International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 4, Pages 631-643

Asymptotic stability of a repairable system with imperfect switching mechanism

Houbao Xu,1,2 Weihua Guo,3 Jingyuan Yu,2 and Guangtian Zhu4

1Department of Mathematics, Beijing Institute of Technology, 16 Fucheng Road, Beijing 100037, China
2Department of System Engineering, The 710 Institute, 16 Fucheng Road, Beijing 100037, China
3Department of Information and Computing Science, Zhengzhou Institute of Light Industry, Henan 450002, China
4Academy of Mathematics and System Science, Chinese Academy of Science, Beijing 100080, China

Received 20 February 2004; Revised 28 May 2004

Copyright © 2005 Houbao Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper studies the asymptotic stability of a repairable system with repair time of failed system that follows arbitrary distribution. We show that the system operator generates a positive C0-semigroup of contraction in a Banach space, therefore there exists a unique, nonnegative, and time-dependant solution. By analyzing the spectrum of system operator, we deduce that all spectra lie in the left half-plane and 0 is the unique spectral point on imaginary axis. As a result, the time-dependant solution converges to the eigenvector of system operator corresponding to eigenvalue 0.