Rendiconti del Circolo Matematico di Palermo

, Volume 49, Issue 3, pp 575–600 | Cite as

Exponential decay for semilinear distributed systems with damping

  • Larbi Berrahmoune


The stability of second order abstract distributed systems with damping and nonlinear perturbations is considered. Sufficient conditions, including unique continuation property assumptions, are formulated to obtain (local, non-uniform and uniform) exponential stability. Applications to the wave and Euler-Bernoulli equations are given.

1991 AMS Subject Classification

35B35 35B40 93D20 

Key words

distributed semilinear systems damping exponential decay 


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  1. [1]
    Ball J. M.,Strong continuous semigroups, weak solutions, and the variation of constants, Proc. Amer; Math. Soc.,63 (1977), 370–373.CrossRefMathSciNetMATHGoogle Scholar
  2. [2]
    Ball J. M.,On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations, J. Diff. Eq.,27 (1978), 244–265.CrossRefGoogle Scholar
  3. [3]
    Ball J., Slemrod M.,Feedback stabilization of distributed semilinear control systems, App. Math. Optim.,5 (1979), 169–179.CrossRefMathSciNetMATHGoogle Scholar
  4. [4]
    Ball J., Slemrod M.,Nonharmonic Fourier series and stabilization of distributed semilinear control systems, C.P.A.M,32, (1979), 555–587.MathSciNetGoogle Scholar
  5. [5]
    Berrahmoune L.,Exponential decay for distributed bilinear control systems with damping, Rend. Circ. Mat. Palermo, Serie II, Tomo XLVIII, (1999), 111–122.Google Scholar
  6. [6]
    Haraux A.,Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portug. Mat.,46 (1989), 245–258.MathSciNetMATHGoogle Scholar
  7. [7]
    Kim J. U.,Exact semi-internal control of an Euler-Bernoulli equation, SIAM J. Contr. Optim.,30 (1992), 1001–1023.CrossRefMATHGoogle Scholar
  8. [8]
    Lions J.-L.,Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, RMA 8, Masson, Paris, 1988.Google Scholar
  9. [9]
    Pazy A.,Semigroups of linear operators and applications to partial differential equations, Springer Verlag, 1983.Google Scholar
  10. [10]
    Ruiz A.,Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures et Appl.,71 (1992), 455–467.MATHGoogle Scholar
  11. [11]
    Simon J.,Compact sets in the space L p (0,T; B). Annali di Matematica Pura e Applicata, (IV)146 (1987), 65–96.MATHGoogle Scholar
  12. [12]
    Tucsnak M.,Semi-internal stabilization for a nonlinear Bernoulli-Euler Equation, Mathematical Methods in the Applied Sciences,19 (1996) 897–907.CrossRefMathSciNetMATHGoogle Scholar
  13. [13]
    Zuazua E.,Exponential decay for the semilinear wave equation with locally distributed damping, Comm. in Partial Differential Equations,15 (1990), 205–235.CrossRefMathSciNetMATHGoogle Scholar
  14. [14]
    Zuazua E.,Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Mat. Pures App.70 (1992), 513–529.MathSciNetGoogle Scholar

Copyright information

© Springer 2000

Authors and Affiliations

  • Larbi Berrahmoune
    • 1
  1. 1.Département de mathématiquesEcole Normale Supérieure de RabatRabatMorocco

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