Exponential stability of a repairable system with imperfect switching mechanism

This paper is devoted to investigating the exponential stability of a kind of repairable system with cold standby units, imperfect switching mechanism and multiple non-critical and critical errors. We have known that the system operator generates a positive C₀-semigroup of contractions and will show its quasi-compactness and irreducibility. Further, 0 is a simple eigenvalue of the system operator. As a result, we obtain that the time-dependent solution of the system converges to the steady-state solution exponentially, which is the positive eigenfunction corresponding to the simple eigenvalue 0.