计算数学 ›› 2023, Vol. 45 ›› Issue (1): 130-140.DOI: 10.12286/jssx.j2022-0909

• 论文 • 上一篇    

半线性抛物最优控制问题全离散插值系数有限元方法的收敛性分析

唐跃龙, 华玉春   

  1. 湖南科技学院, 永州 425199
  • 收稿日期:2022-01-25 出版日期:2023-02-14 发布日期:2023-02-13
  • 基金资助:
    国家自然科学基金(11401201),湖南省自然科学基金(2020JJ4323),湖南省教育厅科学研究项目(20A211,20C0854),湖南科技学院科学研究项目(20XKY059,XKYJ2021020)和湖南科技学院教改项目(XKYJ2022002)资助.
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唐跃龙, 华玉春. 半线性抛物最优控制问题全离散插值系数有限元方法的收敛性分析[J]. 计算数学, 2023, 45(1): 130-140.

Tang Yuelong, Hua Yuchun. CONVERGENCE ANALYSIS OF FULLY DISCRETE INTERPOLATED COEFFICIENT FINITE ELEMENTS FOR SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS[J]. Mathematica Numerica Sinica, 2023, 45(1): 130-140.

CONVERGENCE ANALYSIS OF FULLY DISCRETE INTERPOLATED COEFFICIENT FINITE ELEMENTS FOR SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

Tang Yuelong, Hua Yuchun   

  1. Hunan University of Science and Engineering, Yongzhou 425199, China
  • Received:2022-01-25 Online:2023-02-14 Published:2023-02-13
  • Supported by:
    The project was supported by the National Key Research and Development Program of China (2019YFC1905301);National Natural Science Foundation of China (22078115,21776108,21690083,22008078).
本文考虑全离散插值系数有限元方法求解半线性抛物最优控制问题, 其中控制变量用分片常数函数逼近, 状态变量和对偶状态变量用分片线性函数逼近. 对于方程中的半线性项, 先用插值系数技巧处理, 再用牛顿迭代法求解. 通过引入一些辅助变量和投影算子, 并利用有限元空间的逼近性质, 得到半线性抛物最优控制问题插值系数有限元方法的收敛性结果；数值算例结果验证了理论结果的正确性.
关键词： 收敛性分析, 插值系数有限元, 半线性抛物最优控制问题
In this paper, we consider a fully discrete interpolated coefficient finite element approximation for semilinear parabolic optimal control problems, where the control is approximated by piecewise constant functions, the state and adjoint state are approximated by piecewise linear functions. We first deal with the semilinear term by interpolation coefficient technique, then solve it by Newton iteration method. By introducing some auxiliary variables and projection operator, then utilizing the approximation property of finite element space, we derive the convergence results of interpolated coefficient finite element method for semilinear parabolic optimal control problems. The theoretical results are verified by numerical examples.
Key words: convergence analysis, interpolated coefficient finite elements, semilinear parabolic optimal control problems

MR(2010)主题分类: 

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季刊，创刊于1979年
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