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

Louis Tebou

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 14, Issue: 3, page 561-574
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Tebou, Louis. "A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2007): 561-574. <http://eudml.org/doc/90883>.

@article{Tebou2007,
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. },
author = {Tebou, Louis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Hyperbolic equation; exponential decay; localized damping; Carleman estimates; hyperbolic equation},
language = {eng},
month = {12},
number = {3},
pages = {561-574},
publisher = {EDP Sciences},
title = {A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations},
url = {http://eudml.org/doc/90883},
volume = {14},
year = {2007},
}

TY - JOUR
AU - Tebou, Louis
TI - A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/12//
PB - EDP Sciences
VL - 14
IS - 3
SP - 561
EP - 574
AB - 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.
LA - eng
KW - Hyperbolic equation; exponential decay; localized damping; Carleman estimates; hyperbolic equation
UR - http://eudml.org/doc/90883
ER -

References

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  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.  
  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.  
  3. H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland (1973).  
  4. H. Brezis, Analyse fonctionnelle. Théorie et Applications. Masson, Paris (1983).  
  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.  
  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.  
  7. B. Dehman, Stabilisation pour l'équation des ondes semi-linéaire. Asymptotic Anal.27 (2001) 171–181.  
  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.  
  9. A. Doubova and A. Osses, Rotated weights in global Carleman estimates applied to an inverse problem for the wave equation. Inverse Problems22 (2006) 265–296.  
  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).  
  11. X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Contr. Opt.46 (2007) 1578–1614.  
  12. A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations. J. Diff. Eq.59 (1985) 145–154.  
  13. A. Haraux, Semi-linear hyperbolic problems in bounded domains, in Mathematical Reports3, Hardwood academic publishers (1987) .  
  14. A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math.46 (1989) 245–258.  
  15. A. Haraux, Remarks on weak stabilization of semilinear wave equations. ESAIM: COCV6 (2001) 553–560.  
  16. O.Yu. Imanuvilov, On Carleman estimates for hyperbolic equations. Asympt. Anal.32 (2002) 185–220.  
  17. V. Komornik, Exact controllability and stabilization. The multiplier method. RAM, Masson & John Wiley, Paris (1994).  
  18. J. Lagnese, Control of wave processes with distributed control supported on a subregion. SIAM J. Control Opt.21 (1983) 68–85.  
  19. I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential Integral Equations6 (1993) 507–533.  
  20. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villars, Paris (1969).  
  21. J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués, Vol. 1, RMA 8. Masson, Paris (1988).  
  22. J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I. Springer-Verlag, New York-Heidelberg (1973).  
  23. K. Liu, Locally distributed control and damping for the conservative systems. SIAM J. Control Opt.35 (1997) 1574–1590.  
  24. F. Macià and E. Zuazua, On the lack of observability for wave equations: a gaussian beam approach. Asymptot. Anal.32 (2002) 1–26.  
  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.  
  27. M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math.95 (1996) 25–42.  
  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.  
  29. M. Nakao, Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation. J. Diff. Eq.190 (2003) 81–107.  
  30. M. Nakao and I.H. Jung, Energy decay for the wave equation in exterior domains with some half-linear dissipation. Differential Integral Equations16 (2003) 927–948.  
  31. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences44. Springer-Verlag, New York (1983).  
  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.  
  33. A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures Appl.71 (1992) 455–467.  
  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.  
  35. M. Slemrod, Weak asymptotic decay via a relaxed invariance principle for a wave equation with nonlinear, nonmonotone damping. Proc. Royal Soc. Edinburgh Sect. A113 (1989) 87–97.  
  36. D. Tataru, The X θ s spaces and unique continuation for solutions to the semilinear wave equation. Comm. Partial Differential Equations21 (1996) 841–887.  
  37. L.R. Tcheugoué Tébou, Estimations d'énergie pour l'équation des ondes avec un amortissement nonlinéaire localisé. C. R. Acad. Paris, Sér. I325 (1997) 1175–1179.  
  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.  
  39. L.R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping. J. Diff. Eq.145 (1998) 502–524 
  40. L.R. Tcheugoué Tébou, Well-posedness and energy decay estimates for the damped wave equation with Lr localizing coefficient. Comm. Partial Differential Equations23 (1998) 1839–1855.  
  41. L.R. Tcheugoué Tébou, A direct method for the stabilization of some locally damped semilinear wave equations. C. R. Acad. Sci. Paris, Ser. I342 (2006) 859–864.  
  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.  
  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).  
  45. E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Differential Equations15 (1990) 205–235.  
  46. E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures Appl.15 (1990) 205–235.  

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