Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-25T16:16:16.561Z Has data issue: false hasContentIssue false
  • English
  • Français

A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

Published online by Cambridge University Press:  21 December 2007

Louis Tebou
Louis Tebou*
Affiliation:
Department of Mathematics, Florida International University, Miami FL 33199, USA; teboul@fiu.edu
Get access

Abstract

First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt.46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman estimate [Ruiz, J. Math. Pures Appl.71 (1992) 455–467], we address the same type of problem in an exterior domain for a locally damped semilinear wave equation. For both problems, our method of proof is constructive, and much simpler than those found in the literature. In particular, we improve in some way on earlier results by Dafermos, Haraux, Nakao, Slemrod and Zuazua.

Keywords

Hyperbolic equationexponential decaylocalized dampingCarleman estimates
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 3 , July 2008 , pp. 561 - 574
Copyright
© EDP Sciences, SMAI, 2007

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

Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control and stabilization from the boundary. SIAM J. Control Opt. 30 (1992) 1024–1065. CrossRef
Bradley, M.E. and Lasiecka, I., Global decay rates for the solutions to a von Kármán plate without geometric conditions. J. Math. Anal. Appl. 181 (1994) 254–276. CrossRef
H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland (1973).
H. Brezis, Analyse fonctionnelle. Théorie et Applications. Masson, Paris (1983).
Chen, G., Fulling, S.A., Narcowich, F.J. and Sun, S., Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math. 51 (1991) 266–301. CrossRef
C.M. Dafermos, Asymptotic behaviour of solutions of evolution equations, in Nonlinear evolution equations, M.G. Crandall Ed., Academic Press, New-York (1978) 103–123.
Dehman, B., Stabilisation pour l'équation des ondes semi-linéaire. Asymptotic Anal. 27 (2001) 171–181.
Dehman, B., Lebeau, G. and Zuazua, E., Stabilization and control for the subcritical semilinear wave equation. Ann. Sci. École Norm. Sup. 36 (2003) 525–551. CrossRef
Doubova, A. and Osses, A., Rotated weights in global Carleman estimates applied to an inverse problem for the wave equation. Inverse Problems 22 (2006) 265–296. CrossRef
T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear).
Fu, X., Yong, J. and Zhang, X., Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Contr. Opt. 46 (2007) 1578–1614. CrossRef
Haraux, A., Stabilization of trajectories for some weakly damped hyperbolic equations. J. Diff. Eq. 59 (1985) 145–154. CrossRef
A. Haraux, Semi-linear hyperbolic problems in bounded domains, in Mathematical Reports 3, Hardwood academic publishers (1987) .
Haraux, A., Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math. 46 (1989) 245–258.
Haraux, A., Remarks on weak stabilization of semilinear wave equations. ESAIM: COCV 6 (2001) 553–560. CrossRef
Imanuvilov, O.Yu., Carleman, On estimates for hyperbolic equations. Asympt. Anal. 32 (2002) 185–220.
V. Komornik, Exact controllability and stabilization. The multiplier method. RAM, Masson & John Wiley, Paris (1994).
Lagnese, J., Control of wave processes with distributed control supported on a subregion. SIAM J. Control Opt. 21 (1983) 68–85. CrossRef
Lasiecka, I. and Tataru, D., Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential Integral Equations 6 (1993) 507–533.
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villars, Paris (1969).
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués, Vol. 1, RMA 8. Masson, Paris (1988).
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I. Springer-Verlag, New York-Heidelberg (1973).
Liu, K., Locally distributed control and damping for the conservative systems. SIAM J. Control Opt. 35 (1997) 1574–1590. CrossRef
Macià, F. and Zuazua, E., On the lack of observability for wave equations: a gaussian beam approach. Asymptot. Anal. 32 (2002) 1–26.
P. Martinez, Ph.D. thesis, University of Strasbourg, France (1998).
Martinez, P., A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complut. 12 (1999) 251–283.
Nakao, M., Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math. 95 (1996) 25–42. CrossRef
M. Nakao, Global existence for semilinear wave equations in exterior domains, in Proceedings of the Third World Congress of Nonlinear Analysts, Part 4 (Catania, 2000), Nonlinear Anal. 47 (2001) 2497–2506.
Nakao, M., Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation. J. Diff. Eq. 190 (2003) 81–107. CrossRef
Nakao, M. and Jung, I.H., Energy decay for the wave equation in exterior domains with some half-linear dissipation. Differential Integral Equations 16 (2003) 927–948.
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences 44. Springer-Verlag, New York (1983).
Perla Menzala, G. and Zuazua, E., The energy decay rate for the modified von Kármán system of thermoelastic plates: an improvement. Appl. Math. Lett. 16 (2003) 531–534. CrossRef
Ruiz, A., Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures Appl. 71 (1992) 455–467.
D.L. Russell, Exact boundary value controllability theorems for wave and heat processes in star-complemented regions, in Differential games and control theory (Proc. NSF—CBMS Regional Res. Conf., Univ. Rhode Island, Kingston, R.I., 1973) Dekker, New York. Lect. Notes Pure Appl. Math. 10, Dekker, New York (1974) 291–319.
Slemrod, M., Weak asymptotic decay via a relaxed invariance principle for a wave equation with nonlinear, nonmonotone damping. Proc. Royal Soc. Edinburgh Sect. A 113 (1989) 87–97. CrossRef
Tataru, D., The $X_\theta^s$ spaces and unique continuation for solutions to the semilinear wave equation. Comm. Partial Differential Equations 21 (1996) 841–887. CrossRef
L.R. Tcheugoué Tébou, Estimations d'énergie pour l'équation des ondes avec un amortissement nonlinéaire localisé. C. R. Acad. Paris, Sér. I 325 (1997) 1175–1179.
Tcheugoué Tébou, L.R., On the decay estimates for the wave equation with a local degenerate or nondegenerate dissipation. Portugal. Math. 55 (1998) 293–306.
Tcheugoué Tébou, L.R., Stabilization of the wave equation with localized nonlinear damping. J. Diff. Eq. 145 (1998) 502–524 CrossRef
Tcheugoué Tébou, L.R., Well-posedness and energy decay estimates for the damped wave equation with L r localizing coefficient. Comm. Partial Differential Equations 23 (1998) 1839–1855. CrossRef
L.R. Tcheugoué Tébou, A direct method for the stabilization of some locally damped semilinear wave equations. C. R. Acad. Sci. Paris, Ser. I 342 (2006) 859–864.
J. Vancostenoble, Stabilisation non monotone de systèmes vibrants et Contrôlabilité. Ph.D. thesis, University of Rennes, France (1998).
Vancostenoble, J., Weak asymptotic decay for a wave equation with gradient dependent damping. Asymptot. Anal. 26 (2001) 1–20.
X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains, in Proceedings of the Eleventh International Conference on Hyperbolic Problems: Theory, Numerics and Applications, Lyon (2006) (to appear).
Zuazua, E., Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Differential Equations 15 (1990) 205–235.
Zuazua, E., Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures Appl. 15 (1990) 205–235.