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Research Article

Exponential stability of a rotating Timoshenko beam under thermo-viscoelastic damping

Pages 426-445 | Received 17 Aug 2020, Accepted 01 Apr 2021, Published online: 23 Apr 2021

References

  • F. Alabau-Boussouiran, Asymptotic behaviour for Timoshenko beams subject to a single nonlinear feedback control, Nonlinear Differ. Equ. Appl. 14 (2007), pp. 643–669.
  • F. Ammar-Khodja, A. Racke, and R. Benabdallah, Energy decay for Timoshenko systems of memory type, J. Differ. Equ. 194 (2003), pp. 82–115.
  • D.S. Almeida Júnior, M.L. Santos, and J.E. Muñoz Rivera, Stability to 1-D thermoelastic Timoshenko beam acting on shear force, Z. Angew. Math. Phys. 65 (2014), pp. 1233–1249.
  • W.J. Book, Modeling, design, and control of flexible manipulator arms: a tutorial review. Decision and Control, Proceedings of the 29th IEEE Conference on 5–7 Dec. 1990, pp. 500–506. doi: https://doi.org/10.1109/CDC.1990.203648.
  • A. Berkani, Stabilization of a viscoelastic rotating Euler-Bernoulli beam, Math. Methods Appl. Sci.41 (2018), pp. 2939–2960.
  • A. Berkani and N.E. Tatar, Stabilization of viscoelastic Timoshenko beam fixed in to a moving base, Math. Model. Nat. Phenom. 14 (2019), pp. 501.
  • A. Berkani, N.E. Tatar, and A. Kelleche, Vibration control of a viscoelastic translational Euler-Bernoulli beam, J. Dyn Control Syst. 24 (2018), pp. 167–199.
  • M.M. Cavalcanti, V.N. Domingos Cavalcanti, F.A. Falao Nascimento, I. Lasiecka, and J. H. Rodrigues, Uniform decay rates f or the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Z. Angew. Math. Phys. 65 (2014), pp. 1189–1206.
  • A. Djebabla and N.E. Tatar, Stabilization of the Timoshenko beam by the thermal effect, J. Dyn. Control Syst. 16 (2009), pp. 189–210.
  • A. Djebabla and N.E. Tatar, Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping, Mediterr. J. Math. 99 (2010), pp. 1–13.
  • A. Djebabla and N.E. Tatar, Exponential stabilization of the Timoshenko system by a thermal effect with an oscillating kernel, Math. Comput. Model. 54 (2011), pp. 301–314.
  • D. Feng, D. Shi, and W. Zhang, Boundary feedback stabilization of Timoshenko beam with boundary dissipation, Sci. China 41 (1998), pp. 481–490.
  • X. He, W. He, W. You, and C. Sun, Boundary control design for a flexible robotic manipulator modeled as a Timoshenko beam, 12th World Congress on Intelligent Control and Automation (WCICA). June 12–15, Guilin, China, 2016.
  • X. He, W. He, C. Sun, Robust adaptive vibration control for an uncertain flexible Timoshenko robotic manipulator with input and output constraints, Int. J. Syst. Sci. 48(13) (2017), , pp. 2860–2870. doi: https://doi.org/10.1080/00207721.2017.1360963
  • X. He, W. He, H. Qin, and C. Sun, Boundary vibration control for a flexible Timoshenko robotic manipulator, IET Control Theory Appl. 12(7) (2018), pp. 875–882.
  • B.Z. Guo, Riesz basis approach to the tracking control of a flexible beam with a tip rigid body without dissipativity, Opt. Methods Softw. 17 (2002), pp. 6556681 655–681.
  • B.Z. Guo and Q. Zhang, On harmonic disturbance rejection of an undamped Euler-Bernoulli beam with rigid tip body, ESAIM Contr. Opt. Calc. Var. 10 (2004), pp. 615–623.
  • M. Kafini, General energy decay in a Timoshenko-type system of thermoelasticity of type III with a viscoelastic damping, J. Math. Anal. Appl. 375 (2011), pp. 523–537.
  • J.U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Opt. 25 (1987), pp. 1417–1429.
  • J.L. Lions, Contrôlabilité exacte, pertubations et stabilisation de systèmes distribués, Tome I, Contrôlabilité exacte, RMA. Vol. 8, Masson, Paris, 1988.
  • Ö. Morgül, Orientation and stabilization of a flexible beam attached to a rigid body planar motion, IEEE Trans. Autom. Contr. 36(8) (1991), pp. 953–962.
  • Ö. Morgül, Dynamic boundary control of the Timochenko beam, Automatica 28 (1992), pp. 1255–1260.
  • S.A. Messaoudi and B.S. Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl. 348 (2008), pp. 298–307.
  • S. Messaoudi and M.I. Mustafa, A stability result in a memory-type Timoshenko system, Dyn. Syst. Appl. 18 (2009), pp. 457–468.
  • M.I. Mustafa and S.A. Messaoudi, General energy decay rates for a weakly damped Timoshenko system, J. Dyn. Control Syst. 16 (2010), pp. 211–226.
  • S.A. Messaoudi and M.I. Mustafa, A general stability result in a memory-type Timoshenko system, Commun. Pure Appl. Anal. 12 (2013), pp. 957–972.
  • J.E. Muñoz Rivera and R. Racken, Global stability for damped Timoshenko systems, Discrete Cont. Dyn. Syst. 9 (2003), pp. 1625–1639.
  • T.D Nguyen and O. Egeland, Tracking and observer design for a motorized Euler-Bernoulli beam, Proceeding IEEE International Conference on Decision and Control, Maui, Hawaii, 2003, pp. 3325–3330.
  • R. Racken and J.E. Muñoz Rivera, Mildly dissipative nonlinear Timoshenko systems – global existence and exponential stability, J. Math. Anal. Appl. 276(1) (2002), pp. 248–278.
  • A. Soufyane, Stabilisation de la poutre de Timoshenko, C. R. Acad. Sci. Paris, Sér. I Math. 328 (1999), pp. 731–734.
  • A. Soufyane, Exponential stability of the linearized nonuniform Timoshenko beam, Nonl. Anal. Real World Appl. 10 (2009), pp. 1016–1020.
  • D.-H. Shi and D.-X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, IMA J. Math. Control Inf. 18 (2001), pp. 395–403.
  • A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electr. J. Differ. Equ. 29 (2003), pp. 1–14.
  • D.H. Shi, S.H. Hou, and D.X. Feng, Feedback stabilization of a Timoshenko beam with an end mass, Int. J. Control 69 (1998), pp. 285–300.
  • D-H Shi, S.H. Hou, and D-X. Feng, Feedback stabilization of a Timoshenko beam with an end mass, Int. J. Control 9(2) (1998), pp. 285–300.
  • N.E. Tatar, Viscoelastic Timoshenko beams with occasionally constant relaxation functions, Appl. Math. Opt. 66(1) (2012), pp. 123–145.
  • N.E. Tatar, Stabilization of a viscoelastic Timoshenko beam, Appl. Anal. Int. J. 92 (2013), pp. 27–43.
  • N.E. Tatar, Exponential decay for a viscoelastically damped Timoshenko beam, Acta Math. Sci. 33 (2013), pp. 505–524.
  • G.-Q. Xu, Boundary feedback exponential stabilization of a Timoshenko beam with both ends free, Int. J. Control 78 (2005), pp. 286–297.
  • G.-Q. Xu and S.-P. Yung, Stabilization of Timoshenko beam by means of pointwise controls, ESAIM Control Optim. Calc. Var. 9 (2003), pp. 579–600.
  • G.-Q. Xu, D.-X. Feng, and S.-P. Yung, Riesz basis property of generalized eigenvector system of a Timoshenko beam, IMA J. Math. Control Inf. 21 (2004), pp. 65–83.
  • Q.-X. Yan, S.H. Hou, and S.H. Feng, Asymptotic behaviour of Timoshenko beam with dissipative boundary feedback, J. Math. Anal. Appl. 269 (2002), pp. 556–577.
  • Q. Yan, L. Wan, and D. Feng, On boundary feedback stabilization of Timoshenko beam with rotor inertia at the tip, J. Contr. Theory Appl. 3 (2004), pp. 283–287.
  • S. Zhang and S.S. Ge, Boundary output-feedback stabilization of a Timoshenko beam using disturbance observer, IEEE Trans. Ind. Electr. 60 (2011), pp. 5186–5194.

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