ODE-波动方程耦合系统半离散降阶有限差分格式的一致指数稳定性

Uniformly Exponential Stability for the Semi-Discrete Order Reduction Finite Difference Schemes of ODE-Wave Equation Coupled System

摘要: 本文研究了由线性常微分方程和一维波动方程耦合系统的一致指数稳定性，对其构建了一个半离散有限差分格式，并进一步研究了离散格式的一致指数稳定性。首先采用Backstepping控制法将初始耦合系统转化为一个等价的目标系统，同时推导出使原耦合系统稳定的控制律。接着为目标系统构造适当的Lyapunov函数，采用Lyapunov稳定性分析法证明目标系统的一致指数稳定性。为了更方便地研究目标系统对应的空间半离散有限差分格式，先通过变量代换将目标系统简化为与之等价的低阶耦合系统，并针对该低阶系统构造空间半离散有限差分格式，设计相应的Lyapunov函数，证明了半离散系统的一致指数稳定性，从而表明所提出的基于空间半离散的有限差分格式能够有效保持原连续耦合系统的指数稳定特性。

Abstract: This paper analyzes the uniform exponential stability of a coupled system consisting of linear ordinary differential equations and a one-dimensional wave equation. A semi-discrete finite difference scheme is constructed for this system, and the uniform exponential stability of the discrete scheme is further studied. Firstly, the Backstepping control method is employed to transform the original coupled system into an equivalent target system, while deriving the control law that stabilizes the original coupled system. Subsequently, an appropriate Lyapunov function is constructed for the target system, and the Lyapunov stability analysis method is used to prove the uniform exponential stability of the target system. In order to more conveniently study the spatially semi-discrete finite difference scheme corresponding to the target system, the target system is first simplified into an equivalent lower-order coupled system through variable substitution. A spatially semi-discrete finite difference scheme is then constructed for this lower-order system, and a corresponding Lyapunov function is designed to prove the uniform exponential stability of the semi-discrete system. This demonstrates that the proposed spatially semi-discrete finite difference scheme effectively preserves the exponential stability characteristics of the original continuous coupled system.