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Input-output $ L^2 $-well-posedness, regularity and Lyapunov stability of string equations on networks

  • * Corresponding author: Dongyi Liu

    * Corresponding author: Dongyi Liu 

This research is supported by the Natural Science Foundation of China under grant number: NSFC-61773277

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  • We consider the general networks of elastic strings with Neumann boundary feedbacks and collocated observations in this paper. By selecting an appropriate multiplier, we show that this system is input-output $ L^2 $-well-posed. Moreover, we verify its regularity by calculating the input-output transfer function of system. In the end, by choosing an appropriate multiplier, we give a method to construct a Lyapunov functional and prove the exponential decay of tree-shaped networks with one fixed root under velocity feedbacks acted on all leaf vertices.

    Mathematics Subject Classification: Primary: 93C20, 93D05; Secondary: 35B35, 35L05.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Figure 1.  Networks consisting of strings with one fixed vertex

    Figure 2.  The tree-shaped network consisting of six strings with the fixed root $ p_4 $

    Table 1.  Paths from the root $ p_1 $ to leaves

    $ E_1(G) $ $ V_1(G) $ $ E_2(G) $ $ V_2(G) $ $ E_3(G) $ $ V_3(G) $ $ E_4(G) $ $ V_4(G) $ $ m_p $
    indices $ i_1 $ $ \iota_1 $ $ i_2 $ $ \iota_2 $ $ i_3 $ $ \iota_3 $ $ i_4 $ $ \iota_4 $
    $ P(p_1,p_6) $ $ e_1 $ $ p_2 $ $ e_3 $ $ p_6 $ $ * $ $ * $ $ * $ $ * $ $ 2 $
    $ P(p_1,p_7) $ $ e_1 $ $ p_2 $ $ e_2 $ $ p_3 $ $ e_5 $ $ p_5 $ $ e_9 $ $ p_7 $ $ 4 $
    $ P(p_1,p_8) $ $ e_1 $ $ p_2 $ $ e_2 $ $ p_3 $ $ e_5 $ $ p_5 $ $ e_8 $ $ p_8 $ $ 4 $
    $ P(p_1,p_9) $ $ e_1 $ $ p_2 $ $ e_2 $ $ p_3 $ $ e_6 $ $ p_4 $ $ e_7 $ $ p_9 $ $ 4 $
    $ P(p_1,p_{10}) $ $ e_1 $ $ p_2 $ $ e_2 $ $ p_3 $ $ e_4 $ $ p_{10} $ $ * $ $ * $ $ 3 $
     | Show Table
    DownLoad: CSV

    Table 2.  Paths from the root $ p_4 $ to leaves

    $ E_1(G) $ $ V_1(G) $ $ E_2(G) $ $ V_2(G) $ $ m_p $
    indices $ i_1 $ $ \iota_1 $ $ i_2 $ $ \iota_2 $
    $ P(p_4,p_1) $ $ e_3 $ $ p_3 $ $ e_1 $ $ p_1 $ $ 2 $
    $ P(p_4,p_2) $ $ e_3 $ $ p_3 $ $ e_2 $ $ p_2 $ $ 2 $
    $ P(p_4,p_5) $ $ e_4 $ $ p_6 $ $ e_6 $ $ p_5 $ $ 2 $
    $ P(p_4,p_7) $ $ e_4 $ $ p_6 $ $ e_5 $ $ p_7 $ $ 2 $
     | Show Table
    DownLoad: CSV
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