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SPECTRAL ANALYSIS AND STABILIZATION OF A COUPLED WAVE-ODE SYSTEM

ZHAO Dongxia 1, WANG Junmin2   

  1. 1.Department of Mathematics, North University of China, Taiyuan 030051, China; School of Mathematics, Beijing Institute of Technology, Beijing 100081, China;2.School of Mathematics, Beijing Institute of Technology, Beijing 100081, China.
  • Received:2012-10-15 Online:2014-06-25 Published:2014-08-19
  • Supported by:

    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.

ZHAO Dongxia , WANG Junmin. SPECTRAL ANALYSIS AND STABILIZATION OF A COUPLED WAVE-ODE SYSTEM[J]. Journal of Systems Science and Complexity, 2014, 27(3): 463-475.

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 −1 d , and there are two branches of the asymptotic eigenvalues: The first branch is accumulating towards −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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