Dynamical behavior  of networks of non-uniform Timoshenko
beams system with boundary time-delay inputs

Zhong-Jie Han; Gen-Qi Xu; Zhong-Jie Han; Gen-Qi Xu

doi:10.3934/nhm.2011.6.297

Networks and Heterogeneous Media

2011, Volume 6, Issue 2: 297-327. doi: 10.3934/nhm.2011.6.297

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Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs

Zhong-Jie Han ^{1
,},
Gen-Qi Xu ^{1
,}

1.
Department of Mathematics, Tianjin University, Tianjin 300072

Received: 01 June 2010 Revised: 01 May 2011
Primary: 93D15, 47A75; Secondary: 35B40, 93D20.

The dynamical stability of planar networks of non-uniform Timoshenko beams system is considered. Suppose that the displacement and rotational angle is continuous at the common vertex of this network and the bending moment and shear force satisfies Kirchhoff's laws, respectively. Time-delay terms exist in control inputs at exterior vertices. The feedback control laws are designed to stabilize this kind of networks system. Then it is proved that the corresponding closed loop system is well-posed. Under certain conditions, the asymptotic stability of this system is shown. By a complete spectral analysis, the spectrum-determined-growth condition is proved to be satisfied for this system. Finally, the exponential stability of this system is discussed for a special case and some simulations are given to support these results.
- Stability,
- network,
- Timoshenko beam,
- spectrum,
- feedback control,
- time-delay,
- exponential stability
Citation: Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenkobeams system with boundary time-delay inputs[J]. Networks and Heterogeneous Media, 2011, 6(2): 297-327. doi: 10.3934/nhm.2011.6.297

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Abstract

The dynamical stability of planar networks of non-uniform Timoshenko beams system is considered. Suppose that the displacement and rotational angle is continuous at the common vertex of this network and the bending moment and shear force satisfies Kirchhoff's laws, respectively. Time-delay terms exist in control inputs at exterior vertices. The feedback control laws are designed to stabilize this kind of networks system. Then it is proved that the corresponding closed loop system is well-posed. Under certain conditions, the asymptotic stability of this system is shown. By a complete spectral analysis, the spectrum-determined-growth condition is proved to be satisfied for this system. Finally, the exponential stability of this system is discussed for a special case and some simulations are given to support these results.

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