The Uniformly Exponential Stability Of The Finite Element Semi-discretization Of The 1-D Damped Wave Equation

One dimensional damped wave equation is one of the main objects of the distributed parameter control system research,its exponential stability,exact observability and exact controllability,and their relations have been extensively and intensively studied.In order to facilitate calculation and engineering implementation,since the end of last century,some scholars have studied the controllability,observability and exponential stability of the system from the point of view of numerical analysis,that is the question of whether the semi-discretization numerical approximation system can keep the controllability,observability and stability of the continuous system uniformly.Under such a background,in this paper,the one-dimensional damped wave equation is discretized in finite element space,the exponential decay of the energy of solutions of discrete systems is studied.First of all,through the analysis of discrete systems and their conjugate systems,it is pointed out that the exponential stability of the original discrete system is equivalent to the exact observability of the conjugate system.Secondly,the eigenvalues of the conjugate system are analyzed and the boundary observations of the corresponding eigenvectors are carried out,the discrete system is not uniformly exponential stable.Finally,adding numerical viscosity to discrete systems,the multiplier method is used to further prove that the system has uniform exponential stability.Structure of this paper is as follows: the first chapter is the introduction part,introduces the research background and development of this paper.The second chapter is preliminary knowledge,this paper introduces the exponential stability of one dimensional damped wave equation is equivalent to the exact observability,and also introduces the finite difference and the finite element space semi-discretization of the one dimensional wave equation.In the third chapter,the mathematical model of one dimensional damped wave equation system is established according to the semi discretization of the finite element method,furthermore,we study the conditions of uniform exponential stability and non-uniform exponential stability of the system.Finally,summarize and prospect the development of this paper.