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Asymptotic behavior of second sound thermoelasticity with internal time-varying delay

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Abstract

In this paper, we consider a thermoelastic system of second sound with internal time-varying delay. Under suitable assumption on the weight of the delay, we prove, using the energy method, that the damping effect through heat conduction given by Cattaneo’s law is still strong enough to uniformly stabilize the system even in the presence of time delay.

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References

  1. Abdallah, C., Dorato, P., Benitez-Read, J., Byrne R.: Delayed positive feedback can stabilize oscillatory system. ACC. San Francisco, 3106-3107 (1993)

  2. Ait Benhassi E.M., Ammari K., Boulite S., Maniar L.: Feedback stabilization of a class of evolution equations with delay. J. Evol. Equ. 9, 103–121 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Caraballo T., Real J., Shaikhet L.: Method of Lyapunov functionals construction in stability of delay evolution equations. J. Math. Anal. Appl. 334(2), 1130–1145 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chandrasekharaiah D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729 (1998)

    Article  Google Scholar 

  5. Datko R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim. 26(3), 697–713 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Datko R., Lagnese J., Polis M.P.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24(1), 152–156 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fridman, E., Orlov, Y.: On stability of linear parabolic distributed parameter systems with time-varying delays, CDC 2007, December (2007), New Orleans

  8. Huang C., Vandewalle S.: An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays. SIAM J. Sci. Comp. 25(5), 1608–1632 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jiang, S., Racke, R.: Evolution equations in thermoelasticity, π monographs surveys pure applied mathematics. 112, Chapman & Hall / CRC, Boca Raton (2000)

  10. Kirane M., Said-Houari B., Anwar M.N.: Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Comm. Pure Appl. Anal. 10, 667–686 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Komornik V., Zuazua E.: A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69, 33–54 (1990)

    MathSciNet  MATH  Google Scholar 

  12. Lasiecka I.: Global uniform decay rates for the solution to the wave equation with nonlinear boundary conditions. Appl. Anal. 47, 191–212 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lasiecka I.: Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary. J. Differ. Equ. 79, 340–381 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu K.: Locally distributed control and damping for the conservative systems. SIAM J. Control Optim. 35, 1574–1590 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Messaoudi S.A., Said-Houari B.: Exponential Stability in one-dimensional nonlinear thermoelasticity with second sound. Math. Meth. Appl. Sci. 28, 205–232 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mustafa, M.I.: Boundary stabilization of memory-type thermoelasticity with second sound, Zeitschrift für Angewandte Mathematik und Physik (2012), doi:10.1007/s00033-011-0190-8

  17. 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(5), 1561–1585 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nicaise S., Pignotti C.: Stabilization of the wave equation with boundary or internal distributed delay. Diff. Int. Equ. 21(9-10), 935–958 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Nicaise S., Pignotti C., Valein J.: Exponential stability of the wave equation with boundary time-varying delay. Discrete Contin. Dyn. Syst.Ser. S 4(3), 693–722 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Nicaise S., Valein J., Fridman E.: Stability of the heat and the wave equations with boundary time-varying delays. Discrete Contin. Dyn. Syst. 2(3), 559–581 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Qin Y., Ma Z., Yang X.: Exponential stability for nonlinear thermoelastic equations with second sound. Nonlinear Anal. RWA 11(4), 2502–2513 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Racke R.: Asymptotic behavior of solutions in linear 2- or 3-d thermoelasticity with second sound. Quart. Appl. Math. 61(2), 315–328 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Racke R.: Instability of coupled systems with delay. Commun. Pure Appl. Anal. 11(5), 1753–1773 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Racke R.: Thermoelasticity with second sound - Exponential stability in linear and nonlinear 1-d. Math. Meth. Appl. Sci. 25, 409–441 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Said-Houari B., Laskri Y.: A stability result of a Timoshenko system with a delay term in the internal feedback. Appl. Math. Comp. 217, 2857–2869 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Suh I.H., Bien Z.: Use of time delay action in the controller design. IEEE Trans. Automat. Control. 25, 600–603 (1980)

    Article  MATH  Google Scholar 

  27. Tarabek M.A.: On the existence of smooth solutions in one-dimensional thermoelasticity with second sound. Quart. Appl. Math. 50, 727–742 (1992)

    MathSciNet  MATH  Google Scholar 

  28. Wang T.: Exponential stability and inequalities of solutions of abstract functional differential equations. J. Math. Anal. Appl. 324, 982–991 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yung S.P., Xu C.Q., Li L.K.: Stabilization of the wave system with input delay in the boundary control. ESAIM: Control Optim. Calc. Var. 12, 770–785 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zheng, S.: Nonlinear parabolic equations and hyperbolic-parabolic coupled systems. Pitman monographs surveys pure applied mathematics, vol. 76. Longman: Harlow, (1995)

  31. Zuazua E.: Exponential decay for the semi-linear wave equation with locally distributed damping. Comm. Partial Differ. Equ. 15, 205–235 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zuazua E.: Uniform Stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim. 28(2), 466–477 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Muhammad I. Mustafa

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  1. Muhammad I. Mustafa
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Correspondence to Muhammad I. Mustafa.

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Mustafa, M.I. Asymptotic behavior of second sound thermoelasticity with internal time-varying delay. Z. Angew. Math. Phys. 64, 1353–1362 (2013). https://doi.org/10.1007/s00033-012-0268-y

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  • DOI: https://doi.org/10.1007/s00033-012-0268-y

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