A MEMORY TYPE BOUNDARY STABILIZATION FOR AN EULER-BERNOULLI BEAM UNDER BOUNDARY OUTPUT FEEDBACK CONTROL Kang, Yong-Han; Park, Jong-Yeoul; Kim, Jung-Ae;
In this paper, the memory type boundary stabilization for an Euler-Bernoulli beam with one end fixed and control at the other end is considered. We prove the existence of solutions using the Galerkin method and then investigate the exponential stability of solutions by using multiplier technique.
Control of a viscoelastic translational Euler-Bernoulli beam, Mathematical Methods in the Applied Sciences, 2016
Uniform Decay for Solutions of an Axially Moving Viscoelastic Beam, Applied Mathematics & Optimization, 2016
Control of a riser through the dynamic of the vessel, Applicable Analysis, 2016, 95, 9, 1957
D. Andrade and J. M. Rivera, Exponential decay of non-linear wave equation with a viscoelastic boundary condition, Math. Methods Appl. Sci. 23 (2000), no. 1, 4161.
J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973), 61-90.
F. Conrad and O. Omer, On the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim. 36 (1998), no. 6, 1962-1986.
M. Dahleh and W. Hopkins, Adaptive stabilization of single-input single-output delay systems, IEEE Trans. Automat. Control 31 (1986), no. 6, 577-579.
M. Feckan, Free vibrations of beams on bearings with nonlinear elastic responses, J. Differential Equations 154 (1999), no. 1, 5572.
E. Feireisl, Nonzero time periodic solutions to an equation of Petrovsky type with non-linear boundary condition: slow oscillations of beams on elastic bearings, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), no. 1, 133-146.
B. Z. Guo and W. Guo, Adaptive stabilization for a Kirchhoff-type nonlinear beam under boundary output feedback control, Nonlinear Anal. 66 (2007), no. 2, 427-441.
B. Z. Guo and Z. H. Luo, Initial-boundary value problem and exponential decay for a flexible-beam vibration with gain adaptive direct strain feedback control, Nonlinear Anal. 27 (1996), no. 3, 353-365.
M. Kirane and N. E. Tatar, A memory type boundary stabilization of a mildly damped wave equation, Electron. J. Qual. Theory Differ. Equ. 1999 (1999), no. 6, 1-7.
I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only, J. Differential Equations 95 (1992), no. 1, 169-182.
J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972.
H. Ma, Uniform decay rates for the solutions to the Euler-Bernoulli plate equation with boundary feedback via bending moments, Differential Integral Equations 5 (1992), 1121-1150.
T. F. Ma, Boundary stabilization for a non-linear beam on elastic bearings, Math. Methods Appl. Sci. 24 (2001), 583-594.
J. Y. Park, Y. H. Kang, and J. A. Kim, Existence and exponential stability for a Euler- Bernoulli beam equation with memory and boundary output feedback control term, Acta Appl. Math. 104 (2008), no. 3, 287-301.
J. Y. Park and J. A. Kim, Existence and uniform decay for Euler-Bernoulli beam equa- tion with memory term, Math. Methods Appl. Sci. 27 (2004), no. 14, 1629-1640.
S. K. Patcheu, On a global solution and asymptotic behaviour for the generalized damped extensible beam equation, J. Differential Equations 135 (1997), no. 2, 299314.
M. L. Santos, Asymptotic behaviour of solutions to wave equations with a memory condition at the boundary, Electron. J. Differential Equations 2001 (2001), no. 73, 1-11.