In this Paper, Combining the Nonmonotone and Monotone Line Search ,a Spectral Conjugate Gradient Methods Are Used in this Paper. by the Concept of Spectral, we Construct a Class of Spectral Conjugate Gradient with Different form . we Also Give the Algorithm for Conjugate Gradient Method. the Numerical Result Show the Efficiency of the Method.

The repairable system solution’s exponential asymptotic stability was discussed in this paper, First we prove that the positive contraction strongly continuous semigroup which is generated by the operator corresponding to these equations describing a system with two identical components is a quasi-compact operator. Following the result that 0 is an eigenvalue of the operator with algebraic index one and the strongly continuous semi-group is contraction, we deduce that the spectral bound of the operator is zero. By the above results we obtain easily the exponential asymptotic stability of the solution of the repairable system.