Abstract
This paper discusses the asymptotic stability and Riesz basis generation for a general tree-shaped network of vibrating strings. All exterior vertices are assumed to be fixed and interior vertices are imposed linear damping feedbacks. This paper shows that the system is well-posed and asymptotically stable by C 0-semigroup theory. With some additional conditions, the spectrum of the system is shown to be located in a strip that is parallel to the imaginary axis and the set of all generalized eigenfunctions is completed in the state space. These lead to the conclusion that there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis with parenthesis for the state space.
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This research is supported in part by the Natural Science Foundation of China under Grant No. 60874035 and by the Scientific Research Initiation Foundation of Civil Aviation University of China (08QD09X).
This paper was recommended for publication by Editor Dexing FENG.
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Guo, Y., Xu, G. Asymptotic stability and Riesz basis property for tree-shaped network of strings. J Syst Sci Complex 24, 225–252 (2011). https://doi.org/10.1007/s11424-010-8062-4
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DOI: https://doi.org/10.1007/s11424-010-8062-4