A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 3, p. 561-574

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.

DOI : https://doi.org/10.1051/cocv:2007066
Classification:  93D15,  35L05,  35L70
Keywords: hyperbolic equation, exponential decay, localized damping, Carleman estimates
@article{COCV_2008__14_3_561_0,
     author = {Tebou, Louis},
     title = {A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {3},
     year = {2008},
     pages = {561-574},
     doi = {10.1051/cocv:2007066},
     zbl = {1146.93370},
     mrnumber = {2434066},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2008__14_3_561_0}
}
Tebou, Louis. A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 3, pp. 561-574. doi : 10.1051/cocv:2007066. http://www.numdam.org/item/COCV_2008__14_3_561_0/

[1] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary. SIAM J. Control Opt. 30 (1992) 1024-1065. | MR 1178650 | Zbl 0786.93009

[2] M.E. Bradley and I. Lasiecka, Global decay rates for the solutions to a von Kármán plate without geometric conditions. J. Math. Anal. Appl. 181 (1994) 254-276. | MR 1257968 | Zbl 0806.35180

[3] H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland (1973). | MR 348562 | Zbl 0252.47055

[4] H. Brezis, Analyse fonctionnelle. Théorie et Applications. Masson, Paris (1983). | MR 697382 | Zbl 0511.46001

[5] G. Chen, S.A. Fulling, F.J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math. 51 (1991) 266-301. | MR 1089141 | Zbl 0734.35009

[6] 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. | MR 513814 | Zbl 0499.35015

[7] B. Dehman, Stabilisation pour l'équation des ondes semi-linéaire. Asymptotic Anal. 27 (2001) 171-181. | MR 1852005 | Zbl 1007.35005

[8] B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation. Ann. Sci. École Norm. Sup. 36 (2003) 525-551. | Numdam | MR 2013925 | Zbl 1036.35033

[9] A. Doubova and A. Osses, Rotated weights in global Carleman estimates applied to an inverse problem for the wave equation. Inverse Problems 22 (2006) 265-296. | MR 2194195 | Zbl 1089.35085

[10] 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). | Numdam | MR 2383077 | Zbl pre05247877

[11] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Contr. Opt. 46 (2007) 1578-1614. | MR 2361985 | Zbl 1143.93005

[12] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations. J. Diff. Eq. 59 (1985) 145-154. | MR 804885 | Zbl 0535.35006

[13] A. Haraux, Semi-linear hyperbolic problems in bounded domains, in Mathematical Reports 3, Hardwood academic publishers (1987) . | MR 1078761 | Zbl 0875.35054

[14] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math. 46 (1989) 245-258. | MR 1021188 | Zbl 0679.93063

[15] A. Haraux, Remarks on weak stabilization of semilinear wave equations. ESAIM: COCV 6 (2001) 553-560. | Numdam | MR 1849416 | Zbl 0988.35029

[16] O.Yu. Imanuvilov, On Carleman estimates for hyperbolic equations. Asympt. Anal. 32 (2002) 185-220. | MR 1993649 | Zbl 1050.35046

[17] V. Komornik, Exact controllability and stabilization1994). | MR 1359765 | Zbl 0937.93003

[18] J. Lagnese, Control of wave processes with distributed control supported on a subregion. SIAM J. Control Opt. 21 (1983) 68-85. | MR 688440 | Zbl 0512.93014

[19] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential Integral Equations 6 (1993) 507-533. | MR 1202555 | Zbl 0803.35088

[20] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villars, Paris (1969). | MR 259693 | Zbl 0189.40603

[21] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués, Vol. 1, RMA 8. Masson, Paris (1988). | Zbl 0653.93002

[22] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I. Springer-Verlag, New York-Heidelberg (1973). | MR 350177 | Zbl 0251.35001

[23] K. Liu, Locally distributed control and damping for the conservative systems. SIAM J. Control Opt. 35 (1997) 1574-1590. | MR 1466917 | Zbl 0891.93016

[24] F. Macià and E. Zuazua, On the lack of observability for wave equations: a gaussian beam approach. Asymptot. Anal. 32 (2002) 1-26. | MR 1943038 | Zbl 1024.35062

[25] P. Martinez, Ph.D. thesis, University of Strasbourg, France (1998).

[26] P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complut. 12 (1999) 251-283. | MR 1698906 | Zbl 0940.35034

[27] M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math. 95 (1996) 25-42. | MR 1418287 | Zbl 0860.35072

[28] 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. | MR 1971655 | Zbl 1042.35595

[29] M. Nakao, Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation. J. Diff. Eq. 190 (2003) 81-107. | MR 1970957 | Zbl 1024.35080

[30] M. Nakao and I.H. Jung, Energy decay for the wave equation in exterior domains with some half-linear dissipation. Differential Integral Equations 16 (2003) 927-948. | MR 1988953 | Zbl 1035.35013

[31] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences 44. Springer-Verlag, New York (1983). | MR 710486 | Zbl 0516.47023

[32] G. Perla Menzala and E. Zuazua, The energy decay rate for the modified von Kármán system of thermoelastic plates: an improvement. Appl. Math. Lett. 16 (2003) 531-534. | MR 1983725 | Zbl 1040.74022

[33] A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures Appl. 71 (1992) 455-467. | MR 1191585 | Zbl 0832.35084

[34] 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. | MR 467472 | Zbl 0308.93007

[35] M. Slemrod, 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. | MR 1025456 | Zbl 0699.35023

[36] D. Tataru, The X θ s spaces and unique continuation for solutions to the semilinear wave equation. Comm. Partial Differential Equations 21 (1996) 841-887. | MR 1391526 | Zbl 0853.35017

[37] 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. | MR 1490120 | Zbl 0894.35071

[38] L.R. Tcheugoué Tébou, On the decay estimates for the wave equation with a local degenerate or nondegenerate dissipation. Portugal. Math. 55 (1998) 293-306. | MR 1647163 | Zbl 0918.35079

[39] L.R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping. J. Diff. Eq. 145 (1998) 502-524 | MR 1620983 | Zbl 0916.35069

[40] L.R. Tcheugoué Tébou, Well-posedness and energy decay estimates for the damped wave equation with L r localizing coefficient. Comm. Partial Differential Equations 23 (1998) 1839-1855. | MR 1641733 | Zbl 0918.35078

[41] 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. | MR 2224636 | Zbl 1096.35022

[42] J. Vancostenoble, Stabilisation non monotone de systèmes vibrants et Contrôlabilité. Ph.D. thesis, University of Rennes, France (1998).

[43] J. Vancostenoble, Weak asymptotic decay for a wave equation with gradient dependent damping. Asymptot. Anal. 26 (2001) 1-20. | MR 1829142 | Zbl 0973.35131

[44] 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). | MR 2321657 | Zbl 1134.74030

[45] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Differential Equations 15 (1990) 205-235. | MR 1032629 | Zbl 0716.35010

[46] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures Appl. 15 (1990) 205-235. | MR 1032629 | Zbl 0716.35010