Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T14:08:25.040Z Has data issue: false hasContentIssue false
  • English
  • Français

A new method to obtain decay rate estimates for dissipative systems

Published online by Cambridge University Press:  15 August 2002

Patrick Martinez
Patrick Martinez*
Affiliation:
Département de Mathématiques, ENS Cachan, Antenne de Bretagne and IRMAR, Université Rennes I, Campus de Ker Lann, 35170 Bruz, France; [email protected].
Get access

Abstract

We consider the wave equation damped with a boundary nonlinear velocity feedback p(u'). Under some geometrical conditions, we prove that the energy of the system decays to zero with an explicit decay rate estimate even if the function ρ has not a polynomial behavior in zero. This work extends some results of Nakao, Haraux, Zuazua and Komornik, who studied the case where the feedback has a polynomial behavior in zero and completes a result of Lasiecka and Tataru. The proof is based on the construction of a special weight function (that depends on the behavior of the function ρ in zero), and on a new nonlinear integral inequality.

Keywords

Nonlinear stabilizationasymptotic behavior in zero and at infinity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 419 - 444
Copyright
© EDP Sciences, SMAI, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aassila, M., On a quasilinear wave equation with a strong damping. Funkcial. Ekvac. 41 (1998) 67-78.
V. Barbu, Analysis and control of nonlinear infinite dimensional systems. Academic Press, New York (1993).
Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. CrossRef
Carpio, A., Sharp estimates of the energy for the solutions of some dissipative second order evolution equations. Potential Anal. 1 (1992) 265-289. CrossRef
Chen, G., Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain. J. Math. Pures Appl. 58 (1979) 249-274.
Chen, G. and Wang, H., Asymptotic behavior of solutions of the one dimensional wave equation with a nonlinear boundary stabilizer. SIAM J. Control Optim. 27 (1989) 758-775.
F. Chentouh, Décroissance de l'énergie pour certaines équations hyperboliques semilinéaires dissipatives. Thèse de 3 $\rm ^e$ cycle, Université Pierre et Marie Curie(1984).
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.
Conrad, F. and Rao, B., Decay of solutions of wave equations in a star-shaped domain with non-linear boundary feedback. Asymptotic Analysis 7 (1993) 159-177.
C.M. Dafermos, Asymptotic behavior of solutions of evolutions equations, Nonlinear evolution equations, M.G. Crandall, Ed., Academic Press, New-York (1978) 103-123.
Haraux, A., Comportement à l'infini pour une équation des ondes non linéaire dissipative. C. R. Acad. Sci. Paris Sér. A 287 (1978) 507-509.
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).
Haraux, A. and Zuazua, E., Decay estimates for some semilinear damped hyperbolic problems. Arch. Rat. Mech. Anal . 100 (1988) 191-206. CrossRef
Horn, M.A. and Lasiecka, I., Global stabilization of a dynamic Von Karman plate with nonlinear boundary feedback. Appl. Math. Optim . 31 (1995) 57-84. CrossRef
Horn, M.A. and Lasiecka, I., Nonlinear boundary stabilization of parallelly connected Kirchhoff plates. Dynamics and Control 6 (1996) 263-292. CrossRef
Komornik, V. and Zuazua, E., A direct method for the boundary stabilization of the wave equation. J. Maths Pures Appl . 69 (1990) 33-54.
Komornik, V., On the nonlinear boundary stabilization of the wave equation. Chinese Ann. Math. Ser. B . 14 (1993) 153-164.
V. Komornik, Exact Controllability and Stabilization. RAM: Research in Applied Mathematics. Masson, Paris; John Wiley, Ltd., Chichester (1994).
Kouémou Patcheu, S., On the decay of solutions of some semilinear hyperbolic problems. Panamer. Math. J . 6 (1996) 69-82.
Lagnese, J.E., Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differential Equations 50 (1983) 163-182. CrossRef
J.E. Lagnese, Boundary stabilization of thin plates. SIAM Studies in Appl. Math., Philadelphia, 1989.
Lasiecka, I. and Tataru, D., Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. J. Diff. Integr. Eq . 6 (1993) 507-533.
Lasiecka, I., Uniform stabilizability of a full Von Karman system with nonlinear boundary feedback. SIAM J. Control Optim . 36 (1998) 1376-1422. CrossRef
Lasiecka, I., Boundary stabilization of a 3-dimensional structural acoustic model. J. Math. Pures Appl . 78 (1999) 203-232. CrossRef
J.L. Lions, Contrôlabilité exacte et stabilisation de systèmes distribués, Vol. 1, Masson, Paris (1988).
W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equation, preprint.
Martinez, P., Decay of solutions of the wave equation with a local highly degenerate dissipation. Asymptotic Analysis 19 (1999) 1-17.
P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Compl Madrid, to appear.
Nakao, M., 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. CrossRef
Nakao, M., Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann . 305 (1996) 403-417. CrossRef
Tcheugoué Tébou, L.R., Stabilization of the wave equation with localized nonlinear damping. J. Differential Equations 145 (1998) 502-524. CrossRef
J. Vancostenoble, Optimalité d'estimations d'énergie pour une équation des ondes amortie. C. R. Acad. Sci. Paris Sér. A, to appear.
J. Vancostenoble and P. Martinez, Optimality of energy estimates for a damped wave equation with polynomial or non polynomial feedbacks, submitted.
Zuazua, E., Stability and decay for a class of nonlinear hyperbolic problems. Asymptotic Analysis 1 (1988) 1-28.
Zuazua, E., Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control and Optim . 28 (1990) 466-478. CrossRef