Exponential Stability Analysis of a Long Chain of Coupled Vibrating Strings with Dissipative Linkage

• Published in: SIAM journal on applied mathematics Vol. 49; no. 6; pp. 1694 - 1707
• Main Authors: Liu, Kang-Sheng, Huang, Fa-Lun, Chen, Goong
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.12.1989
• Subjects: 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; Wave equation; Vibrating string; Continuum mechanics; Exponential stability; Hyperbolic equation
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/0149102

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.