International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 4, Pages 631-643
doi:10.1155/IJMMS.2005.631
Asymptotic stability of a repairable system with imperfect switching mechanism
Houbao Xu1
, Weihua Guo2
, Jingyuan Yu3
and Guangtian Zhu4
1Department of Mathematics, Beijing Institute of Technology, 16 Fucheng Road, Beijing 100037, China
2Department of Information and Computing Science, Zhengzhou Institute of Light Industry, Henan 450002, China
3Department of System Engineering, The 710 Institute, 16 Fucheng Road, Beijing 100037, China
4Academy of Mathematics and System Science, Chinese Academy of Science, Beijing 100080, China
Abstract
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.