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《山东大学学报(理学版)》 ›› 2024, Vol. 59 ›› Issue (8): 56-66.doi: 10.6040/j.issn.1671-9352.0.2023.232

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热传导方程的非线性传热定律识别问题

杜乐(),杨柳*(),张涛   

  1. 兰州交通大学数理学院, 甘肃 兰州 730070
  • 收稿日期:2023-05-24 出版日期:2024-08-20 发布日期:2024-07-31
  • 通讯作者: 杨柳 E-mail:l_du1678@163.com;l_yang218@163.com
  • 作者简介:杜乐(1997—), 男, 硕士研究生, 研究方向为数学物理反问题. E-mail: l_du1678@163.com
  • 基金资助:
    国家自然科学基金资助项目(61663018);国家自然科学基金资助项目(11961042);甘肃省自然科学基金资助项目(22JR5RA341)

Identification of nonlinear heat transfer law of heat conduction equation

Le DU(),Liu YANG*(),Tao ZHANG   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
  • Received:2023-05-24 Online:2024-08-20 Published:2024-07-31
  • Contact: Liu YANG E-mail:l_du1678@163.com;l_yang218@163.com

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摘要：

考虑利用边界控制识别热传导方程中与温度相关的传热系数问题。基于最优控制理论, 将热传导方程参数反演问题转化为变分问题, 进而讨论了极小值的存在性以及必要条件, 最终利用能量模估计的方法, 在假设终端时刻较小的情况下, 证得极小值的唯一性和稳定性。

关键词: 热传导方程, 反问题, 非线性传热定律, 非线性边界, 最优控制

Abstract:

The identification of temperature-dependent heat transfer coefficients in heat conduction equations by boundary control is considered. Based on the optimal control theory, the general heat conduction equation parameter inversion problem is transformed into a variational problem, and then the existence and necessary conditions of the minimum value are discussed. Finally, by using the energy norm estimation method, the uniqueness and stability of the minimum value are proved under the assumption that the terminal time is small.

Key words: heat conduction equation, inverse problem, nonlinear heat transfer law, nonlinear boundary, optimal control

中图分类号: 

  • O175.26
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