Well-Posedness of an N-Unit Series System with Finite Number of Vacations

Abstract

We investigate the solution of an N-unit series system with finite number of vacations. By using -semigroup theory of linear operators, we prove well-posedness and the existence of the unique positive dynamic solution of the system.

Asymptotic Stability of the Dynamic Solution of an N-Unit Series System with Finite Number of Vacations

“…In [5], the authors transformed the system (1), (2) and 3into the following abstract Cauchy problem ( [6], Def.II.6.1) on the Banach space ( )…”

“…But they did not prove the existence of the dynamic solution and the asymptotic stability of the dynamic solution. Motivated by this, A. Osman and A. Haji proved in [5] the existence of a unique positive dynamic solution of the system by using C 0 -semigroup theory of linear operators. In this paper, we further study this system and prove that the dynamic solution of the system converges strongly to its steady state solution by analyzing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator; thus we obtain the asymptotic stability of the dynamic solution of this system.…”

“…The rest of this paper is organized as follows: In Section 2, we present the mathematical model of the system and give some results obtained in [5]. In Section 3, we obtain main result on stability of the system by analyzing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator.…”