Semi-discrete Uniform Exponential Stability Approximation For Two Kinds Of PDE-ODE Boundary Coupled Systems

In recent years,the PDE-ODE boundary coupling system has been a worthwhile research topic,and its theory and research methods have penetrated into other disciplinary fields Due to the difficulty in obtaining accurate solutions for such equations in practical applications,the study of numerical approximation methods for PDE-ODE boundary coupled systems is crucial This article studies the spatial semi discrete uniform stability approximation of two types of PDE-ODE boundary coupled systems,mainly including the following two research contents:Firstly,the uniform exponential stability approximation of the coupling system of transport equation and ordinary differential equation is studied.The Lyapunov function is constructed to study the exponential stability of the coupled system,and then a standard spatial semi-discrete finite difference scheme is constructed for the equivalent target system,and the eigenvalue problem of the discrete system is studied.Then the discrete of Lyapunov function is further studied to prove the uniform exponential stability of the equivalent semi-discrete target system,and then the uniform exponential stability approximation of the coupled system with control is obtained.Secondly,the uniform exponential stability approximation of heat equations and ordinary differential equations is studied.By constructing a standard spatial semi discrete finite difference scheme for the target system,when studying discrete Lyapunov functions,redundant terms will be generated during the calculation process.In order to better study the reduced order method,we use the reduced order method to convert the discrete form coupled system into a reduced order equivalent system,and then further study the discrete form Lyapunov function for the reduced order equivalent system.The reduced order method can effectively eliminate redundant terms,This makes it easier to prove the Lyapunov function of discrete systems.Finally,numerical experiments further illustrate the stability of these two kinds of PDEODE boundary coupled systems.