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September & December  2018, 8(3&4): 789-808. doi: 10.3934/mcrf.2018035

## Weak stability of a laminated beam

 1 College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China 2 Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812, USA 3 College of Information Science and Technology, Donghua University, School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China

* Corresponding authorr: Yang Wan

Received  October 2017 Revised  May 2018 Published  September 2018

In this paper, we consider the stability of a laminated beam equation, derived by Liu, Trogdon, and Yong [6], 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.

Citation: Yanfang Li, Zhuangyi Liu, Yang Wang. Weak stability of a laminated beam. Mathematical Control & Related Fields, 2018, 8 (3&4) : 789-808. doi: 10.3934/mcrf.2018035
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