一类带漂移-扩散项的非线性抛物型方程组的能控性

数学物理学报 ›› 2023, Vol. 43 ›› Issue (6): 1744-1758.

一类带漂移-扩散项的非线性抛物型方程组的能控性

张志朋(),张亮^*()

武汉理工大学理学院数学系武汉 430070

收稿日期:2023-03-09 修回日期:2023-08-16 出版日期:2023-12-26 发布日期:2023-11-16
通讯作者: *张亮，E-mail: zhangl@whut.edu.cn
作者简介:张志朋，E-mail: 1005829259@qq.com
基金资助:
国家自然科学基金(61573012)

Controllability of a Nonlinear Parabolic Systems with Drift-Diffusion Term

Zhang Zhipeng(),Zhang Liang^*()

Department of Mathematics, Wuhan University of Technology, Wuhan 430070

Received:2023-03-09 Revised:2023-08-16 Online:2023-12-26 Published:2023-11-16
Supported by:
NSFC(61573012)

摘要/Abstract

摘要：

该文研究一类非线性抛物型方程组的能控性问题及其时间最优控制的存在性问题. 该方程具有漂移-扩散项和非线性项 $h(u,v)$, $g(u,v)$. 文中利用半群的 $L^{p}$-$L^{q}$ 估计和近年来发展的最大正则化研究方程组的正则性和控制函数的估计, 利用不动点定理证明了非线性系统的局部精确能控性, 并将该结论应用于证明时间最优控制的存在性.

关键词: 抛物型方程组, 能观性估计, 精确能控性, Kakutani不动点

Abstract:

This paper studies the controllability and the existence of time optimal control for a class of nonlinear parabolic equations, which has drift-diffusion term with nonlinear terms $h(u,v)$ and $g(u,v)$. It applies the $L^{p}$-$L^{q}$ estimate of semigroups and the recently developed maximal regularity theory to study the regularity and the cost estimate of control functions. Moreover, this paper establishes the local exact controllability of the nonlinear control system by utilizing the Kakutani's fixed point theorem. Therefore, it is applied to the existence of time optimal controls.

Key words: Parabolic equations, Observability estimate, Exact controllability, Kakutani's fixed point

中图分类号:

O231.4

张志朋, 张亮. 一类带漂移-扩散项的非线性抛物型方程组的能控性[J]. 数学物理学报, 2023, 43(6): 1744-1758.

Zhang Zhipeng, Zhang Liang. Controllability of a Nonlinear Parabolic Systems with Drift-Diffusion Term[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1744-1758.

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