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

Dongyi Liu; Genqi Xu; Dongyi Liu; Genqi Xu

doi:10.3934/nhm.2022007

Networks and Heterogeneous Media

2022, Volume 17, Issue 4: 519-545. doi: 10.3934/nhm.2022007

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

Dongyi Liu ^,,
Genqi Xu

School of Mathematics, Tianjin University, Tianjin 300354, China

Received: 01 August 2021 Revised: 01 January 2022 Published: 21 March 2022
Primary: 93C20, 93D05; Secondary: 35B35, 35L05

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.
- Networks of strings,
- Input-output well-posedness,
- Regularity,
- Exponential stability,
- Multiplier method,
- Lyapunov functional
Citation: Dongyi Liu, Genqi Xu. Input-output $ L^2 $-well-posedness, regularity and Lyapunov stability of string equations on networks[J]. Networks and Heterogeneous Media, 2022, 17(4): 519-545. doi: 10.3934/nhm.2022007

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Abstract

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.

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