| 引用本文: | 王丽梅,郭宝珠.非线性波动方程半离散格式的一致指数稳定性[J].控制理论与应用,2026,43(3):451~459.[点击复制] |
| WANG Li-mei,GUO Bao-zhu.On uniform exponential stability of semi-discrete scheme for nonlinear wave equation[J].Control Theory & Applications,2026,43(3):451~459.[点击复制] |
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| 非线性波动方程半离散格式的一致指数稳定性 |
| On uniform exponential stability of semi-discrete scheme for nonlinear wave equation |
| 摘要点击 839 全文点击 145 投稿时间：2024-03-20 修订日期：2025-10-29 |
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| DOI编号 10.7641/CTA.2024.40160 |
| 2026,43(3):451-459 |
| 中文关键词 波动方程 半离散有限差分格式 能量乘子法 降阶法 一致指数稳定性 |
| 英文关键词 wave equation semi-discrete finite difference scheme energy multiplier method order reduction method uniform exponential stability |
| 基金项目 国家自然科学基金项目(12131008)资助. |
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| 中文摘要 |
| 本文讨论一维非线性波动方程半离散有限差分格式的一致指数稳定性.首先,利用能量乘子法证明了偏微
分方程描述的连续系统的指数稳定性.引入辅助变量,利用降阶法将原系统转化为奇异偏微分方程(PDE)系统;再
用有限差分法对空间变量离散,在消除引入的辅助变量后,得到原系统的半离散有限差分格式;最后,平行于连续系
统, 利用能量乘子法证明了离散系统的一致指数稳定性,并通过数值模拟进行验证. |
| 英文摘要 |
| This paper investigates the uniform exponential stability of semi-discrete finite difference schemes applied to
one-dimensional nonlinear wave equations. Firstly, the energy multiplier method is employed to establish the exponential
stability of the continuous system governed by the partial differential equation (PDE). This involves introducing auxiliary
variables and employing the reduction technique to convert the original system into a singular PDE system. Subsequently,
the spatial variable is discretized using the finite difference method, and upon eliminating the auxiliary variables, the semi
discrete finite difference scheme for the original system is derived. Finally, mirroring the approach for the continuous
system, the energy multiplier method is utilized to prove the uniform exponential stability of the discrete system, which is
further validated through numerical simulations. |