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Well-posedness and stability of the repairable system with \(N\) failure modes and one standby unit. (English) Zbl 1207.35065

Summary: The well-posedness and stability of the repairable system with \(N\) failure modes and one standby unit were discussed by applying the \(c_0\) semigroups theory of function analysis. Firstly, the integro-differential equations described the system were transformed into some abstract Cauchy problem of Banach space. Secondly, the system operator generates positive contractive \(c_0\) semigroups \(T(t)\) and so the well-posedness of the system was obtained. Finally, the spectral distribution of the system operator was analyzed. It was proven that zero is strictly dominant eigenvalue of the system operator and the dynamic solution of the system converges to the steady-state solution. The steady-state solution was shown to be the eigenvector of the system operator corresponding to the eigenvalue zero. At the same time the dynamic solution exponentially converges to the steady-state solution.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35R09 Integro-partial differential equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
47D06 One-parameter semigroups and linear evolution equations
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References:

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