Eigenvalues And Eigenvectors Of The Semi Discrete Wave Equation

From the beginning of this century,the numerical calculation of distributed parameter control system is favored by many researchers.In this process,the researchers are two steps to complete the numerical calculation,the first semi discretization of the spatial variables,and then the discretization of the time variable.It is an important foundation for the follow-up work to maintain the consistency of the various performance of the control system.Based on this,the researchers study the uniform exponential stability,uniform exact observability and exact controllability of distributed parameter control systems.It is an important step to obtain and analyze the eigenvalues and eigenvectors of the semi discrete system in the study of the consistency of the above three kinds of performance.In this paper,we give the methods for the eigenvalue and eigenvector of a variety of semi discrete wave equation.The key idea of this method is to solve the eigenvalue problem of the discrete system.The model is focused on one-dimensional wave equation and wave network system,and the spatial discretization method includes the linear element and finite difference.All these factors lead to the change of eigenvalues and eigenvectors.The structure of this paper is as follows: the first chapter is the introduction.The second chapter is the preliminary knowledge.It introduces the definitions,theorems,properties and lemmas used in the study of the semi discrete wave equation.In the third chapter,according to the mathematical model of the semi discrete wave equation,we obtain the characteristic value and characteristic vector.The fourth chapter summarizes the application of the equation.