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Spectral analysis and stabilization of a coupled wave-ODE system

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Abstract

In this paper, an interconnected wave-ODE system with K-V damping in the wave equation and unknown parameters in the ODE is considered. It is found that the spectrum of the system operator is composed of two parts: Point spectrum and continuous spectrum. The continuous spectrum consists of an isolated point \(- \tfrac{1} {d}\), and there are two branches of the asymptotic eigenvalues: The first branch is accumulating towards \(- \tfrac{1} {d}\), and the other branch tends to −∞. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. As a consequence, the spectrum-determined growth condition and exponential stability of the system are concluded.

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Authors and Affiliations

  1. Department of Mathematics, North University of China, Taiyuan, 030051, China

    Dongxia Zhao

  2. School of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    Dongxia Zhao & Junmin Wang

Authors
  1. Dongxia Zhao
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  2. Junmin Wang
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Corresponding author

Correspondence to Dongxia Zhao.

Additional information

This research was supported by Shanxi Youth Foundation under Grant No. 2013021002-1 and the National Natural Science Foundation of China under Grant Nos. 61074049 and 61273130.

This paper was recommended for publication by Editor FENG Dexing.

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Zhao, D., Wang, J. Spectral analysis and stabilization of a coupled wave-ODE system. J Syst Sci Complex 27, 463–475 (2014). https://doi.org/10.1007/s11424-014-2219-5

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  • DOI: https://doi.org/10.1007/s11424-014-2219-5

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