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Stability analysis of a new kind series system. (English) Zbl 1193.93150
Summary: An \(n\)-unit series repairable system with a repairman doing other work is discussed in this paper. We are devoted to studying the unique existence of the solution and the exponential stability of the system. \(C_0\)-semigroup theory is used to prove the existence of a unique non-negative solution of the system. Then by analysing the spectra distribution of the system operator and the quasi-compactness of the system, we derive that the transient solution of the system exponentially converges to the non-negative steady-state solution, which is the eigenvector corresponding to eigenvalue 0 of the system operator. Some reliability indices of the system and the strategy of the repairman doing other work in his idle time are presented at the end of the paper.

93D20 Asymptotic stability in control theory
93B28 Operator-theoretic methods
93C25 Control/observation systems in abstract spaces
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