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Weak stability of a laminated beam. (English) Zbl 1421.35023

Summary: In this paper, we consider the stability of a laminated beam equation, derived by Z. Liu et al. [Math. Comput. Modelling 30, No. 1–2, 149–167 (1999; Zbl 1043.74513)], subject to viscous or Kelvin-Voigt damping. The model is a coupled system of two wave equations and one Euler-Bernoulli beam equation, which describes the longitudinal motion of the top and bottom layers of the beam and the transverse motion of the beam. We first show that the system is unstable if one damping is only imposed on the beam equation. On the other hand, it is easy to see that the system is exponentially stable if direct damping are imposed on all three equations. Hence, we investigate the system stability when two of the three equations are directly damped. There are a total of seven cases from the combination of damping locations and types. Polynomial stability of different orders and their optimality are proved. Several interesting properties are revealed.

MSC:

35B35 Stability in context of PDEs
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
47D06 One-parameter semigroups and linear evolution equations
35B40 Asymptotic behavior of solutions to PDEs
35L57 Initial-boundary value problems for higher-order hyperbolic systems

Citations:

Zbl 1043.74513
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Full Text: DOI

References:

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