# A new method to obtain decay rate estimates for dissipative systems

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 4, page 419-444
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topMartinez, Patrick. "A new method to obtain decay rate estimates for dissipative systems." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 419-444. <http://eudml.org/doc/116559>.

@article{Martinez2010,

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.
},

author = {Martinez, Patrick},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Nonlinear stabilization; asymptotic behavior in zero and at infinity.; nonlinear stabilization; asymptotic behavior in zero and at infinity; nonlinear integral inequality},

language = {eng},

month = {3},

pages = {419-444},

publisher = {EDP Sciences},

title = {A new method to obtain decay rate estimates for dissipative systems},

url = {http://eudml.org/doc/116559},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Martinez, Patrick

TI - A new method to obtain decay rate estimates for dissipative systems

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 4

SP - 419

EP - 444

AB -
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.

LA - eng

KW - Nonlinear stabilization; asymptotic behavior in zero and at infinity.; nonlinear stabilization; asymptotic behavior in zero and at infinity; nonlinear integral inequality

UR - http://eudml.org/doc/116559

ER -

## References

top- M. Aassila, 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).
- 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.
- A. Carpio, Sharp estimates of the energy for the solutions of some dissipative second order evolution equations. Potential Anal.1 (1992) 265-289.
- 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.
- 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.
- F. Chentouh, Décroissance de l'énergie pour certaines équations hyperboliques semilinéaires dissipatives. Thèse de 3${}^{\mathrm{e}}$ 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.
- F. Conrad and B. Rao, Decay of solutions of wave equations in a star-shaped domain with non-linear boundary feedback. Asymptotic Analysis7 (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.
- A. Haraux, Comportement à l'infini pour une équation des ondes non linéaire dissipative. C. R. Acad. Sci. Paris Sér. A287 (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).
- A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems. Arch. Rat. Mech. Anal. 100 (1988) 191-206.
- 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.
- M.A. Horn and I. Lasiecka, Nonlinear boundary stabilization of parallelly connected Kirchhoff plates. Dynamics and Control6 (1996) 263-292.
- V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation. J. Maths Pures Appl. 69 (1990) 33-54.
- V. Komornik, 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).
- S. Kouémou Patcheu, On the decay of solutions of some semilinear hyperbolic problems. Panamer. Math. J. 6 (1996) 69-82.
- J.E. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differential Equations50 (1983) 163-182.
- J.E. Lagnese, Boundary stabilization of thin plates. SIAM Studies in Appl. Math., Philadelphia, 1989.
- I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. J. Diff. Integr. Eq. 6 (1993) 507-533.
- I. Lasiecka, Uniform stabilizability of a full Von Karman system with nonlinear boundary feedback. SIAM J. Control Optim. 36 (1998) 1376-1422.
- I. Lasiecka, Boundary stabilization of a 3-dimensional structural acoustic model. J. Math. Pures Appl. 78 (1999) 203-232.
- 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.
- P. Martinez, Decay of solutions of the wave equation with a local highly degenerate dissipation. Asymptotic Analysis19 (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.
- 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.
- M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305 (1996) 403-417.
- L.R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping. J. Differential Equations145 (1998) 502-524.
- 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.
- E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems. Asymptotic Analysis1 (1988) 1-28.
- E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control and Optim. 28 (1990) 466-478.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.