Well-posedness and Stability of the Repairable System with Three Units and Vacation

Han Xiaoshuang (), Teng Mingyan () and Fang Ming ()
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Han Xiaoshuang: College of Science & Technology, Yanbian University, Yanji133002, China
Teng Mingyan: Department of Mathematics, Bohai University, Jinzhou121002, China
Fang Ming: Department of Mathematics, Science College, Yanbian University, Yanji133002, China

Journal of Systems Science and Information, 2014, vol. 2, issue 1, 54-76

Abstract: The stability of the repairable system with three units and vacation was investigated by two different methods in this note. The repairable system is described by a set of ordinary differential equation coupled with partial differential equations with initial values and integral boundaries. To apply the theory of positive operator semigroups to discuss the repairable system, the system equations were transformed into an abstract Cauchy problem on some Banach lattice. The system equations have a unique non-negative dynamic solution and positive steady-state solution and dynamic solution strongly converges to steady-state solution were shown on the basis of the detailed spectral analysis of the system operator. Furthermore, the Cesáro mean ergodicity of the semigroup T(t) generated by the system operator was also shown through the irreducibility of the semigroup.

Keywords: repairable system; positive operator semigroups; spectral analysis; stability (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:bpj:jossai:v:2:y:2014:i:1:p:54-76:n:6

DOI: 10.1515/JSSI-2014-0054

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