Exponential Stability Analysis of a Long Chain of Coupled Vibrating Strings with Dissipative Linkage
Consider a long chain of coupled vibrating strings, where a stabilizer is installed at each internal node and perhaps also at a boundary point. The exponential stability of the stabilizers' arrangement for this large dynamic structure will be determined. Through a careful transformation of the...
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Published in | SIAM journal on applied mathematics Vol. 49; no. 6; pp. 1694 - 1707 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Philadelphia, PA
Society for Industrial and Applied Mathematics
01.12.1989
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Abstract | Consider a long chain of coupled vibrating strings, where a stabilizer is installed at each internal node and perhaps also at a boundary point. The exponential stability of the stabilizers' arrangement for this large dynamic structure will be determined. Through a careful transformation of the coupled wave equations for this structure into an equivalent hyperbolic system and analysis of the eigendeterminant, it will be proven that the energy of the system decays uniformly exponentially if there is a stabilizer installed at a boundary point. If the stabilizers are installed only at internal nodes, it will be proven that the energy may decay either uniformly exponentially or nonuniformly, or may not decay at all, depending on the different wave speeds and the stabilizers' arrangement. All possible outcomes have been classified. |
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AbstractList | Consider a long chain of coupled vibrating strings, where a stabilizer is installed at each internal node and perhaps also at a boundary point. The exponential stability of the stabilizers' arrangement for this large dynamic structure will be determined. Through a careful transformation of the coupled wave equations for this structure into an equivalent hyperbolic system and analysis of the eigendeterminant, it will be proven that the energy of the system decays uniformly exponentially if there is a stabilizer installed at a boundary point. If the stabilizers are installed only at internal nodes, it will be proven that the energy may decay either uniformly exponentially or nonuniformly, or may not decay at all, depending on the different wave speeds and the stabilizers' arrangement. All possible outcomes have been classified. |
Author | Huang, Fa-Lun Chen, Goong Liu, Kang-Sheng |
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Keywords | Wave equation Vibrating string Continuum mechanics Exponential stability Hyperbolic equation |
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References | R2 R4 Marion J. B. (R9) 1965 Huang Fa Lun (R6) 1983; 28 Kuipers L. (R7) 1974 R8 Huang Fa Lun (R5) 1985; 1 Pazy Amnon (R10) 1974 Quinn John P. (R11) 1977; 77 R1 |
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Snippet | Consider a long chain of coupled vibrating strings, where a stabilizer is installed at each internal node and perhaps also at a boundary point. The exponential... |
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SubjectTerms | Applied mathematics Boundary conditions Boundary points Classical and quantum physics: mechanics and fields Classical mechanics of continuous media: general mathematical aspects Energy Exact sciences and technology Hilbert space Hilbert spaces Mechanical stabilizers Ordinary differential equations Partial differential equations Physics Wave equations |
Title | Exponential Stability Analysis of a Long Chain of Coupled Vibrating Strings with Dissipative Linkage |
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