The Observability Of Schr(?)dinger Equation For Discrete System

In recent years,the control system of Schr(?)dinger equation has been widely used in various fields.With the rapid development of modern information technology,more and more scholars study the discrete system of Schrodinger equation.This paper focuses on the one-dimensional Schr(?)dinger equation and its observability for discrete systems.Firstly,the exponential stability of one-dimensional Schr(?)dinger equation continuous system is studied.By constructing Lyapunov function,Cauchy inequality,Poincaré inequality and Gronwall lemma are used to obtain the exponential stability of the system,so that the system is considerable.In addition,we also verify the observability inequality by constructing a new Lyapunov function,and directly judge the observability of the system through the observability inequality.Then the discrete system with Schr(?)dinger equation is studied by finite difference method,and the exponential stability of the discrete system is obtained by multiplier method and Lyapunov function,so the observability of the discrete system is obtained.Finally,the observability inequality of the discrete system is verified.The main structure of this paper is as follows.The first chapter is the introduction part,which introduces the research content of Schr(?)dinger equation in recent years and the research background of this paper.The second chapter is the preparatory knowledge,introduces the control system related theory and the related inequalities used in this paper.The third chapter is the main content of this paper,which studies the observability of the Schr(?)dinger equation governing system and its discrete system.The finite difference method is also used to discretization system,and the energy of the discrete system is decreasing.The fourth chapter is the summary and prospect of this paper.