Skip to main content

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


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.

This is a preview of subscription content, access via your institution.


  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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  3. Baiocchi, C., Komornik, V., Loreti, P.: Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hungar. 97, 55–95 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  4. Beale, J.T.: Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25(9), 895–917 (1976)

    MathSciNet  MATH  Article  Google 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)

  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)

    MathSciNet  Google Scholar 

  9. Morgül, Ö.: A dynamic control law for the wave equation. Automatica J. IFAC 30(11), 1785–1792 (1994)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

  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)

    MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  16. Wehbe, A.: Rational energy decay rate for a wave equation with dynamical control. Appl. Math. Lett. 16(3), 357–364 (2003)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to S. Nicaise.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Abbas, Z., Nicaise, S. Polynomial decay rate for a wave equation with general acoustic boundary feedback laws. SeMA 61, 19–47 (2013).

Download citation


  • Acoustic boundary control
  • Stability
  • Polynomial decay

Mathematics Subject Classification (2000)

  • 35L05
  • 93D15