项目名称: 状态受限最优控制问题的有限元方法
项目编号: No.11201464
项目类型: 青年科学基金项目
立项/批准年度: 2013
项目学科: 数理科学和化学
项目作者: 龚伟
作者单位: 中国科学院数学与系统科学研究院
项目金额: 22万元
中文摘要: 偏微分方程约束的最优控制问题是近年来国际上研究的热点问题,引起了人们从最优控制理论、偏微分方程约束的优化以及偏微分方程数值解等各个角度的研究兴趣。状态受限的最优控制问题是其中一类非常重要的问题,在大气污染控制、产品加工成型等实际问题中有重要的应用,但状态约束条件的引入给相应最优控制问题解的正则性、误差分析及数值计算等带来了困难,目前有很多问题尚未解决。本项目拟研究状态受限的最优控制问题的有限元方法并争取得到创新性成果。首先对于半线性椭圆方程约束的状态受限问题,拟采用直接有限元离散或正则化方法来处理状态约束,在合适的二阶充分性条件下对局部解得到问题的先验误差估计。其次,研究具有不同种类状态约束的线性抛物最优控制问题的有限元逼近,争取得到最优的先验误差估计。最后,对线性椭圆和抛物方程约束的状态受限控制问题的有限元逼近进行后验误差分析,争取得到有效和可靠的后验误差估计子用以指导自适应有限元计算。
中文关键词: 最优控制问题;有限元方法;先验误差估计;后验误差估计;
英文摘要: PDE-constrained optimal control problem is a hot research topic in the decades, a lot of achievements have been made from the viewpoint of optimal control theory, PDE-constrained optimization and numerical approximations for PDEs. State-constrained optimal control problem finds many real applications such as air pollution control in environmental science and product process in industry, it is among the most important problems in the study of optimal control problems which are not fully solved yet. Due to the existence of state constraints the solutions of state-constrained optimal control problems have less smoothness, which introduce many difficulties in both error analysis and numerical calculation,there are many problems remaining open. The aim of this project is to study some unsolved problems in this field and achieve some original results. At first, we will study the semilinear optimal control problems with state constraints based on direct finite element approximation or regularization method to deal with the state constraints, and derive the a priori error estimates for local solutions based on proper second order sufficient optimality conditions. Secondly, we will study the finite element approximations of parabolic optimal control problems with different types of state constraints and obtain optimal a
英文关键词: Optimal control problem;finite element method;a priori error estimate;a posteriori error estimate;