Skip to main content
SpringerLink
Log in

On Some Recent Advances on Stabilization for Hyperbolic Equations

  • Chapter
  • First Online:
Control of Partial Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2048))

Abstract

The purpose of these Notes is to present some recent advances on stabilization for wave-like equations together with some well-known methods of stabilization. This course will give several references on the subject but do not pretend to exhaustivity. The spirit of these Notes is more that of a research monograph. We aim to give a simplified overview of some aspects of stabilization, on the point of view of energy methods, and insist on some of the methodological approaches developed recently. We will focus on nonlinear stabilization, memory-damping and indirect stabilization of coupled PDE’s and present recent methods and results. Energy methods have the advantage to handle and deal with physical quantities and properties of the models under consideration. For nonlinear stabilization, our purpose is to present the optimal-weight convexity method introduced in (Alabau-Boussouira, Appl. Math. Optim. 51(1):61–105, 2005; Alabau-Boussouira, J. Differ. Equat. 248:1473–1517, 2010) which provides a whole methodology to establish easy computable energy decay rates which are optimal or quasi-optimal, and works for finite as well as infinite dimensions and allow to treat, in a unified way different PDE’s, as well as different types of dampings: localized, boundary. Another important feature is that the upper estimates can be completed by lower energy estimates for several examples, and these lower estimates can be compared to the upper ones. Optimality is proved in finite dimensions and in particular for one-dimensional semi-discretized wave-like PDE’s. These results are obtained through energy comparison principles (Alabau-Boussouira, J. Differ. Equat. 248: 1473–1517, 2010), which are, to our knowledge, new. This methodology can be extended to the infinite dimensional setting thanks to still energy comparison principles supplemented by interpolation techniques. The optimal-weight convexity method is presented with two approaches: a direct and an indirect one. The first approach is based on the multiplier method and requires the assumptions of the multiplier method on the zone of localization of the feedback. The second one is based on an indirect argument, namely that the solutions of the corresponding undamped systems satisfy an observability inequality, the observation zone corresponding to the damped zone for the damped system. The advantage is that, this observability inequality holds under the sharper optimal Geometric Control Condition of Bardos et al. (SIAM J. Contr. Optim. 30:1024–1065, 1992). The optimal-weight convexity method also extends to the case memory-damping, for which the damping effects are nonlocal, and leads to nonautonomous evolution equations. We will only state the results in this latter case. Indirect stabilization of coupled systems have received a lot of attention recently. This subject concerns stabilization questions for coupled PDE’s with a reduced number of feedbacks. In practice, it is often not possible to control all the components of the vector state, either because of technological limitations or cost reasons. From the mathematical point of view, this means that some equations of the coupled system are not directly stabilized. This generates mathematical difficulties, which requires to introduce new tools to study such questions. In particular, it is important to understand how stabilization may be transferred from the damped equations to the undamped ones. We present several recent results of polynomial decay for smooth initial data. These results are based on energy methods, a nondifferential integral inequality introduced in (Alabau, Compt. Rendus Acad. Sci. Paris I 328:1015–1020, 1999) [see also (Alabau-Boussouira, SIAM J. Contr. Optim. 41(2):511–541, 2002; Alabau et al. J. Evol. Equat. 2:127–150, 2002)] and coercivity properties due to the coupling operators.

In memory of my father Abdallah Boussouira.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. F. Alabau-Boussouira, M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications (submitted)

    Google Scholar 

  2. F. Alabau-Boussouira, M. Léautaud, Indirect stabilization of locally coupled wave-type systems. ESAIM Contr. Optim. Calc. Var. Prépublication HAL, hal-00476250 (in press) DOI: 10.1051/cocv/2011106

    Google Scholar 

  3. F. Alabau-Boussouira, P. Cannarsa, R. Guglielmi, Indirect stabilization of weakly coupled systems with hybrid boundary conditions. Mathematical Control and Related Fields 1, 413–436 (2011)

    Google Scholar 

  4. F. Alabau-Boussouira, Strong lower energy estimates for nonlinearly damped Timoshenko beams and Petrowsky equations. Nonlinear Differ. Equat. Appl. 18, 571–597 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Alabau-Boussouira, M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions. Compt. Rendus Math. Acad. Sci. Paris I 349, 395–400 (2011)

    Article  MATH  Google Scholar 

  6. F. Alabau-Boussouira, K. Ammari, Sharp energy estimates for nonlinearly locally damped PDE’s via observability for the associated undamped system. J. Funct. Anal. 260, 2424–2450 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Alabau-Boussouira, New trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems. J. Differ. Equat. 249, 1145–1178 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems. J. Differ. Equat. 248, 1473–1517 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Alabau-Boussouira, K. Ammari, Nonlinear stabilization of abstract systems via a linear observability inequality and application to vibrating PDE’s. Compt. Rendus Math. Acad. Sci. Paris 348, 165–170 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Alabau-Boussouira, P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations. Compt. Rendus Math. Acad. Sci. Paris 347, 867–872 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Alabau-Boussouira, J. Prüss and R. Zacher, Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels. Comptes Rendus Mathématique, Compt. Rendus Acad. Sci. Paris I 347, 277–282 (2009)

    Google Scholar 

  12. F. Alabau-Boussouira, P. Cannarsa, in Control of Partial Differential Equations. Springer Encyclopedia of Complexity and Systems Science (Springer, New-York, 2009), pp. 1485–1509

    Google Scholar 

  13. F. Alabau-Boussouira, Asymptotic stability of wave equations with memory and frictional boundary dampings. Appl. Math. 35, 247–258 (2008)

    MathSciNet  MATH  Google Scholar 

  14. F. Alabau-Boussouira, P. Cannarsa, D. Sforza, Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254, 1342–1372 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control. Nonlinear Differ. Equat. Appl. 14(5–6), 643–669 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equations with nonlinear dissipation. J. Evol. Equ. 6(1), 95–112 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 51(1), 61–105 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Alabau-Boussouira, Une formule générale pour le taux de décroissance des systèmes dissipatifs non linéaires. Compt. Rendus Acad. Sci. Paris I Math. 338, 35–40 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Alabau-Boussouira, A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems. SIAM J. Contr. Optim. 42, 871–906 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Alabau, P. Cannarsa, V. Komornik, Indirect internal damping of coupled systems. J. Evol. Equat. 2, 127–150 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled systems. SIAM J. Contr. Optim. 41(2), 511–541 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Alabau, Observabilité et contrôlabilité frontière indirecte de deux équations des ondes couplées. Compt. Rendus Acad. Sci. Paris I Math. 333, 645–650 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés. Compt. Rendus Acad. Sci. Paris I 328, 1015–1020 (1999)

    MathSciNet  MATH  Google Scholar 

  24. K. Ammari, M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM Contr. Optim. Calc. Var. 6, 361–386 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. K. Ammari, M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM. J. Contr. Optim. 39, 1160–1181 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Bader, F. Ammar Khodja, Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force. SIAM J. Contr. Optim. 39, 1833–1851 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Contr. Optim. 30, 1024–1065 (1992)

    Google Scholar 

  28. A. Bátkai, K.J. Engel, J. Prüss, R. Schnaubelt, Polynomial stability of operator semigroups. Math. Nachr. 279(13, 14), 1425–1440 (2006)

    Google Scholar 

  29. C. Batty, T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equat. 8, 765–780 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Berrimi, S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 64, 2314–2331 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Beyrath, Stabilisation indirecte localement distribué de systèmes faiblement couplés. Compt. Rendus Acad. Sci. Paris I Math. 333, 451–456 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Beyrath, Indirect linear locally distributed damping of coupled systems. Bol. Soc. Parana. Mat. 22, 17–34 (2004)

    MathSciNet  MATH  Google Scholar 

  33. A. Borichev, Y. Tomilov, Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347, 455–478 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. H. Brézis, in Analyse Fonctionnelle. Théorie et Applications (Masson, Paris, 1983)

    Google Scholar 

  35. N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers. Asymptot. Anal. 14, 157–191 (1997)

    MathSciNet  MATH  Google Scholar 

  36. N. Burq, P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. Compt. Rendus Acad. Sci. Paris I Math. 325, 749–752 (1997)

    Article  MATH  Google Scholar 

  37. N. Burq, Decay of the local energy of the wave equation for the exterior problem and absence of resonance near the real axis. Acta Math. 180, 1–29 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  38. N. Burq, G. Lebeau, Mesures de défaut de compacité, application au système de Lamé. Ann. Sci. Ecole Norm. Sup. (4) 34(6), 817–870 (2001)

    Google Scholar 

  39. N. Burq, M. Hitrik, Energy decay for damped wave equations on partially rectangular domains. Math. Res. Lett. 14(1), 35–47 (2007)

    MathSciNet  MATH  Google Scholar 

  40. A. Carpio, Sharp estimates of the energy for the solutions of some dissipative second order evolution equations. Potential Anal. 1, 265–289 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  41. M.M. Cavalcanti, H.P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Contr. Optim. 42, 1310–1324 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. G. Chen, A note on boundary stabilization of the wave equation. SIAM J. Contr. Optim. 19, 106–113 (1981)

    Article  MATH  Google Scholar 

  43. I. Chueshov, I. Lasiecka, D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete Contin. Dyn. Syst. 20, 459–509 (2008)

    MathSciNet  MATH  Google Scholar 

  44. F. Conrad, B. Rao, Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback. Asymptot. Anal. 7, 159–177 (1993)

    MathSciNet  MATH  Google Scholar 

  45. J.-M. Coron, in Control and Nonlinearity. Mathematical Surveys and Monographs, 136 (American mathematical society, Providence, 2007)

    Google Scholar 

  46. C. Dafermos, in Asymptotic Behavior of Solutions of Evolution Equations. Nonlinear Evolution Equations (Proc. Sympos. Univ. Wisconsin, Madison, Wis.), Publ. Math. Res. Center University of Wisconsin, 40, (Academic, New York, 1978), pp. 103–123

    Google Scholar 

  47. C. Dafermos, Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  48. C. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity. J. Differ. Equat. 7, 554–569 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  49. R. Dáger, E. Zuazua, in Wave Propagation, Observation and Control in 1 − d Flexible Multi-structures. Mathematics & Applications, 50 (Springer, Berlin, 2006)

    Google Scholar 

  50. M. Daoulatli, I. Lasiecka, D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth conditions. Discrete Contin. Dyn. Syst. S 2, 67–94 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. M. Eller, J. Lagnese, S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping. Special issue in memory of Jacques-Louis Lions. Comput. Appl. Math. 21, 135–165 (2002)

    MathSciNet  MATH  Google Scholar 

  52. S. Ervedoza, E. Zuazua, Uniformly exponentially stable approximation for a class of damped systems. J. Math. Pure. Appl. 91, 20–48 (2008)

    MathSciNet  Google Scholar 

  53. M. Fabrizio, A. Morro, Viscoelastic relaxation functions compatible with thermodynamics. J. Elasticity 19, 63–75 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  54. X. Fu, J. Yong, X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Contr. Optim. 46, 1578–1614 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  55. R. Glowinski, C.H. Li, J.-L. Lions, A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods. Jpn. J. Appl. Math. 7(1), 1–76 (1990)

    MathSciNet  MATH  Google Scholar 

  56. A. Haraux Semi-groupes linéaires et équations d’évolutions linéaires périodiques. Publications du Laboratoire d’Analyse Numérique 78011, Université Pierre et Marie Curie, Paris 1978

    Google Scholar 

  57. A. Haraux, L p estimates of solutions to some nonlinear wave equation in one space dimension. Publications du laboratoire d’analyse numérique, Université Paris VI, CNRS, Paris 1995

    Google Scholar 

  58. A. Haraux, in Nonlinear Evolution Equations—Global Behavior of Solutions. Lecture Notes in Mathematics 841 (Springer, Berlin, 1981)

    Google Scholar 

  59. A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations. J. Differ. Equat. 59, 145–154 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  60. A. Haraux, in Systèmes Dynamiques Dissipatifs et Applications. Collection Recherches en Mathématiques Appliquées (Masson, Paris, Milan, Barcelone, 1991)

    Google Scholar 

  61. A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math. 46, 245–258 (1989)

    MathSciNet  MATH  Google Scholar 

  62. L.F. Ho, Observabilité frontière de l’équation des ondes. Compt. Rendus Acad. Sci. Paris I Math. 302, 443–446 (1986)

    MATH  Google Scholar 

  63. J.A. Infante, E. Zuazua, Boundary observability for the space semi-discretizations of the 1-D wave equation. M2AN Math. Model. Numer. Anal. 33(2), 407–438 (1999)

    Google Scholar 

  64. V. Komornik, E. Zuazua, A direct method for the boundary stabilization of the wave equation. J. Math. Pure. Appl. 69(1), 33–54 (1990)

    MathSciNet  MATH  Google Scholar 

  65. V. Komornik, in Exact Controllability and Stabilization; The Multiplier Method. Collection RMA, vol 36 (Wiley, Masson, Paris, 1994)

    Google Scholar 

  66. J. Lagnese, Decay of solutions to wave equations in a bounded region. J. Differ. Equat. 50, 163–182 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  67. J.-P. Lasalle, The extent of asymptotic stability. Proc. Symp. Appl. Math. 13, Amunozsalva, 299–307 (1962)

    Google Scholar 

  68. J.-P. Lasalle, Asymptotic stability criteria. Proc. Nat. Acad. Sci. USA 46, 363–365 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  69. I. Lasiecka, Stabilization of waves and plate like equations with nonlinear dissipation on the boundary. J. Differ. Equat. 79, 340–381 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  70. I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integr. Equat. 8, 507–533 (1993)

    MathSciNet  Google Scholar 

  71. I. Lasiecka I, J.-L. Lions, R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pure. Appl. 65(2), 149–192 (1986)

    Google Scholar 

  72. I. Lasiecka I, R. Triggiani, in Control Theory for Partial Differential Equations: Continuous and Approxiamtion Theories. I. Encyclopedia of Mathematics and its Applications, vol 74 (Cambridge University Press, Cambridge, 2000)

    Google Scholar 

  73. I. Lasiecka I, R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. II. Encyclopedia of Mathematics and its Applications, vol 75 (Cambridge University Press, Cambridge, 2000)

    Google Scholar 

  74. G. Lebeau, in Equations des ondes amorties, ed. by A. Boutet de Monvel et al. Algebraic and Geometric Methods in Mathematical Physics. Math. Phys. Stud. 19, 73–109 (Kluwer Academic Publishers, Dordrecht, 1996)

    Google Scholar 

  75. G. Lebon, C. Perez-Garcia, J. Casas-Vazquez, On the thermodynamic foundations of viscoelasticity. J. Chem. Phys. 88, 5068–5075 (1988)

    Article  MathSciNet  Google Scholar 

  76. J.-L. Lions, in Contrôlabilité exacte et stabilisation de systèmes distribués, vol 1 (Masson, Paris, 1988)

    Google Scholar 

  77. J.-L. Lions, W. Strauss, Some non linear evolution equations. Bull. Soc. Math. Fr. 93, 43–96 (1965)

    MathSciNet  MATH  Google Scholar 

  78. K. Liu, Locally distributed control and damping for the conservative systems. SIAM J. Contr. Optim. 35, 1574–1590 (1997)

    Article  MATH  Google Scholar 

  79. Z. Liu, S. Zheng, in Semigroups Associated with Dissipative Systems. Chapman Hall CRC Research Notes in Mathematics, 398 (Chapman Hall/CRC, Boca Raton, 1999)

    Google Scholar 

  80. W.-J. Liu, E. Zuazua, Decay rates for dissipative wave equations. Ricerche. Matemat. 48, 61–75 (1999)

    MathSciNet  MATH  Google Scholar 

  81. Z. Liu, B. Rao, Frequency domain approach for the polynomial stability of a system of partially damped wave equations. J. Math. Anal. Appl. 335, 860–881 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  82. P. Loreti, B. Rao B, Optimal energy decay rate for partially damped systems by spectral compensation. SIAM J. Contr. Optim. 45, 1612–1632 (2006)

    Google Scholar 

  83. A. Lunardi, in Interpolation Theory. Edizioni della Normale, Pisa (Birkhaüser, Basel, 2009)

    Google Scholar 

  84. P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complut. 12, 251–283 (1999)

    MathSciNet  MATH  Google Scholar 

  85. P. Martinez, A new method to obtain decay rate estimates for dissipative systems. ESAIM Contr. Optim. Calc. Var. 4, 419–444 (1999)

    Article  MATH  Google Scholar 

  86. L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation. SIAM J. Contr. Optim. 41(5), 1554–1566 (2002)

    Article  Google Scholar 

  87. A. Munch, A. Pazoto, Uniform stabilization of a viscous numerical approximation for a locally damped wave equation. ESAIM Contr. Optim. Calc. Var. 13, 265–293 (2007)

    Article  MathSciNet  Google Scholar 

  88. J.E. Muñoz Rivera, A. Peres Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials Quart. Appl. Math. 59, 557–578 (2001)

    MATH  Google Scholar 

  89. M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305, 403–417 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  90. J. Prüss, in Evolutionary Integral Equations and Applications. Monographs in Mathematics, 87 (Birkhäuser, Basel, 1993)

    Google Scholar 

  91. J. Rauch, M. Taylor, Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains. Indiana Univ. Math. J. 24, 79–86 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  92. M. Renardy, W.J. Hrusa, J.A. Nohel, in Mathematical Problems in Viscoelasticity. Pitman Monographs in Pure and Applied Mathematics, vol 35 (Longman Scientific and Technical, Harlow, 1988)

    Google Scholar 

  93. R.T. Rockafellar, in Convex Analysis (Princeton University Press, Princeton, 1970)

    Google Scholar 

  94. A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures Appl. 71, 455–467 (1992)

    MathSciNet  MATH  Google Scholar 

  95. D.L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems. J. Math. Anal. Appl. 173, 339–358 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  96. L.R. Tebou, E. Zuazua, Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math. 95, 563–598 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  97. L.R. Tebou, A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations. ESAIM Contr. Optim. Calc. Var. 14, 561–574 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  98. G. Todorova, B. Yordanov, The energy decay problem for wave equation with nonlinear dissipative terms in \({\mathbb{R}}^{n}\). Indiana Univ. Math. J. 56, 389–416 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  99. M. Tucsnak, G. Weiss, in Observation and Control for Operator Semigroups. Birkhuser Advanced Texts (Birkhäuser, Basel, 2009)

    Google Scholar 

  100. J. Valein, E. Zuazua, Stabilization of the wave equation on 1-D networks. SIAM J. Contr. Optim. 48, 2771–2797 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  101. J. Vancostenoble, P. Martinez, Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks. SIAM J. Contr. Optim. 39, 776–797 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  102. J. Vancostenoble, Optimalité d’estimation d’énergie pour une équation des ondes amortie. Compt. Rendus Acad. Sci. Paris I 328, 777–782 (1999)

    MathSciNet  MATH  Google Scholar 

  103. W.H. Young, On classes of summable functions and their Fourier series. Proc. Royal Soc. (A) 87, 225–229 (1912)

    Article  Google Scholar 

  104. A. Wehbe, W. Youssef, Stabilization of the uniform Timoshenko beam by one locally distributed feedback. Appli. Anal. 88, 1067–1078 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  105. E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Part. Differ. Equat. 15, 205–235 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  106. E. Zuazua, Uniform stabilization of the wave equation by nonlinear feedbacks. SIAM J. Contr. Optim. 28, 265–268 (1989)

    MathSciNet  Google Scholar 

  107. E. Zuazua, Propagation, observation and control of wave approximation by finite difference methods. SIAM Rev. 47, 197–243 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Paul Verlaine-Metz, Ile du Saulcy, 57045, Metz, Cedex 1, France

    Fatiha Alabau-Boussouira

Authors
  1. Fatiha Alabau-Boussouira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fatiha Alabau-Boussouira .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Alabau-Boussouira, F. (2012). On Some Recent Advances on Stabilization for Hyperbolic Equations. In: Control of Partial Differential Equations. Lecture Notes in Mathematics(), vol 2048. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27893-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation