具有对称初始数据的二维反应扩散方程的边界镇定

齐洁; 齐金鹏

doi:10.16383/j.aas.2015.c140108

具有对称初始数据的二维反应扩散方程的边界镇定

doi: 10.16383/j.aas.2015.c140108

齐洁^1,2,
齐金鹏^1,2, ,

1.
东华大学信息科学与技术学院上海 201620;
2.
东华大学数字化纺织服装技术教育部工程研究中心上海 201620

基金项目:

国家自然科学基金重点项目(61134009);国家自然科学基金(61104154);中央高校基本科研业务费专项资金资助

详细信息

作者简介:
齐洁东华大学信息科学与技术学院副教授.主要研究方向为分布式参数系统控制,多智能体协同控制与优化.E-mail:jieqi@dhu.edu.cn

通讯作者:
齐金鹏东华大学信息科学与技术学院副教授.主要研究方向为系统分析与建模,分布式参数系统,数据分析与智能算法.本文通信作者.E-mail:qipengkai@126.com

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- 被引次数: 0
出版历程
- 收稿日期: 2014-02-21
- 修回日期: 2014-06-30
- 刊出日期: 2015-01-20

Boundary Stabilization for a 2-D Reaction-diffusion Equation with Symmetrical Initial Data

QI Jie^1,2,
QI Jin-Peng^{1,2
, ,}

1.
School of Information Science and Technology, Donghua University, Shanghai 201620;
2.
Engineering Research Center of Digitized Textile and Fashion Technology, Ministry of Education, Donghua University, Shanghai 201620

Funds:

Supported by National Natural Science Foundation Key Program of China (61134009), National Natural Science Foundation of China (61104154), and Fundamental Research Funds for the Central Universities

摘要

摘要: 研究了二维圆盘上具有对称初始数据的反应扩散方程的边界控制. 由于初始条件和边界条件关于圆心旋转对称, 系统可以转化为等价的极坐标系下的一维抛物方程. 此时, 极点的奇异性成为了控制器设计中的难点. 本文设计了一系列方程变换, 消除了核函数方程中极点奇异性的影响, 将其转化为修正的Bessel方程, 求出了显式的核函数表达式和精确的边界反馈控制律, 扩展了偏微分方程的backstepping方法. 系统的收敛速度可通过改变控制器中的一个参数来调节. 然后用Lyapunov函数法证明了闭环系统在H1范数下指数稳定, 表明了系统对初值的连续依赖. 最后用数值仿真验证了方法的有效性.
- 边界控制 /
- 镇定 /
- 反步法 /
- 扩散反应方程 /
- 李雅普诺夫函数
Abstract: This paper designs a boundary controller for a 2-D reaction-diffusion equation with symmetrical initial data. The system can be transformed to an equivalent 1-D parabolic equation in the polar coordinates due to the circle rotationally symmetric initial and boundary conditions. The singularity of the pole brings a great challenge to the design. This paper designs a series function transformations to destroy the impact of the singularity of the pole on the kernel function. Finally, the kernel function is transformed to a modified Bessel function, which produces an explicit kernel as well as an explicit boundary control law. The design process expands the existing backstepping methods for partial differential equations. Also, the convergence rate can be changed by adjusting a parameter in the controller. The exponential stability in H1 norm is proved, which implies that the system is continuous in terms of the initial conditions. The effectiveness of the method is illustrated with numerical simulations.
- Boundary control /
- stabilization /
- backstepping /
- advection-diffusion equation /
- Lyapunov function

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参考文献(22)

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具有对称初始数据的二维反应扩散方程的边界镇定

doi: 10.16383/j.aas.2015.c140108

作者简介:
齐洁东华大学信息科学与技术学院副教授.主要研究方向为分布式参数系统控制,多智能体协同控制与优化.E-mail:jieqi@dhu.edu.cn

通讯作者:
齐金鹏东华大学信息科学与技术学院副教授.主要研究方向为系统分析与建模,分布式参数系统,数据分析与智能算法.本文通信作者.E-mail:qipengkai@126.com

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Boundary Stabilization for a 2-D Reaction-diffusion Equation with Symmetrical Initial Data

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具有对称初始数据的二维反应扩散方程的边界镇定

doi: 10.16383/j.aas.2015.c140108

作者简介: 齐洁 东华大学信息科学与技术学院副教授.主要研究方向为分布式参数系统控制,多智能体协同控制与优化.E-mail:jieqi@dhu.edu.cn

通讯作者: 齐金鹏 东华大学信息科学与技术学院副教授.主要研究方向为系统分析与建模,分布式参数系统,数据分析与智能算法.本文通信作者.E-mail:qipengkai@126.com

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出版历程

Boundary Stabilization for a 2-D Reaction-diffusion Equation with Symmetrical Initial Data

计量

出版历程

目录

作者简介:
齐洁东华大学信息科学与技术学院副教授.主要研究方向为分布式参数系统控制,多智能体协同控制与优化.E-mail:jieqi@dhu.edu.cn

通讯作者:
齐金鹏东华大学信息科学与技术学院副教授.主要研究方向为系统分析与建模,分布式参数系统,数据分析与智能算法.本文通信作者.E-mail:qipengkai@126.com