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Existence and Exponential Stability for a Euler-Bernoulli Beam Equation with Memory and Boundary Output Feedback Control Term

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Abstract

In this paper, we consider the Euler-Bernoulli beam equation with memory and boundary output feedback control term. We prove the existence of solutions using the Galerkin method and then investigate the exponential stability of solutions by using multiplier technique.

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References

  1. Andrade, D., Rivera, J.M.: Exponential Decay of Non-Linear Wave Equation with Viscoelastic Boundary Condition. Mathematical Methods in the Applied Sciences, vol. 23, pp. 41–60 (2000)

  2. Conrad, F., Omër, O.: On the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim. 36, 1962–1986 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Feireisl, E.: Nonzero time periodic solutions to an equation of Petrovsky type with nonlinear boundary condition: slow oscillations of beams on elastic bearings. Ann. Sc. Norm. Super. Pisa 20, 133–146 (1993)

    MATH  MathSciNet  Google Scholar 

  4. Fekan, M.: Free vibrations of beams on bearings with nonlinear elastic responses. J. Differ. Equ. 154, 55–72 (1999)

    Article  Google Scholar 

  5. Guo, B.Z., Guo, W.: Adaptive stabilization for a Kirchhoff-type nonlinear beam under boundary output feedback control. Nonlinear Anal. 66(2), 427–441 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kirane, M., Tatar, N.E.: A memory type boundary stabilization of a mildly damped wave equation. Electron. J. Qual. Theory Differ. Equ. 6, 1–7 (1999)

    MathSciNet  Google Scholar 

  7. Lagnese, J.E., Leugering, G.: Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. J. Differ. Equ. 91, 355–388 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lasiecka, I.: Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only. J. Differ. Equ. 95, 169–182 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ma, H.: Uniform decay rates for the solutions to the Euler-Bernoulli plate equation with boundary feedback via bending moments. Differ. Integral Equ. 5, 1121–1150 (1992)

    Google Scholar 

  10. Ma, T.F.: Existence results for a model of nonlinear beam on elastic bearings. Appl. Math. Lett. 13, 11–15 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ma, T.F.: Boundary stabilization for a non-linear beam on elastic bearings. Math. Methods Appl. Sci. 24, 583–594 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Nicaise, S., Pignotti, C.: Stabilization of the wave equation with variable coefficients and boundary condition of memory type. Asymptot. Anal. 50, 31–67 (2006)

    MATH  MathSciNet  Google Scholar 

  13. Park, J.Y., Kim, J.A.: Existence and uniform decay for Euler-Bernoulli beam equation with memory term. Math. Methods Appl. Sci. 27(14), 1629–1640 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Patcheu, S.K.: On a global solution and asymptotic behaviour for the generalized damped extensible beam equation. J. Differ. Equ. 135, 299–314 (1997)

    Article  MATH  Google Scholar 

  15. Santos, M.L.: Asymptotic behavior of solutions to wave equations with a memory condition at the boundary. Electron. J. Differ. Equ. 73, 1–11 (2001)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics College of Science, Pusan National University, Kumjung, Pusan, 609-735, South Korea

    Jong Yeoul Park

  2. Department of Mathematics, University of Ulsan, Ulsan, 680-749, South Korea

    Yong Han Kang

  3. National Institute for Mathematical Sciences, 3F TowerKoreana 628 Daeduk-Boulevard, Yuseong-Gu Daejeon, South Korea

    Jung Ae Kim

Authors
  1. Jong Yeoul Park
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  2. Yong Han Kang
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  3. Jung Ae Kim
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Corresponding author

Correspondence to Jung Ae Kim.

Additional information

This work was supported by grant number KRF-2005-202-C00030 from the Korea Research Foundation. This work was supported by National Institute for Mathematical Sciences.

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Park, J.Y., Kang, Y.H. & Kim, J.A. Existence and Exponential Stability for a Euler-Bernoulli Beam Equation with Memory and Boundary Output Feedback Control Term. Acta Appl Math 104, 287–301 (2008). https://doi.org/10.1007/s10440-008-9257-8

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  • DOI: https://doi.org/10.1007/s10440-008-9257-8

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