Optimal Control Of A Kind Of Parallel Repairable System With Warm Standby

First,by choosing state space X,defining norm,defining major operator B and do-main D(B)and operator of system B+C in space X.We transfer the equations of the parallel repairable system with warm standby into an abstract Cauchy problem in Banach space X.Second,by using resolvent positive operator theory and cofinal theory to proof the properties of the major operator B,we prove the major operator is a densely defined resolvent positive operator,the positive cone of domain of major operator's adjoint op-erator is cofinal in positive cone of conjugate space.Then we got the spectral bound of major operator to boundary parameter.The following proof system operator B + C also have the same nature.We estimate the spectral bound and growth bound of the major operator and system operator.At last,we prove the parallel repairable system with warm standby of time depen-dent solution for the exponential stability.We also prove the existence and uniqueness of the optimal control element.