Boundary Stabilization of a More General Kirchhoff-Type Beam Equation
ABSTRACT
Simultaneously, considering the viscous effect of material, damping of medium, geometrical nonlinearity, physical nonlinearity, we set up a more general equation of beam subjected to axial force and external load. We prove the existence and uniqueness of global solutions under non-linear boundary conditions which the model is added one damping mechanism at l end. What is more, we also prove the exponential decay property of the energy of above mentioned system.
Simultaneously, considering the viscous effect of material, damping of medium, geometrical nonlinearity, physical nonlinearity, we set up a more general equation of beam subjected to axial force and external load. We prove the existence and uniqueness of global solutions under non-linear boundary conditions which the model is added one damping mechanism at l end. What is more, we also prove the exponential decay property of the energy of above mentioned system.
Cite this paper
J. Zhang and D. Wang, "Boundary Stabilization of a More General Kirchhoff-Type Beam Equation," International Journal of Modern Nonlinear Theory and Application, Vol. 1 No. 3, 2012, pp. 97-101. doi: 10.4236/ijmnta.2012.13014.
J. Zhang and D. Wang, "Boundary Stabilization of a More General Kirchhoff-Type Beam Equation," International Journal of Modern Nonlinear Theory and Application, Vol. 1 No. 3, 2012, pp. 97-101. doi: 10.4236/ijmnta.2012.13014.
References
[1] S. Woinowsky-Krieger, “The Effect of Axial Force on the Vibration of Hinged Bars,” Journal of applied Mechanics, Vol. 17, 1950, pp. 35-36. [2] J. M. Ball, “Initial-Boundary Value Problems for an Extensible Beam,” Journal of Mathematical Analysis and Applications, Vol. 42, No. 1, 1973, pp. 61-90. doi:10.1016/0022-247X(73)90121-2 [3] L. A. Mederios, “On a New Class of Nonlinear Wave Equations,” Journal of Mathematical Analysis and Applications, Vol. 69, No. 1, 1979, pp. 252-262. [4] M. Tucsnak, “Semi-Internal Stabilization for a Nonlinear Euler-Bernoulli Equation,” Mathematical Methods in the Applied Sciences, Vol. 19, No. 11, 1996, pp. 897-907. doi:10.1002/(SICI)1099-1476(19960725)19:11<897::AID-MMA801>3.0.CO;2-# [5] S. Kouemou Patcheu, “On a Global Solution and Asymptotic Behavior for the Generalized Damped Extensible Beam Equation,” Journal of Differential Equations, Vol. 135, No. 2, 1997, pp. 299-314. doi:10.1006/jdeq.1996.3231 [6] F. M. To, “Boundary Stabilization for a Non-Linear Beam on Elastic Bearings,” Mathematical Methods in the Applied Sciences, Vol. 24, No. 8, 2001, pp. 583-594. doi:10.1002/mma.230 [7] J. M. Ball, “Stability Theory for an Extensible Beam,” Journal of Differential Equations, Vol. 14, No. 3, 1973, pp. 61-90. doi:10.1016/0022-0396(73)90056-9
[1] S. Woinowsky-Krieger, “The Effect of Axial Force on the Vibration of Hinged Bars,” Journal of applied Mechanics, Vol. 17, 1950, pp. 35-36. [2] J. M. Ball, “Initial-Boundary Value Problems for an Extensible Beam,” Journal of Mathematical Analysis and Applications, Vol. 42, No. 1, 1973, pp. 61-90. doi:10.1016/0022-247X(73)90121-2 [3] L. A. Mederios, “On a New Class of Nonlinear Wave Equations,” Journal of Mathematical Analysis and Applications, Vol. 69, No. 1, 1979, pp. 252-262. [4] M. Tucsnak, “Semi-Internal Stabilization for a Nonlinear Euler-Bernoulli Equation,” Mathematical Methods in the Applied Sciences, Vol. 19, No. 11, 1996, pp. 897-907. doi:10.1002/(SICI)1099-1476(19960725)19:11<897::AID-MMA801>3.0.CO;2-# [5] S. Kouemou Patcheu, “On a Global Solution and Asymptotic Behavior for the Generalized Damped Extensible Beam Equation,” Journal of Differential Equations, Vol. 135, No. 2, 1997, pp. 299-314. doi:10.1006/jdeq.1996.3231 [6] F. M. To, “Boundary Stabilization for a Non-Linear Beam on Elastic Bearings,” Mathematical Methods in the Applied Sciences, Vol. 24, No. 8, 2001, pp. 583-594. doi:10.1002/mma.230 [7] J. M. Ball, “Stability Theory for an Extensible Beam,” Journal of Differential Equations, Vol. 14, No. 3, 1973, pp. 61-90. doi:10.1016/0022-0396(73)90056-9