Acta Mathematica Hungarica

, Volume 123, Issue 1–2, pp 1–10 | Cite as

Stabilization of coupled systems

  • K. AmmariEmail author
  • M. Mehrenberger


We characterize the stabilization for some coupled infinite dimensional systems. The proof of the main result uses the methodology introduced in Ammari and Tucsnak [2], where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined to a boundedness property of the transfer function of the associated open loop system and a result in [11].

Key words and phrases

coupled system stabilization dissipatif system observability 

2000 Mathematics Subject Classification

93B07 93C20 93D15 93D25 35L05 47D03 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F. Alabau, Observabilité frontière de systèmes faiblement couplés, C.R. Acad. Sci. Paris Sér. I, 333 (2001), 645–650.zbMATHMathSciNetGoogle Scholar
  2. [2]
    K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM COCV., 6 (2001), 361–386.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, vol. 1, Birkhäuser (Boston, 1992).Google Scholar
  5. [5]
    A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math., 46 (1989), 245–258.zbMATHMathSciNetGoogle Scholar
  6. [6]
    V. Komornik and P. Loreti, Fourier series in control theory, Springer Monographs in Mathematics (New York, 2005).zbMATHGoogle Scholar
  7. [7]
    V. Komornik and P. Loreti, Observability of compactly perturbed systems, J. Math. Anal. Appl., 243 (2000), 409–428.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    G. Lebeau, Equation des ondes amorties, Boutet de Monvel, Anne (ed.) et al., in: Algebraic and Geometric Methods in Mathematical Physics, Kluwer Academic Publishers. Math. Phys. Stud., 19 (1996), pp. 73–109.Google Scholar
  9. [9]
    J. L. Lions, Contrôlabilité exacte des systèmes distribués, Masson (Paris, 1998).Google Scholar
  10. [10]
    J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod (Paris, 1968).zbMATHGoogle Scholar
  11. [11]
    M. Mehrenberger, Observability of coupled systems, Acta Math. Hungar., 4 (2004), 321–348.CrossRefMathSciNetGoogle Scholar
  12. [12]
    A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer Verlag (New York, 1983).Google Scholar
  13. [13]
    M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air. I. Well-posedness and energy balance, ESAIM COCV., 9 (2003), 247–274.MathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2009

Authors and Affiliations

  1. 1.Département de MathématiquesFaculté des Sciences de MonastirMonastirTunisie
  2. 2.Institut de Recherche Mathématique AvancéeUniversité Louis PasteurStrasbourgFrance

Personalised recommendations