Stabilization of a nonlinear Euler-Bernoulli beam. (English) Zbl 1502.35014
Summary: In this work, we study the vibration control of a flexible mechanical system. The dynamic of the problem is modeled as a viscoelastic nonlinear Euler-Bernoulli beam. To suppress the undesirable transversal vibrations of the beam, we adopt a control at the right boundary of the beam. This control law is simple to implement. We prove uniform stability of the system using a viscoelastic material, the multiplier method and some ideas introduced in [N.-E. Tatar, Appl. Math. Comput. 215, No. 6, 2298–2306 (2009; Zbl 1422.74022)]. It is shown that a large range of rates of decay of the energy can be achieved through a determined class of kernels. Unlike most of the existing classes in the market, ours are not necessarily strictly decreasing.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L35 | Initial-boundary value problems for higher-order hyperbolic equations |
| 35L76 | Higher-order semilinear hyperbolic equations |
| 35R09 | Integro-partial differential equations |
| 35R10 | Partial functional-differential equations |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
Keywords:
vibration control; viscoelastic nonlinear Euler-Bernoulli beam; control at the right boundaryCitations:
Zbl 1422.74022References:
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