TY - JOUR
T1 - Weak stability of a laminated beam
AU - Li, Yanfang
AU - Liu, Zhuangyi
AU - Wang, Yang
N1 - Publisher Copyright:
© 2018, American Institute of Mathematical Sciences. All rights reserved.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2018
Y1 - 2018
N2 - 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.
AB - 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.
KW - Exponential stability
KW - Laminated beam
KW - Optimal decay rate
KW - Polynomial stability
KW - Semigroup
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U2 - 10.3934/mcrf.2018035
DO - 10.3934/mcrf.2018035
M3 - Article
AN - SCOPUS:85056520729
VL - 8
SP - 789
EP - 808
JO - Mathematical Control and Related Fields
JF - Mathematical Control and Related Fields
SN - 2156-8472
IS - 3-4
ER -