We discuss exponential asymptotic property of the solution of a parallel repairable system with warm standby under commoncause failure. This system can be described by a group of partial differential equations with integral boundary. First we show that the positive contraction C 0 -semigroup T (t) [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm standby under common-cause failure, Acta Anal. Funct. Appl. 8 (1) (2006) 5-20] which is generated by the operator corresponding to these equations is a quasi-compact operator. Then by using [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm standby under common-cause failure, Acta Anal. Funct. Appl. 8 (1) (2006) 5-20] that 0 is an eigenvalue of the operator with algebraic index one and the C 0 -semigroup T (t) is contraction, we conclude that the spectral bound of the operator is zero. By using the above results the exponential asymptotical stability of the time-dependent solution of the system follows easily.
Shen et al. REF explored exponential asymptotic property of a parallel repairable system with common cause failure.
Exponential asymptotic property of a parallel repairable system with warm standby under common-cause failure
Journal of Mathematical Analysis and Applications
Mathematics