Research On The Repairable System With Fragile Units Under Common Failure

In recent years, Theoretical Study of the repairable system is concerned by more and more scholars. The repairable system is a important kind of systems, the state, which the system is in, is a random process. By using the probability transfer matrix, we can build the system state distribution of differential-integral equations. When the repair rate of the system changes over the time, the system is called distributed parameter systems. In this paper,we use the functional analysis as tool to research the system posedness, stability, and semi-discretization, the main work is as follow:(1)To study the existence and uniqueness of the system.Transform the differential-integral equation into Volterra equation form inBanach space.Research the existence and uniqueness of system solution by functional analysis theory.(2)To research the system stability and reliability. Through discuss the dis-tribution of system operators, we get that system operators locate in the left half plane in the complex plane. Imaginary axis has no other spectral points than 0. Eigenvectors,which are corresponding to the eigenvalues 0, are the stable solutions of system.When tâ†'∞, the system solution converges to stable asymptotic so-lution, we obtain the stability of solutions. Moreover, when the repair rate and failure rate are constant, study the monotonicity of po(t), further, we prove stability of system.(3)To discuss the optimal control problem for the system. By setting the norm index and control the variablesμj(x), makes the norm index minimum at the time t=T of the control variableμj(x) to research the optimal control problem.(4)To research the semi-discrete problem of system problem. With two ele-mentary step functions approximate toμj(x), obtain the the semi-discrete model for the system, and with the theory of the Co semi-group, prove that the feasibility of the algorithm for further numerical basis.