Abstract
The exponential stability of a multi-state device is discussed in this paper. We present that the C 0-semigroup generated by the system operator is quasi-compact and irreducible. It is known that 0 is a simple eigenvalue of the system operator. In combination with this, we obtain that the time-dependent solution exponentially converges to the steady-state solution, which is the positive eigenfuction corresponding to the simple eigenvalue 0.
Similar content being viewed by others
References
W. K. Chung, A reparable multistate device with arbitrarily distributed times, Micro. Reliab., 1981, 21(2): 255–256.
H. B. Xu, J. Y. Yu, and G. T. Zhu, Asymptotic property of a reparable multi-state device, Quarterly of Applied Mathematics, 2005, 63(4): 779–789.
G. Gupur, X. Z. Li, and G. T. Zhu, Functional Analysis Method in Queueing Theory, Research Information Ltd, Hertfordshire, United Kingdom, 2001.
R. Nagel, One-Parameter Semigroup of Positive Operator, Lecture Notes in Mathematics, Springer-Verlag, New York, 1986.
Ph. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroup, North-Holland, Amsterdam·New York·Oxford·Tokyo, 1987.
G. Gupur and B. Z. Guo, Asymptotic property of the time-dependent solution of a reliability model, Journal of System Science and Complexity, 2005, 18(3): 319–339.
G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research is supported by Beijing Institute of Technology Foundation under Grant No. 20060742011.
Rights and permissions
About this article
Cite this article
Hu, W., Xu, H., Yu, J. et al. Exponential Stability of a Reparable Multi-State Device. Jrl Syst Sci & Complex 20, 437–443 (2007). https://doi.org/10.1007/s11424-007-9039-9
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11424-007-9039-9