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Networks and Heterogeneous Media

2010, Volume 5, Issue 2: 315-334. doi: 10.3934/nhm.2010.5.315
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Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks

  • 1.
    Department of Mathematics, Tianjin University, Tianjin 300072
  • Received: 01 September 2009 Revised: 01 April 2010

  • Primary: 35B40, 47A75; Secondary: 93D20, 93D15.

  • A kind of planar network of strings with non-collocated terms in boundary feedback controls is considered. Suppose that the network is constituted by n non-uniform strings, connected by one vibrating point mass. The displacements of these strings are continuous at the common vertex. The non-collocated terms are contained in feedback controls at exterior nodes. The well-posedness of the corresponding closed-loop system is proved. A complete spectral analysis is carried out and the asymptotic expression of the spectrum of this system operator is obtained, which implies that the asymptotic behavior of the spectrum is independent of these non-collocated terms. Then the Riesz basis property of the (generalized) eigenvectors of the system operator is proved. Thus, the spectrum determined growth condition holds. Finally, the exponential stability of a special case of this kind of network is gotten under certain conditions. In order to support these results, a numerical simulation is given.

    Citation: Zhong-Jie Han, Gen-Qi Xu. Spectrum and dynamical behavior of a kind of planar network of non-uniformstrings with non-collocated feedbacks[J]. Networks and Heterogeneous Media, 2010, 5(2): 315-334. doi: 10.3934/nhm.2010.5.315

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  • A kind of planar network of strings with non-collocated terms in boundary feedback controls is considered. Suppose that the network is constituted by n non-uniform strings, connected by one vibrating point mass. The displacements of these strings are continuous at the common vertex. The non-collocated terms are contained in feedback controls at exterior nodes. The well-posedness of the corresponding closed-loop system is proved. A complete spectral analysis is carried out and the asymptotic expression of the spectrum of this system operator is obtained, which implies that the asymptotic behavior of the spectrum is independent of these non-collocated terms. Then the Riesz basis property of the (generalized) eigenvectors of the system operator is proved. Thus, the spectrum determined growth condition holds. Finally, the exponential stability of a special case of this kind of network is gotten under certain conditions. In order to support these results, a numerical simulation is given.


  • This article has been cited by:

    1. Shen Wang, Zhong-Jie Han, Zhi-Xue Zhao, Anti-disturbance stabilization for a hybrid system of non-uniform elastic string, 2021, 358, 00160032, 9653, 10.1016/j.jfranklin.2021.10.018
    2. Ya-Xuan Zhang, Zhong-Jie Han, Gen-Qi Xu, Stability and Spectral Properties of General Tree-Shaped Wave Networks with Variable Coefficients, 2019, 164, 0167-8019, 219, 10.1007/s10440-018-00236-y
    3. Zhong-Jie Han, Enrique Zuazua, Decay rates for elastic-thermoelastic star-shaped networks, 2017, 12, 1556-181X, 461, 10.3934/nhm.2017020
    4. Yanni Guo, Genqi Xu, 2012, Stabilization of wave equations with boundary delays and non-collocated feedback controls, 978-1-4577-2074-1, 3142, 10.1109/CCDC.2012.6244495
    5. Zhong-Jie Han, Enrique Zuazua, Decay rates for 1−d heat-wave planar networks, 2016, 11, 1556-1801, 655, 10.3934/nhm.2016013
    6. Zhong-Jie Han, Gen-Qi Xu, Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs, 2011, 6, 1556-181X, 297, 10.3934/nhm.2011.6.297
    7. Zhong-Jie Han, Gen-Qi Xu, Output feedback stabilisation of a tree-shaped network of vibrating strings with non-collocated observation, 2011, 84, 0020-7179, 458, 10.1080/00207179.2011.561441
    8. Yaru Xie, Genqi Xu, The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls, 2016, 11, 1556-1801, 527, 10.3934/nhm.2016008
    9. Jia-Xian Guang, Zhong-Jie Han, Decay rates for star-shaped degenerate heat-wave coupled networks, 2025, 443, 00220396, 113505, 10.1016/j.jde.2025.113505
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  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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