Asymptotic behavior of a delayed wave equation without displacement term

Article
  • 63 Downloads

Abstract

This paper is dedicated to the investigation of the asymptotic behavior of a delayed wave equation without the presence of any displacement term. First, it is shown that the problem is well-posed in the sense of semigroups theory. Thereafter, LaSalle’s invariance principle is invoked in order to establish the asymptotic convergence for the solutions of the system to a stationary position which depends on the initial data. More importantly, without any geometric condition such as BLR condition (Bardos et al. in SIAM J Control Optim 30:1024–1064, 1992; Lebeau and Robbiano in Duke Math J 86:465–491, 1997) in the control zone, the logarithmic convergence is proved by using an interpolation inequality combined with a resolvent method.

Keywords

Wave equation Time-delay Asymptotic behavior Logarithmic stability 

Mathematics Subject Classification

34B05 34D05 70J25 93D15 

References

  1. 1.
    Adams, R.A.: Sobolev Spaces, Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975)Google Scholar
  2. 2.
    Ammari, K., Nicaise, S.: Stabilization of Elastic Systems by Collocated Feedback. Lecture Notes in Mathematics, vol. 2124. Springer, Cham (2015)MATHGoogle Scholar
  3. 3.
    Ammari, K., Nicaise, S., Pignotti, C.: Feedback boundary stabilization of wave equations with interior delay. Syst. Control Lett. 59, 623–628 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bardos, C., Lebau, G., Rauch, J.: Sharp sufficient conditions for the observation, controllability ans stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1064 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Batty, C.J.K., Duyckaerts, T.: Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8, 765–780 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brezis, H.: Analyse Fonctionnelle, Théorie et Applications. Masson, Paris (1992)MATHGoogle Scholar
  7. 7.
    Burq, N.: Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180, 1–29 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, G.: Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain. J. Math. Pures Appl. 58, 249–274 (1979)MathSciNetMATHGoogle Scholar
  9. 9.
    Chen, G.: Control and stabilization for the wave equation in a bounded domain, part I. SIAM J. Control Optim. 17, 66–81 (1979)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, G.: Control and stabilization for the wave equation in a bounded domain, part II. SIAM J. Control Optim. 19, 114–122 (1981)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen, G.: A note on boundary stabilization of the wave equations. SIAM J. Control Optim. 19, 106–113 (1981)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chentouf, B., Boudellioua, M.S.: A New Approach to Stabilization of the Wave Equation with Boundary Damping Control, vol. 7, p. 9. Sultan Qaboos University Science, Oman (2004)Google Scholar
  13. 13.
    Chentouf, B., Boudellioua, M.S.: On the stabilization of the wave equation with dynamical control. In: Proceedings of the 16th International Symposium on Mathematical Theory of Networks and Systems, Leuven, Belgium, pp. 6 (2004)Google Scholar
  14. 14.
    Chentouf, B., Guesmia, A.: Neumann-boundary stabilization of a non-homogeneous \(n\)-dimensional wave equation with damping control. In: Proceedings of the Fifth Asian Mathematical Conference, Kuala Lampur, Malaysia, pp. 6 (2009)Google Scholar
  15. 15.
    Chentouf, B., Guesmia, A.: Neumann-boundary stabilization of the wave equation with damping control and applications. Commun. Appl. Anal. 14, 541–566 (2010)MathSciNetMATHGoogle Scholar
  16. 16.
    Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Haraux, A.: Systèmes Dynamiques Dissipatifs et Applications, Paris (1991)Google Scholar
  18. 18.
    Kato, T.: Perturbation Theory of Linear Operators. Springer, New York (1976)MATHGoogle Scholar
  19. 19.
    Lagnese, J.: Decay of solutions of the wave equation in a bounded region with boundary dissipation. J. Diff. Equ. 50, 163–182 (1983)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lagnese, J.: Note on boundary stabilization of wave equations. SIAM J. Control Optim. 26, 1250–1256 (1988)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lebeau, G., Robbiano, L.: Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86, 465–491 (1997)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lions, J.L.: Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30(1), 1–68 (1988)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lions, J.L.: Contrôlabilité exacte et stabilisation de systèmes ditribués, vol. 1. Masson, Paris (1988)Google Scholar
  24. 24.
    Majda, A.: Disappearing solutions for the dissipative wave equations. Indiana Univ. Math. J. 24, 1119–1133 (1975)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Morawetz, C.S.: Decay of solutions of the exterior problem for the wave equations. Commun. Pure Appl. Math. 28, 229–264 (1975)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45, 1561–1585 (2006)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefMATHGoogle Scholar
  28. 28.
    Quinn, J.P., Russell, D.L.: Asymptotic stability and energy decay rate for solutions of hyperbolic equations with boundary damping. Proc. R. Soc. Edinb. Sect. A 77, 97–127 (1977)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Rauch, J., Taylor, M.: Exponential decay of solutions to hyperbolic equations in bounded domain. Indiana Univ. Math. J. 24, 79–86 (1974)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Russell, D.L.: Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20, 639–739 (1978)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Triggiani, R.: Wave equation on a bounded domain with boundary dissipation: an operator approach. J. Math. Anal. Appl. 137, 438–461 (1989)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.UR Analysis and Control of PDEs UR13ES64, Department of Mathematics Faculty of Sciences of MonastirUniversity of MonastirMonastirTunisia
  2. 2.Department of Mathematics Faculty of ScienceKuwait UniversitySafatKuwait

Personalised recommendations