几类非线性最优控制问题混合有限元高效算法研究

Optimal control problems play a very important role in many science and engineering applications. The numerical simulation of these optimal control problems is one of important areas in scientific and engineering computing. The finite element methods are widely used in numerically solving optimal control problems. There have been extensive studies on this aspect. However, in many control problems, the objective functional contains gradient of the state variables. Thus the accuracy of gradient is important in numerical approximation of the state equations. In finite element methods, mixed finite elements are widely used to approximate flux variables, although there is only very limited research work on analyzing such elements for optimal control problems. Mixed finite element methods have many advantages. It has much significance to extend the standard finite element method to mixed finite element methods for optimal control problems..Nonlinear optimal control problems widely exist in many practical applications, such as air pollution control problems, complicated fluids control problems, engineering design problems, reservoir numerical simulation and so on. Therefore, the study of the numerical methods for the nonlinear optimal control problems has significant theoretical value and application prospect. This project will do research on semilinear elliptic and parabolic optimal control problems, quasilinear elliptic and parabolic optimal control problems and strongly nonlinear elliptic optimal control problems by using the mixed finite element methods and efficient algorithms. The state and the co-state will be approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control will be approximated by piecewise constant functions. Combining the regularity of nonlinear equations, duality argument, weighted Clement-type interpolation operator, integral average operator, gradient recovery operator and interpolation postprocessing technique, as well as the application of Helmholtz decomposition and Bubble function, this study will obtain a priori error estimates, residual and recovery a posteriori error estimates and local and global superconvergence of mixed finite element solution. This research will achieve a finite element discretization for some complex optimal control problems and expand people's recognition and understanding for optimal control problems which has great significance to reveal the internal laws of nonlinear problems. What's more, it provides the theoretical and practical basis for solving some more complex practical problems.

非线性最优控制问题在很多实际应用领域中广泛存在,如空气污染控制问题、复杂流体控制问题、工程设计问题、油藏数值模拟问题等，因此研究非线性最优控制问题的数值计算方法具有重要的理论价值和应用前景。本项目利用混合有限元高效算法研究半线性椭圆和抛物最优控制问题、拟线性椭圆和抛物最优控制问题和强非线性椭圆最优控制问题，采用最低阶Raviart-Thomas混合有限元逼近状态变量、分片常数函数逼近控制变量,结合非线性方程解的正则性、对偶论证、加权Clement型插值算子、积分平均算子、梯度重构算子和插值后处理技巧，应用Helmoholtz分解和Bubble函数思想，获得混合有限元解的先验误差估计、残量型及重构型后验误差估计和局部及全局超收敛，实现对一些复杂最优控制问题的有限元离散，拓展人们对最优控制问题的认识和理解，对揭示非线性问题内部规律有重要意义，为一些更复杂实际问题的解决奠定理论和实践基础。

项目的背景: .随着科学技术的飞速发展，非线性最优控制问题已经被广泛应用于众多学科领域，如物理学、生物学、医学、工程设计、航空航天、流体力学、大气污染控制、水污染控制、金融、社会经济学等，因此非线性最优控制问题是一个比较普遍和很有实际应用前景的研究热点之一。..主要研究内容:.1. 研究了非线性椭圆和抛物最优控制问题的最优性条件和混合有限元离散格式，利用一些处理非线性方程的技巧，获得了混合有限元解的先验误差估计和后验误差估计，并获得了混合有限元解的超收敛； .2. 研究了双线性最优控制问题和双曲最优控制问题的混合有限元逼近格式，并给出了混合有限元解的先验误差估计、后验误差估计和超收敛，并构造了相应的自适应有限元算法，最后利用数值算例验证了相应的理论结果；.3. 对于非线性抛物最优控制问题，建立了相应的最优性条件，获得了该问题的有限元离散解，最后给出了有限元离散解的后验误差估计。研究了积分微分型最优控制问题的有限元方法，构造了有限元离散格式和相应的最优性条件，证明了离散解的存在唯一性，并获得了有限元离散解的后验误差估计。..重要结果:.在Computers and Mathematics with Applications，Journal of Mathematical Inequalities等国内外重要刊物上发表和接收学术论文12篇，SCI期刊论文11篇；主编国家信息与计算科学丛书《偏微分方程数值解法》1部（科学出版社出版，获“十二五”国家重点图书出版规划项目资助），主编的英文专著《High efficient and accuracy numerical methods for optimal control problems》1部（科学出版社出版，获国家重点图书出版规划项目资助），立项中国博士后科学基金面上一等资助项目1项、教育部春晖计划1项、重庆市科委项目1项，结题中国博士后科学基金面上二等资助项目、重庆市科委项目和重庆市教委项目各1项。..关键数据及其科学意义:.项目发表和接收学术论文12篇，SCI期刊论文11篇，主要研究了几类非线性最优控制问题的混合有限元解，获得了离散解的先验误差估计、后验误差估计和超收敛，实现了对一些非线性最优控制问题的有限元逼近，拓展人们对最优控制问题和混合有限元方法的理解，为进一步解决更复杂实际问题奠定坚实的理论基础。