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  

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

作　　者：唐跃龙[1] 华玉春[1] Tang Yuelong;Hua Yuchun(Hunan University of Science and Engineering,Yongzhou 425199,China)

机构地区：[1]湖南科技学院,永州425199

出　　处：《计算数学》2023年第1期130-140,共11页Mathematica Numerica Sinica

基　　金：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);湖南科技学院教改项目(XKYJ2022002)资助。

摘　　要：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.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.

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 

分 类 号：O241.82[理学—计算数学] O232[理学—数学]