 1999) 1-17. | MR | Zbl

[28] P. Martinez, A new method to obtain decay rate estimates for dissipative Systems with localized damping. Rev. Mat. Compl Madrid, to appear. | MR | Zbl

[29] M. Nakao, Asymptotic stability of the bounded or almost periodic solution of the wave equation with a nonlinear dissipative term. J. Math. Anal. Appl. 58 ( 1977) 336-343. | MR | Zbl

[30] M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305 ( 1996) 403-417. | MR | Zbl

[31] L.R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping. J. Differential Equations 145 ( 1998) 502-524. | MR | Zbl

[32] J. Vancostenoble, Optimalité d'estimations d'énergie pour une équation des ondes amortie. C. R. Acad. Sci. Paris Sér. A, to appear. | Zbl

[33] J. Vancostenoble and P. Martinez, Optimality of energy estimates for a damped wave equation with polynomial or non polynomial feedbacks, submitted.

[34] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems. Asymptotic Analysis 1 ( 19881-28. | MR | Zbl

[35] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control and Optim. 28 ( 1990) 466-478. | MR | Zbl