一类状态受限最优控制问题谱方法的后验误差估计

项目名称： 一类状态受限最优控制问题谱方法的后验误差估计

项目编号： No.11201212

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： 周建伟

作者单位： 临沂大学

项目金额： 22万元

中文摘要： 本项目主要研究基于谱方法的状态受限最优控制问题及其后验误差估计问题。状态变量受限最优控制问题是偏微分方程约束的最优控制理论领域中一个热门方向,并且其理论研究已经获得长足发展。申请人已经讨论了基于混合有限元方法控制受限最优控制问题的超收敛结果，研究了低维空间谱方法的后验误差估计和状态积分受限最优控制问题的最优性条件。目前，状态受限最优控制问题谱方法的后验误差估计研究还很少。在已有工作基础上，本项目计划深入探讨如下问题。首先通过改进两类正交多项式一阶导数逆估计的阶，研究模型问题的后验误差估计，构造(拟)最优后验误差估计子，尽可能精确的给出误差上界对应常数的表达式。其次研究多种类型的状态受限最优控制问题的谱方法分析，如积分受限、能量模受限和梯度模受限等，推演最优性条件，并且分析先验和后验误差估计。同时讨论显格式的后验误差估计子，并证明构造算法的收敛性。

中文关键词： 谱方法；后验误差估计；最优控制问题；状态受限；

英文摘要： This project is mainly focused on the spectral method for state constrained optimal control problems and its a-posteriori error estimations. Recently more and more scholars are increasingly paying attention to state constrained optimal control problems governed by partial differential equations, and much headway has been made. Many numerical simulations have been provided and investigated. The applicant has deduced the superconvergence of mixed finite elements for optimal control problems with control constraints, investigated the a-posteriori error estimator in one dimension and the optimal conditions for state integral constrained optimal control problems. Currently, there is less research on the a-posteriori error estimations of state constrained optimal control problems with spectral methods. Firstly, motivated by preliminary works of spectral methods, members of this team will derive an improved inverse estimation of two types of polynomials with the first derivative and analyze the a-posteriori error estimations and the (pseudo-)optimal a-posteriori error estimator for the model problems. Members will also construct an explicit constant expression in upper bound estimation and study various types of state constrained optimal control problems, such as integral constraint, energy norm constraint and gradient

英文关键词： Spectral method；A posteriori error estimate；Optimal control problem；State-constraint；