一类非线性抛物最优控制问题的有限元误差估计

数学物理学报 ›› 2010, Vol. 30 ›› Issue (6): 1542-1554.

一类非线性抛物最优控制问题的有限元误差估计

付红斐¹|芮洪兴²

1.中国石油大学数学与计算科学学院山东东营 257061|2.山东大学数学学院济南 250100

收稿日期:2008-08-27 修回日期:2009-10-16 出版日期:2010-12-25 发布日期:2010-12-25
基金资助:
国家基础研究项目(2007CB814906)和国家自然科学基金(10771124)资助

Error Estimates for the Finite Element Approximation of Nonlinear Parabolic Optimal Control Problems

FU Hong-Fei¹, RUI Hong-Xing²

1.School of Mathematics and Computational Science, China University of Petroleum, Shandong Dongying 257061|2.School of Mathematics, Shandong University, Jinan 250100

Received:2008-08-27 Revised:2009-10-16 Online:2010-12-25 Published:2010-12-25
Supported by:
国家基础研究项目(2007CB814906)和国家自然科学基金(10771124)资助

摘要/Abstract

摘要：

该文对一类非线性抛物最优控制问题给出了有限元逼近格式, 并讨论了两种不同类型的控制约束集. 文中对状态和伴随状态变量采用了线性连续函数离散, 而控制变量则由分片常函数近似. 得到了控制和状态逼近的先验误差估计O(h_U+h+k), 这里h_U与h 分别表示控制和状态的空间网格步长, k 表示时间步长. 数值试验表明了算法的有效性.

关键词: 有限元逼近, 非线性抛物最优控制, 先验误差估计

Abstract:

In this paper, the finite element approximation to a class of nonlinear optimal control problems with two different kinds of control constrained sets is investigated, where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by piecewise constant functions. Some a priori error estimates are derived for both control and state approximations.
It is proven that these approximations have convergence order O(h_U+h+k), where h_U and h are the spatial mesh-sizes for control and state, respectively, and k is the time increment. Numerical examples are given to show the efficiency of the present scheme.

Key words: Finite element approximation, Nonlinear optimal control, Priori error estimates

中图分类号:

49J20

付红斐, 芮洪兴. 一类非线性抛物最优控制问题的有限元误差估计[J]. 数学物理学报, 2010, 30(6): 1542-1554.

FU Hong-Fei, RUI Hong-Xing. Error Estimates for the Finite Element Approximation of Nonlinear Parabolic Optimal Control Problems[J]. Acta mathematica scientia,Series A, 2010, 30(6): 1542-1554.

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