Stability Analysis For A Cold System With Single Vacation And A Timoshenko Beam

The rise and development of reliability theory are due to the development of scienceand technology. As a result of the development of the reliability theory, Reliability Math-ematics which is constituted by mathematical modeling and model analysis is becomingmore and more popular in academic research. On the other hand, the mathematical modelprovides convenience for the analysis of system reliability indices. The theoretical analysisand numerical calculation of the mathematical model provides the basic tools for reliabilityanalysis.We consider two systems in this paper. Firstly, we study the system which consistsof two dissimilar components and one repairman, in which the component1is the majorworking unit and the component2is the cold stand-by unit, the component1has priori-ty in use and repair; the repairman takes a single vacation. Under these assumptions, wederive a model of diferential-integral equations. Then the diferential-integral equationsare transformed into some abstract Cauchy problem on some suitable state space. Further-more, using functional analysis method, we discuss the dynamic asymptotical behavior ofthe system and get some reliability indices, such as availability and the failure frequency.Finally we analyze the variety of the indices of reliability and the profit of the system withthe change of the other parameters of the system by numerical simulation. Then, we mainlydiscuss the stabilization of a Timoshenko beam system with delay in the boundary control.In the case of single beam, some properties (such as well-posedness, asymptotic stability,exponential stability) were usually discussed by using multiplier method which is still animportant way to study the system stability. Under this case the eigenvalues usually can becalculated and the properties of them are simple. However, for serially connected system, itis difcult to find a multiplier for the system. The eigenvalues of system are very complexand usually can not be obtained. So another way should be found to solve this problem.In the present paper, we introduce a new control strategy for the Timoshenko beam. Andthis control strategy makes the system stable exponentially in the more general conditions.Moreover, the control strategy in the present paper can be used in other low-dimensionalmodels.