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Semigroup Forum

, Volume 96, Issue 2, pp 357–376 | Cite as

Stabilization of the wave equation with acoustic and delay boundary conditions

  • Gilbert R. Peralta
Research Article
  • 44 Downloads

Abstract

In this paper, we consider the wave equation on a bounded domain with mixed Dirichlet-impedance type boundary conditions coupled with oscillators on the Neumann boundary. The system has either a delay in the pressure term of the wave component or the velocity of the oscillator component. Using the velocity as a boundary feedback it is shown that if the delay factor is less than that of the damping factor then the energy of the solutions decays to zero exponentially. The results are based on the energy method, a compactness-uniqueness argument and an appropriate weighted trace estimate. In the critical case where the damping and delay factors are equal, it is shown using variational methods that the energy decays to zero asymptotically.

Keywords

Wave equations Acoustic boundary conditions Feedback delays Stabilization Energy method 

Notes

Acknowledgements

The author would like to thank Georg Propst for his helpful comments and suggestions. This research is partially supported by the Austrian Science Fund (FWF) under SFB grant Mathematical Optimization and Applications in Biomedical Sciences.

References

  1. 1.
    Ammari, K., Nicaise, S., Pignotti, C.: Feedback boundary stabilisation of wave equations with interior delay. Syst. Control Lett. 59, 623–628 (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Beale, J.T.: Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25, 895–917 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cornilleau, P., Nicaise, S.: Energy decay of solutions of the wave equation with general memory boundary conditions. Differ Integral Equ. 22(11/12), 1173–1192 (2009)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Datko, R.: Not all feedback stabilised hyperbolic systems are robust with respect to small time delays in their feedback. SIAM J. Control Optim. 26, 697–713 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Datko, R., Lagnese, J., Polis, P.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim 24, 152–156 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Desch, W., Fas̆angová, E., Milota, J., Propst, G.: Stabilization through viscoelastic boundary damping: a semigroup approach. Semigroup Forum 80, 405–415 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, 2nd edn. Springer, Berlin (2000)zbMATHGoogle Scholar
  8. 8.
    Gerbi, S., Said-Houari, B.: Existence and exponential stability for a damped wave equation with dynamic boundary conditions and a delay term. Appl. Math. Comput. 218(24), 11900–11910 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 21, Pitman, Boston-London-Melbourne (1985)Google Scholar
  10. 10.
    Lasiecka, I., Triggiani, R., Yao, P.F.: Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235, 13–57 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary of internal feedbacks. Siam J. Control Optim. 45, 1561–1585 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21(9–10), 801–900 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Pignotti, C.: A note on stabilization of locally damped wave equations with time delay. Syst. Control Lett. 61, 92–97 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Simon, J.: Compact sets in the space \(L^p(0,T;B)\). Annali di Matematica pura ed applicata (IV) CXLVI, 65–96 (1987)zbMATHGoogle Scholar
  15. 15.
    Tucsnak, M., Weiss, G.: Observation and Control for Operator Semigroups. Birkhäuser, Basel (2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    Xu, G.Q., Yung, S.P., Li, L.K.: Stabilzation of wave systems with input delay in the boundary control. ESAIM Control Optim Calc. Var. 12, 770–785 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of the Philippines BaguioBaguioPhilippines

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