Uniform Polynomial Stability Of Wave Equations With Dynamic Boundary Control

One dimensional obstacle wave equation is one of the main research objects of second order hyperbolic system.Its exponential stability,precise observability,precise controllability and the relationship between them have been studied extensively and deeply.However,in order to facilitate the calculation and engineering realization,from the end of last century,some scholars have studied the controllability,observability and exponential stability of the system from the perspective of numerical analysis,that is,whether the semi discrete numerical approximation system can keep the controllability,observability and stability of the continuous system uniformly.It has important theoretical significance and practical value for its research.Although many scholars have made some progress in the uniform controllability,observability and exponential stability of wave equations,the study of uniform stability is limited to the study of uniform exponential stability,and the study of uniform polynomial stability is rare.In addition,the methods to verify the uniform exponential stability are multiplier method,Lyapunov function method and so on.In this paper,a new verification method is proposed to study the uniform polynomial stability of one-dimensional obstacle wave equation.Due to the semi discretization of space variables by the finite element method,it will be very difficult to use the multiplier method and Lyapunov function method to verify.Firstly,for the wave equation with dynamic boundary control,the spatial variables are discretized by the finite element method to obtain the discretized system;secondly,the discretized system is transformed into a family of abstract Cauchy problems in the state space by selecting appropriate inner product,state space and operators;finally,the dissipativity and spectral division of the system operators are verified by using the theory of operator semigroup The uniform polynomial stability of the discretized system is obtained by analyzing and estimating the boundlessness of its resolvent on the virtual axis.