SeMA Journal

, Volume 61, Issue 1, pp 19–47 | Cite as

Polynomial decay rate for a wave equation with general acoustic boundary feedback laws

  • Z. Abbas
  • S. NicaiseEmail author


We consider the stabilization of the wave equation by a general class of acoustic boundary feedback laws at one extremity. This system is not uniformly stable but we furnish sufficient conditions that guarantee a polynomial stability. Our method combines the use of an observability inequality for the associated undamped problem obtained via sharp spectral results with regularity results of the solution of the undamped problem with a specific right-hand side. In some particular cases, the optimality of the decay is shown again with the help of precise spectral results of the operator associated with the damped problem. We illustrate our general results by some examples.


Acoustic boundary control Stability Polynomial decay 

Mathematics Subject Classification (2000)

35L05 93D15 


  1. Ammari, K., Tucsnak, M.: Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM J. Control Optim. 39(4), 1160–1181 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  2. Arendt, W., Batty, C.J.K.: Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 305(2), 837–852 (1988)MathSciNetCrossRefGoogle Scholar
  3. Baiocchi, C., Komornik, V., Loreti, P.: Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hungar. 97, 55–95 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  4. Beale, J.T.: Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25(9), 895–917 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  5. Belinsky, B.P.: Wave propagation in the ice-covered ocean wave guide and operator polynomials. In: Proceedings of the Second ISAAC Congress, vol. 2 (Fukuoka, 1999), volume 8 of Int. Soc. Anal. Appl. Comput., pages 1319–1333, Dordrecht, Kluwer Acad. Publ (2000)Google Scholar
  6. Conrad, F., Morgül, Ö.: On the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim. 36(6):1962–1986 (electronic) (1998)Google Scholar
  7. Littman, W., Liu, B.: On the spectral properties and stabilization of acoustic flow. SIAM J. Appl. Math. 59(1):17–34 (electronic) (1999)Google Scholar
  8. Mercier, D., Nicaise, S.: Polynomial decay rate for a wave equation with weak dynamic boundary feedback laws. J. Abstr. Differ. Equ. Appl. 2(1), 29–53 (2011)MathSciNetGoogle Scholar
  9. Morgül, Ö.: A dynamic control law for the wave equation. Automatica J. IFAC 30(11), 1785–1792 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  10. Munoz Rivera, J.E., Qin, Y.: Polynomial decay for the energy with an acoustic boundary condition. Appl. Math. Lett. 16(2), 249–256 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  11. Nicaise, S., Valein, J.: Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. Netw. Heterog. Media 2(3), 425–479 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. Pazy, A.: Semigroups of linear operators and applications to partial differential equations, volume 44 of In: Applied Math. Sciences. vol 44, Springer, New York (1983)Google Scholar
  13. Rao, B.: Stabilization of elastic plates with dynamical boundary control. SIAM J. Control Optim. 36(1):148–163 (electronic) (1998)Google Scholar
  14. Russell, D.L.: Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods. J. Differ. Equ. 19(2), 344–370 (1975)zbMATHCrossRefGoogle Scholar
  15. Russell, D.L., Zhang, B.Y.: Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31(3), 659–676 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  16. Wehbe, A.: Rational energy decay rate for a wave equation with dynamical control. Appl. Math. Lett. 16(3), 357–364 (2003)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2013

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et ses Applications de Valenciennes, FR CNRS 2956, Institut des Sciences et Techniques de ValenciennesUniversité de Valenciennes et du Hainaut-CambrésisValenciennes Cedex 9France

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