Article Contents
Article Contents

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

• * Corresponding author: Dongyi Liu

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

• 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:

• 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 $i_1$ $\iota_1$ $i_2$ $\iota_2$ $i_3$ $\iota_3$ $i_4$ $\iota_4$ $P(p_1,p_6)$ $e_1$ $p_2$ $e_3$ $p_6$ $*$ $*$ $*$ $*$ $2$ $P(p_1,p_7)$ $e_1$ $p_2$ $e_2$ $p_3$ $e_5$ $p_5$ $e_9$ $p_7$ $4$ $P(p_1,p_8)$ $e_1$ $p_2$ $e_2$ $p_3$ $e_5$ $p_5$ $e_8$ $p_8$ $4$ $P(p_1,p_9)$ $e_1$ $p_2$ $e_2$ $p_3$ $e_6$ $p_4$ $e_7$ $p_9$ $4$ $P(p_1,p_{10})$ $e_1$ $p_2$ $e_2$ $p_3$ $e_4$ $p_{10}$ $*$ $*$ $3$

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 $i_1$ $\iota_1$ $i_2$ $\iota_2$ $P(p_4,p_1)$ $e_3$ $p_3$ $e_1$ $p_1$ $2$ $P(p_4,p_2)$ $e_3$ $p_3$ $e_2$ $p_2$ $2$ $P(p_4,p_5)$ $e_4$ $p_6$ $e_6$ $p_5$ $2$ $P(p_4,p_7)$ $e_4$ $p_6$ $e_5$ $p_7$ $2$
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