The Uniform Exponential Stability Of The Order Reduction Finite Difference Approach Of Wave Equation With Dynamical Boundary Damping

Because the uniform exponential stabilities with respect to the discretized parameter play key roles in the computing of optimal control and the inverse problem of observability,they were broadly and intensively discussed.It is well known that,for the continuous wave equation,it is exponentially stable.If the continuous system is discretized in spacial variable by finite difference method,the numerical scheme yields spurious high frequency oscillations which induce the deficiency of the uniform exponential stability.To restore the uniform qualitative behaviors,researchers introduced the methods of vanishing viscosity terms and filtering.However,there are rare results on the uniform exponential stability of the wave equation with dynamical boundary condition.In this paper,the reduced order difference method is used to semi discretize the system for the equations with dynamic boundary damping.The main work is as follows:In the second chapter,the reduced order difference method is introduced firstly.The so-called reduced order difference method is to reduce the order of time and space variables by introducing appropriate intermediate variables,and then discretize the space variables by using the finite difference method to get the discretization form of the system.Secondly,the theory of Linear Operator Semigroups on Hilbert space is introduced by introducing some operators and the properties of semigroups they generate.In the third chapter,we first introduce the state space and the inner product of the continuous system,then write the continuous system into the form of its equivalent abstract differential equation,then prove the existence and uniqueness of the solution of theequation by the theory of operator semigroup,and then introduce the appropriate Lyapunov function and energy multiplier to verify the exponential stability of the continuous system.Finally,the proof of continuous system is extended to discrete system and error system in discrete process.