Skip to main content
SpringerLink
Log in

General decay of energy to a nonlinear viscoelastic two-dimensional beam

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

A viscoelastic beam in a two-dimensional space is considered with nonlinear tension. A boundary feedback is applied at the right boundary of the beam to suppress the undesirable vibration. The well-posedness of the problem is established. With the multiplier method, a uniform decay result is proven.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. COLEMAN, B. D. and NOLL, W. Foundations of linear viscoelasticity. Reviews of Modern Physics, 33, 239–249 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  2. COLEMAN, B. D. and MIZEL, V. J. On the general theory of fading memory. Archive for Rational Mechanics and Analysis, 29, 18–31 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  3. FABRIZIO, M. and MORRO, A. Mathematical problems in linear viscoelasticity. Mathematical Problems in Linear Viscoelasticity, Society for Industrial and Applied Mathematics Philadelphia, Pennsylvania (1992)

    Chapter  Google Scholar 

  4. GE, S. S., HE, W., HOW, B. V. E., and CHOO, Y. S. Boundary control of a coupled nonlinear exible marine riser. IEEE Transactions on Control Systems and Technology, 18, 1080–1091 (2010)

    Article  Google Scholar 

  5. HE, W., GGE, S. S., and ZHANG, S. Adaptive boundary control of a exible marine installation system. Automatica, 47, 2728–2734 (2011)

    Article  MATH  Google Scholar 

  6. HE, W., ZHANG, S., and GE, S. S. Boundary control of a exible riser with the application to marine installation. Transactions on Industrial Electronics, 60, 5802–5810 (2013)

    Article  Google Scholar 

  7. NQUYEN, T. L., DO, K. D., and PAN, J. Boundary control of coupled nonlinear three dimensional marine risers. Journal of Marine Sciences and Application, 12, 72–88 (2013)

    Article  Google Scholar 

  8. PARK, J. Y., KANQ, H. K., and KIM, J. A. Existence and exponential stability for a Euler-Bernoulli beam equation with memory and boundary output feedback control term. Acta Applicandae Mathematicae, 104, 287–301 (2008)

    Article  MathSciNet  Google Scholar 

  9. LAZZARI, B. and NIBBI, R. On the exponential decay of the Euler-Bernoulli beam with boundary energy dissipation. Journal of Mathematical Analysis and Applications, 389, 1078–1085 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. KANG, Y. H., PARK, J. Y., and KIM, J. A. A memory type boundary stabilization for an Euler-Bernoulli beam under boundary output feedback control. Journal of the Korean Mathematical Society, 49, 947–964 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. CAVALCANTI, M. M. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete and Continuous Dynamical Systems, 8, 675–695 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. CAVALCANTI, M. M., DOMINGOS-CAVALCANTI, V. N., and MA, T. F. Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains. Differential and Integral Equations, 17, 495–510 (2004)

    MathSciNet  MATH  Google Scholar 

  13. CAVALCANTI, M. M., DOMINGOS-CAVALCANTI, V. N., and SORIANO, J. A. Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation. Communications in Contemporary Mathematics, 6, 705–731 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. CAVALCANTI, M. M., DOMINGOS-CAVALCANTI, V. N., and FERREIRA, J. Existence and uniform decay for a non-linear viscoelastic equation with strong damping. Mathematical Methodes in the Applied Sciences, 24, 1043–1053 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. CAVALCANTI, M. M., DOMINGOS-CAVALCANTI, V. N., and MARTINAZ, P. General decay rate estimates for viscoelastic dissipative systems. Nonlinear Analysis: Theory, Methods & Applications, 68, 177–193 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. BERKANI, A., TATAR, N. E., and KHEMMOUDJ, A. Control of a viscoelastic translational Euler-Bernoulli beam. Mathematical Methodes in the Applied Sciences, 40, 237–254 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. SEGHOUR, L., KHEMMOUDJ, A., and TATAR, N. E. Control of a riser through the dynamic of the vessel. Applicable Analysis, 95, 1957–1973 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. KELLECHE, A., TATAR, N. E., and KHEMMOUDJ, A. Uniform stabilization of an axially moving Kirchhoff string by a boundary control of memory type. Journal of Dynamical Control Systems, 23, 237–247 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. KELLECHE, A., TATAR, N. E., and KHEMMOUDJ, A. Stability of an axially moving viscoelastic beam. Journal of Dynamical Control Systems, 23, 283–299 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. ZHANG, Z. Y., MIAO, X. J., and YU, D. On solvability and stabilization of a class of hyperbolic hemivariational inequalities in elasticity. Funkcialaj Ecvacioj, 54, 297–314 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. ZHANG, Z. Y. and HUANG, J. On solvability of the dissipative Kirchhoff equation with nonlinear boundary damping. Bulletin Korean Mathematical Society, 51, 189–206 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. ZHANG, Z. Y. and MIAO, X. J. Global existence and uniform decay for wave equation with dissipative term and boundary damping. Computers and Mathematics with Applications, 59, 1003–1018 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. KOMORNIK, V. and ZUAZUA, E. A direct method for boundary stabilization of the wave equation. Journal de Mathématiques Pures et Appliquées, 69, 33–54 (1990)

    MathSciNet  MATH  Google Scholar 

  24. ZHANG, Z. Y., LIU, Z. H., MIAO, X. J., and CHEN, Y. Z. Stability analysis of heat flow with boundary time-varying delay effect. Nonlinear Analysis, 73, 1878–1889 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. ZHANG, Z. Y., LIU, Z. H., MIAO, X. J., and CHEN, Y. Z. A note on decay properties for the solutions of a class of partial differential equation with memory. International Journal of Applied Mathematics and Computer, 37, 85–102 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. ZHANG, Z. Y., LIU, Z. H., MIAO, X. J., and CHEN, Y. Z. Global existence and uniform stabilization of a generalized dissipative Klein-Gordon equation type with boundary damping. Journal of Mathematical Physics, 52, 023502 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. DAFERMOS, C. M. On abstract Volterra equations with applications to linear viscoelasticity. Journal of Differential Equations, 7, 554–569 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  28. DAFERMOS, C. M. Asymptotic stability in viscoelasticity. Archive for Rational Mechanics and Analysis, 37, 297–308 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  29. JIN, K. P., LIANG, J., and XIAO, T. J. Coupled second order evolution equations with fading memory: optimal energy decay rate. Journal of Differential Equations, 257, 1501–1528 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. BERRIMI, S. and MESSAOUDI, S. A. Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Analysis, 64, 2314–2331 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. CABANILLAS, E. L. and MUNOZ-RIVERA, J. E. Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decaying kernels. Communications in Mathematical Physics, 177, 583–602 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. HRUSA, W. J. Global existence and asymptotic stability for a semilinear Volterra equation with large initial data. SIAM Journal of Mathematical Analysis, 16, 110–134 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  33. MESSAOUDI, S. A. and TATAR, N. E. Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Mathematical Methodes in the Applied Sciences, 30, 665–680 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. MUNOZ-RIVERA, J. E. Asymptotic behavior in linear viscoelasticity. Quarterly of Applied Mathematics, 52(4), 628–648 (1994)

    Article  Google Scholar 

  35. ALABAU-BOUSSOUIRA, F., CANNARSA, P., and SFORZA, D. Decay estimates for the second order evolution equation with memory. Journal of Functional Analysis, 245, 1342–1372 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. CANNARSA, P. and SFORZA, D. Integro-differential equations of hyperbolic type with positive definite kernels. Journal of Differential Equations, 250, 4289–4335 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. MUNOZ-RIVERA, J. E. and NASO, M. G. On the decay of the energy for systems with memory and indefinite dissipation. Asymptotic Analysis, 49, 189–204 (2006)

    MathSciNet  MATH  Google Scholar 

  38. MUNOZ-RIVERA, J. E., NASO, M. G., and VEGNI, F. M. Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory. Journal of Mathematical Analysis and Applications, 286, 692–704 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. CAVALCANTI, M. M., DOMINGOS-CAVALCANTI, V. N., and FERREIRA, J. Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Mathematical Methodes in the Applied Sciences, 24, 1043–1053 (2001)

    Article  MATH  Google Scholar 

  40. MESSAOUDI, S. A. General decay of solutions of a viscoelastic equation. Journal of Mathematical Analysis and Applications, 341, 1457–1467 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. MESSAOUDI, S. A. General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Analysis, 69, 2589–2598 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. HAN, X. I. and WANG, M. General decay of energy for a viscoelastic equation with nonlinear damping. Mathematical Methodes in the Applied Sciences, 32, 346–358 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. LIU, W. J. General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. Journal of Mathematical Physics, 50, 113506 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. LIU, W. J. General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Analysis, 73, 1890–1904 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. LASIECKA, I. and TATARU, D. Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differential and Integral Equations, 8, 507–533 (1993)

    MathSciNet  MATH  Google Scholar 

  46. ALABAU-BOUSSOUIRA, F. and CANNARSA, P. A general method for proving sharp energy decay rates for memory-dissipative evo-lution equations. Comptes Rendus de l’Académie des Sciences Paris, Serie I, 347, 867–872 (2009)

    MATH  Google Scholar 

  47. MUSTAFA, M. I. and MESSAOUDI, S. A. General stability result for viscoelastic wave equations. Journal of Mathematical Physics, 53, 053702 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. CAVALCANTI, M. M., CAVALCANTI, A. N. D., LASIECKA, I., and WANG, X. Existence and sharp decay rate estimates for a von Karman system with long memory. Nonlinear Analysis Real World Applications, 22, 289–306 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. CAVALCANTI, M. M., CAVALCANTI, A. N. D., LASIECKA, I., and Nascimento, F. A. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete and Continuous Dynamical Systems. Series B, 19, 1987–2012 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  50. CAVALCANTI, M. M., CAVALCANTI, A. N. D., LASIECKA, I., and WEBLER, C. M. Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density. Advances in Nonlinear Analysis, 6, 121–145 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  51. GUESMIA, A. Asymptotic stability of abstract dissipative systems with infinite memory. Journal of Mathematical Analysis and Applications, 382, 748–760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. LASIECKA, I., MESSAOUDI, S. A., and MUSTAFA, M. I. Note on intrinsic decay rates for abstract wave equations with memory. Journal of Mathematical Physics, 54, 031504 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. LASIECKA, I. and 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, Springer, Chambridge (2014)

    Google Scholar 

  54. MESSAOUDI, S. A. General decay of solutions of a viscoelastic equation. Journal of Mathematical Analysis and Applications, 341, 1457–1467 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  55. MESSAOUDI, S. A. General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Analysis, 69, 2589–2598 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. DO, K. D. and PAN, J. Boundary control of transverse motion of marine risers with actuator dynamics. Journal of Sound and Vibration, 318, 768–791 (2008)

    Article  Google Scholar 

  57. HARDY, G. H., LITTLEWOOD, J. E., and POLYA, G. Inequalities, Cambridge University Press, Cambridge (1959)

    MATH  Google Scholar 

  58. RAHN, C. D. Mechantronic Control of Distributed Noise and Vibration, Springer, New York (2001)

    Book  MATH  Google Scholar 

  59. AUBIN, J. P. Un théorème de compacité. Comptes Rendus de l’Académie des Sciences Paris, 256, 5042–5044 (1963)

    MATH  Google Scholar 

  60. LIONS, J. L. Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris (1969)

    MATH  Google Scholar 

  61. ZHENG, S. Nonlinear evolution equations. Pitman series Monographs and Survey in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton (2004)

    Google Scholar 

  62. ARARUNA, F. D., BRAZ, E., SILVA, P., and ZUAZUA, E. Asymptotic limits and stabilization for the 1d nonlinear Mindlin-Timoshenko system. Journal of Systems Science and Complexity, 23, 1–17 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for giving constructive and fruitful suggestions.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Sciences and Technology Houari Boumedienne, P.O.Box 32, El-Alia 16111, Bab Ezzouar, Algiers, Algeria

    B. Lekdim & A. Khemmoudj

Authors
  1. B. Lekdim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Khemmoudj
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Khemmoudj.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lekdim, B., Khemmoudj, A. General decay of energy to a nonlinear viscoelastic two-dimensional beam. Appl. Math. Mech.-Engl. Ed. 39, 1661–1678 (2018). https://doi.org/10.1007/s10483-018-2389-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-018-2389-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation