Analysis Of Stability Of System And Control

The purpose of this paper is to solve the stability of dissipative system and control of vibrating system.For the dissipative system, we discuss the existence and uniqueness of the solution of the k-out-of-N:G redundant system with repair and multiple critical and non-critical errors given by Chung. The behavior of the system is govened by partial differential equations coupled in equation and its boundary. We shall solve problem of the existence and uniqueness of solution of the complex system with method of functional analysis. It gives rise to Cauchy problem, and then, we may handle this problem using the semigroup theory. It is too complicated to verify the condition of Yosida theorem. Here, we use the conjugate method to prove the existence and uniqueness of solution of the system, which is to prove that operator A of system is dissipative and 1 ∈ ρ(A) by showing that 1 is not the eigenvalue of the conjugate operator A* recursively. We prove 0 is the simple eigenvalue of A and approximating each term shows that there is no spectrum of A when solving the stability of the system.For the vibrating system, we study the string system and our only job is to make an improvement based on what have been done. Before, people only needed to design the control to make sure that the closed loop system decays exponentially. Here, we also have to guarantee that the rate at which the system decays exponentially is less than the given 7. We may choose the appropriate feedback to make the asymptote Re λ = β of the system less than γ, and then, there exist finitely many eigenvalues of the system. Then we can use the second control to move these points in this half plane Re λ ≥γ to the desired position in the plane Reλ＜ γ. We basically apply the property of Reisz basis of eigenvectors to the idea described above.Even though the purpose of this paper is to study two special systems, this method can be generalized in the study of other models.