Abstract: A kind of single-module repairable system with warning state was investigated， which can be expressed as a set of differential equations. By selecting the appropriate state space and the definition domain of the system operator， the equations were transformed into an abstract Cauchy problem in a Banach space. With the functional analysis method and linear operator semigroup theory， the eigenvalue 0 was proved to be the strictly dominant simple eigenvalue of the system. It shows the solution of the system is asymptotically stable. Then by studying the spectrum distribution of the main operator of the system， it was proved that the essential spectrum bound of the main operator of the system is negative. Finally， the variation of essential spectrum bound of the main operator under compact perturbation were discussed. The results demonstrate that the dynamic solution of the system is exponentially stable.

Key words: C₀ semigroup, exponential stability, essential spectrum bound, compact operator, perturbation