• 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.

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.
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.

MR(2010)主题分类: 

()
[1] Lions J. Optimal Control of Systems Governed by Partial Differential Equations[M]. Berlin:Springer-Verlag, 1971.
[2] Liu W, Yan N. Adaptive Finite Element Methods for Optimal Control Governed by PDEs[M]. Beijing:Science Press, 2008.
[3] Chen Y, Lu Z. High Efficient and Accuracy Numerical Methods for Optimal Control Problems[M]. Beijing:Science Press, 2015.
[4] Liu W, Yan N. A posteriori error estimates for control problems governed by nonlinear elliptic equations[J]. Appl. Numer. Math., 2003, 43:173-187.
[5] Chen Y, Dai Y. Superconvergence for optimal control problems governed by semi-linear elliptic equations[J]. J. Sci. Comput., 2009, 39:206-221.
[6] Tang Y, Chen Y. Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems[J]. Front. Math. China, 2013, 8(2):443-464.
[7] Zlamal M. Finite element methods for nonlinear parabolic equations[J]. RAIRO Model. Anal. Numer., 1997, 11:93-107.
[8] Larsson S, Thomee V, Zhang N. Interpolation of coefficients and transformation of the dependent variable in finite element methods for the non-linear heat equation[J]. Math. Meth. Appl. Sci., 1989, 11(1):105-124.
[9] Xiong Z, Chen Y. A rectangular finite volume element for a semilinear elliptic equation[J]. J. Sci. Comput., 2008, 36:177-179.
[10] Xiong Z, Chen Y. Finite volume element method with interpolation coefficients for two-point boundary value problem of semilinear differential equation[J]. Comput. Meth. Appl. Mech. Engry., 2007, 196:3798-3804.
[11] Lu Z, Cao L, Li L. Interpolation coefficients mixed finite element methods for general semilinear Dirichlet boundary elliptic optimal control problems[J]. Appl. Anal., 2018, 97(14):2496-2509.
[12] 曹龙舟, 鲁祖亮, 李林. 非线性抛物最优控制问题插值系数混合有限元解的先验误差估计[J]. 云南民族大学学报:自然科学版, 2017, 26(4):299-305.
[13] Liu W, Tiba T. Error estimates for the finite element approximation of a class of nonlinear optimal control problems[J]. J. Numer. Funct. Optim., 2001, 22:953-972.
[14] Tang Y, Hua Y. Superconvergence of splitting positive definite mixed finite element for parabolic optimal control problems[J]. Anal. Appl., 2018, 97(16):2778-2793.
[15] Chen Y, Lu Z, Huang Y. Superconvergence of triangular Raviart-Thomas mixed finite element methods for bilinear constrained optimal control problem[J]. Comput. Math. Appl., 2013, 66(8):1498-1513.
[16] Li R, Liu W, Yan N. A posteriori error estimates of recovery type for distributed convex optimal control problems[J]. J. Sci. Comput., 2002, 41(5):1321-1349.
[1] 杨学敏, 牛晶, 姚春华. 椭圆型界面问题的破裂再生核方法[J]. 计算数学, 2022, 44(2): 217-232.
[2] 古振东. 非线性弱奇性Volterra积分方程的谱配置法[J]. 计算数学, 2021, 43(4): 426-443.
[3] 李旭, 李明翔. 连续Sylvester方程的广义正定和反Hermitian分裂迭代法及其超松弛加速[J]. 计算数学, 2021, 43(3): 354-366.
[4] 古振东, 孙丽英. 非线性第二类Volterra积分方程的Chebyshev谱配置法[J]. 计算数学, 2020, 42(4): 445-456.
[5] 王志强, 文立平, 朱珍民. 时间延迟扩散-波动分数阶微分方程有限差分方法[J]. 计算数学, 2019, 41(1): 82-90.
[6] 陈圣杰, 戴彧虹, 徐凤敏. 稀疏线性规划研究[J]. 计算数学, 2018, 40(4): 339-353.
[7] 古振东, 孙丽英. 一类弱奇性Volterra积分微分方程的级数展开数值解法[J]. 计算数学, 2017, 39(4): 351-362.
[8] 刘丽华, 马昌凤, 唐嘉. 求解广义鞍点问题的一个新的类SOR算法[J]. 计算数学, 2016, 38(1): 83-95.
[9] 黄娜, 马昌凤, 谢亚君. 求解非对称代数Riccati 方程几个新的预估-校正法[J]. 计算数学, 2013, 35(4): 401-418.
[10] 任志茹. 三阶线性常微分方程Sinc方程组的结构预处理方法[J]. 计算数学, 2013, 35(3): 305-322.
[11] 陈绍春, 梁冠男, 陈红如. Zienkiewicz元插值的非各向异性估计[J]. 计算数学, 2013, 35(3): 271-274.
[12] 张亚东, 石东洋. 各向异性网格下抛物方程一个新的非协调混合元收敛性分析[J]. 计算数学, 2013, 35(2): 171-180.
[13] 陈争, 马昌凤. 求解非线性互补问题一个新的 Jacobian 光滑化方法[J]. 计算数学, 2010, 32(4): 361-372.
[14] 来翔, 袁益让. 一类三维拟线性双曲型方程交替方向有限元法[J]. 计算数学, 2010, 32(1): 15-36.
[15] 蔚喜军. 非线性波动方程的交替显-隐差分方法[J]. 计算数学, 1998, 20(3): 225-238.
阅读次数
全文


摘要