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Stabilization of viscoelastic wave equation with variable coefficients and a delay term in the internal feedback

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In this paper, we consider the stabilization of the viscoelastic wave equation with variable coefficients in a bounded domain with smooth boundary, subject to linear dissipative internal feedback with a delay. Our stabilization result is mainly based on the use of the Riemannian geometry methods and Lyapunov functional techniques.

Abstract

In this paper, we consider the stabilization of the viscoelastic wave equation with variable coefficients in a bounded domain with smooth boundary, subject to linear dissipative internal feedback with a delay. Our stabilization result is mainly based on the use of the Riemannian geometry methods and Lyapunov functional techniques.

참고문헌 (26)

  1. C. Abdallah, "Delayed positive feedback can stabilize oscillatory system", Proceedings of the American Control Conference, pp.3106~3107 (1993)
  2. F. Alabau-Boussouira, "Decay estimates for second order evolution equations with memory", J. Funct. Anal, 254(5), pp.1342~1372 (2008)
  3. J. A. D. Appleby, "On exponential asymp-totic stability in linear viscoelasticity", Math. Models Methods Appl. Sci, 16(10), pp.1677~1694 (2006)
  4. M.M. Cavalcanti, "Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions", J. Math. Anal. Appl., 328(2), pp.900~930 (2007)
  5. M. M. Cavalcanti, "Frictional versus viscoelastic damping in a semilin-ear wave equation", SIMA J. Control Optim, 42(4), pp.1310~1324 (2003)
  6. C. M. Dafermos, "An abstract Volterra equation with application to linear viscoelasticity", J. Differential Equations, 7, pp.554~589 (1970)
  7. R. Datko, "An example on the of time delays in boundary feedback stabilization of wave equations", SIAM J. Control Optim, 24, pp.152~156 (1986)
  8. R. Gulliver, "Geometric Methods in Inverse Problems and PDE Control", pp.73~181 (2004), Springer
  9. B. Z. Guo, "On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback", Nonlinear Anal, 71(12), pp.5961~5978 (2009)
  10. M. Kirane, "Existence and asymptotic stability of a viscoelastic wave equation with a delay", Z. Angew. Math. Phys, 62(6), pp.1065~1082 (2011)
  11. V. Komornik, "A direct method for the boundary stabilization of the wave equation", J. Math. Pures Appl, 60(1), pp.33~54 (1990)
  12. W. J. Liu, "Asymptotic behavior of solutions of time-delayed Burgers’ equation", Discrete Contin. Dyn. Syst. Ser. B, 2(1), pp.47~56 (2002)
  13. J. E. M˜unoz Rivera, "Existence and uniform rates of decay for contact problems in viscoelasticity", Appl. Anal, 67(3-4), pp.175~199 (1997)
  14. S. Nicaise, "Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks", SIAM J. Control Optim, 45(5), pp.1561~1585 (2006)
  15. S. Nicaise, "Internal and boundary observability estimates for the heterogeneous Maxwells system", Appl. Math. Optim, 54(1), pp.47~70 (2006)
  16. S. Nicaise, "Stabilization of the wave equation with boundary or internal distributed delay", Differential Integral Equations, 21(9-10), pp.935~958 (2008)
  17. Z. H. Ning, "Stabilization of the wave equation with variable coefficients and a delay in dissipative internal feedback", J. Math. Anal. Appl, 405(1), pp.148~155 (2013)
  18. Z. H. Ning, "Nonlinear boundary stabilization of the wave equations with variable coefficients and time depen-dent delay", Appl. Math. Comput, 232, pp.511~520 (2014)
  19. J. E. M. Rivera, "Asymptotic behaviour in linear viscoelasticity", Quart. Appl. Math, 52, pp.629~648 (1994)
  20. J. E. M. Rivera, "Asymptotic behaviour of the energy in partially viscoelastic materials", Quart. Appl. Math, 59, pp.557~578 (2001)
  21. I. H. Suh, "Use of time delay action in the controller design", IEEE Trans. Autom. Control, 25, pp.600~603 (1980)
  22. C. Q. Xu, "Stabilization of wave systems with input delay in the boundary control", ESAIM Control Optim. Calc. Var, 12(4), pp.770~785 (2006)
  23. P. F. Yao, "On the observability inequalities for exact controllability of wave equations with variable coefficients", SIAM J. Contr. Optim, 37(5), pp.1568~1599 (1999)
  24. P. F. Yao, "Global smooth solutions for the quasilinear wave equation with boundary dissi-pation", J. Differential Equations, 241(1), pp.62~93 (2007)
  25. P. F. Yao, "Boundary controllability for the quasilinear wave equation", Appl. Math. Optim, 61(2), pp.191~233 (2010)
  26. CRC Applied Mathematics and Nonlinear Science Series, "Modeling and Control in Vibrational and Structural Dynamics, A Differential Geometric Approach, Chapman and Hall", pp.2011, P. F. Yao

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