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On the Control of Dissipative Viscoelastic Timoshenko Beams

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Abstract

In this paper we consider a Timoshenko system subject to a complementary effect of frictional damping and viscoelastic damping. Under very general assumptions on the relaxation function and the frictional damping term, we establish, for the first time as per our knowledge, explicit and optimal energy decay rates for this system. Our result generalizes and improves earlier related results in the literature.

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References

  1. Alabau-Boussouira, F.: Asymptotic behavior for Timoshenko beams subject to a single non-linear feedback control. Nonlinear Differ. Equ. Appl. 14, 643–669 (2007)

    Article  MathSciNet  Google Scholar 

  2. Alabau-Boussouira, F., Cannarsa, P.: A general method for proving sharp energy decay rates for memory-dissipative evolution equations. C. R. Acad. Sci. Paris Ser. I 347, 867–872 (2009)

    Article  MathSciNet  Google Scholar 

  3. Al-Mahdi, A.M.: Optimal decay result for Kirchhoff plate equations with nonlinear damping and very general type of relaxation functions. Bound. Value Probl. 2019, 82 (2019). https://doi.org/10.1186/s13661-019-1196-y

    Article  MathSciNet  Google Scholar 

  4. Almeida Júnior, D.S., Santos, M.L., Muñoz Rivera, J.E.: Stability to weakly dissipative Timoshenko systems. Math. Methods Appl. Sci. 36(14), 1965–1976 (2013)

    Article  MathSciNet  Google Scholar 

  5. Ammar-Khodja, F., Benabdallah, A., Muñoz Rivera, J.E., Racke, R.: Energy decay for Timoshenko systems of memory type. J. Differ. Equ. 194(1), 82–115 (2003)

    Article  MathSciNet  Google Scholar 

  6. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)

    Book  Google Scholar 

  7. Cavalcanti, M.M., Cavalcanti, V.N.D., Lasiecka, I., Nascimento, F.A.: Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discr. Contin. Dyn. Syst. Ser. B 19(7), 1987–2012 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Cavalcanti, M.M., Oquendo, H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)

    Article  MathSciNet  Google Scholar 

  9. Fabrizio, M., Polidoro, S.: Asymptotic decay for some differential systems with fading memory. Appl. Anal. 81(6), 1245–1264 (2002)

    Article  MathSciNet  Google Scholar 

  10. Guesmia, A., Messaoudi, S.A.: General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math. Methods Appl. Sci. 32, 2102–2122 (2009)

    Article  MathSciNet  Google Scholar 

  11. Kim, J.U., Renardy, Y.: Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25(6), 1417–1429 (1987)

    Article  MathSciNet  Google Scholar 

  12. Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integral Equ. 8, 507–533 (1993)

    MathSciNet  MATH  Google Scholar 

  13. Lasiecka, I., Wang, X.: Intrinsic decay rate estimates for semilinear abstract second order equations with memory, New prospects in direct, inverse and control problems for evolution equations, 271303, Springer INdAM Ser., 10, Springer, Cham (2014)

  14. Liu, W.J.: General decay of solutions to a viscoelastic wave equation with nonlinear localized damping. Ann. Acad. Sci. Fenn. Math. 34(1), 291–302 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Messaoudi, S.A., Mustafa, M.I.: A general stability result in a memory-type Timoshenko system. Commun. Pure Appl. Anal. 12(2), 957–972 (2013)

    Article  MathSciNet  Google Scholar 

  16. Messaoudi, S.A., Mustafa, M.I.: A stability result in a memory-type Timoshenko system. Dyn. Syst. Appl. 18, 457–468 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Messaoudi, S.A., Soufyane, A.: Boundary stabilization of a nonlinear system of Timoshenko type. Nonlinear Anal. 67, 2107–212 (2007)

    Article  MathSciNet  Google Scholar 

  18. Muñoz Rivera, J.E., Racke, R.: Global stability for damped Timoshenko systems. Discrete Contin. Dyn. Syst. 9(6), 1625–1639 (2003)

    Article  MathSciNet  Google Scholar 

  19. Mustafa, M.I.: Laminated Timoshenko beams with viscoelastic damping. J. Math. Anal. Appl. 466, 619–641 (2018)

    Article  MathSciNet  Google Scholar 

  20. Mustafa, M.I.: The control of Timoshenko beams by memory-type boundary conditions. App. Anal. (2019). https://doi.org/10.1080/00036811.2019.1602724

  21. Mustafa, M.I.: Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations. Nonlinear Anal. RWA 13, 452–463 (2012)

    Article  MathSciNet  Google Scholar 

  22. Mustafa, M.I., Messaoudi, S.A.: Energy decay rates for a Timoshenko system with viscoelastic boundary conditions. Appl. Math. Comput. 218, 9125–9131 (2012)

    Article  MathSciNet  Google Scholar 

  23. Mustafa, M.I., Messaoudi, S.A.: General energy decay rates for a weakly damped Timoshenko system. J. Dyn. Control Syst. 16, 211–226 (2010)

    Article  MathSciNet  Google Scholar 

  24. Raposo, C.A., Ferreira, J., Santos, M.L., Castro, N.N.O.: Exponential stability for the Timoshenko system with two weak dampings. Appl. Math. Lett. 18, 535–541 (2005)

    Article  MathSciNet  Google Scholar 

  25. Santos, M.: Decay rates for solutions of a Timoshenko system with a memory condition at the boundary. Abstr. Appl. Anal. 7(10), 531–546 (2002)

    Article  MathSciNet  Google Scholar 

  26. Soufyane, A., Wehbe, A.: Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron. J. Differ. Equ. 29, 1–14 (2003)

    MathSciNet  Google Scholar 

  27. Timoshenko, S.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philios. Magn. 41, 744–746 (1921)

    Google Scholar 

  28. Xiao, T.J., Liang, J.: Coupled second order semilinear evolution equations indirectly damped via memory effects. J. Differ. Equ. 254(5), 2128–2157 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by MASEP Research Group in the Research Institute of Sciences and Engineering at University of Sharjah.

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  1. Department of Mathematics, University of Sharjah, P.O.Box 27272, Sharjah, United Arab Emirates

    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. On the Control of Dissipative Viscoelastic Timoshenko Beams. Mediterr. J. Math. 18, 49 (2021). https://doi.org/10.1007/s00009-020-01680-7

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