状态受限对流扩散最优控制问题的边界稳定有限元法数值模拟
发布时间：2018-08-14 11:51
【摘要】：由对流扩散方程所描述的最优控制问题被广泛应用于很多领域,如:大气污染控制问题,流体控制问题等,研究此类最优控制问题的数值模拟具有重要的理论意义与应用价值.本文主要研究了求解状态受限的对流扩散最优控制问题的边界稳定有限元法,建立了残量型后验误差估计.主要工作如下:一.采用Moreau-Yosida正则化和Lavrentiev正则化方法,建立状态受限对流扩散最优控制问题的正则化问题,利用拉格朗日乘子法推导连续的一阶最优性条件.二.采用边界稳定有限元法离散正则化后的状态方程,采用变分离散法离散控制变量,利用拉格朗日乘子法推导离散格式的一阶最优性条件,运用凸分析、对偶论证等技术建立控制变量和状态变量的后验误差估计.三.根据理论分析,基于原始对偶策略给出求解状态受限最优控制问题的数值迭代算法,并在此基础上基于后验误差估计子构造自适应算法.
[Abstract]:The optimal control problem described by convection-diffusion equation is widely used in many fields, such as air pollution control problem, fluid control problem and so on. The numerical simulation of this kind of optimal control problem has important theoretical significance and application value. In this paper, the boundary stability finite element method for solving the convection-diffusion optimal control problem with state constraints is studied, and a residual type posteriori error estimate is established. The main work is as follows: 1. The regularization problem of state constrained convection-diffusion optimal control problem is established by means of Moreau-Yosida regularization and Lavrentiev regularization. The continuous first-order optimality conditions are derived by Lagrange multiplier method. II. The state equation is discretized by the boundary stable finite element method, the control variables are discretized by the variational discrete method, the first-order optimality conditions of the discrete scheme are derived by the Lagrangian multiplier method, and the convex analysis is used. A posteriori error estimation of control variables and state variables is established by dual argument and other techniques. Three Based on the theoretical analysis, a numerical iterative algorithm for the state constrained optimal control problem is presented based on the primal duality strategy, and an adaptive algorithm is constructed based on the posteriori error estimator.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82;O232
本文编号：2182774
[Abstract]:The optimal control problem described by convection-diffusion equation is widely used in many fields, such as air pollution control problem, fluid control problem and so on. The numerical simulation of this kind of optimal control problem has important theoretical significance and application value. In this paper, the boundary stability finite element method for solving the convection-diffusion optimal control problem with state constraints is studied, and a residual type posteriori error estimate is established. The main work is as follows: 1. The regularization problem of state constrained convection-diffusion optimal control problem is established by means of Moreau-Yosida regularization and Lavrentiev regularization. The continuous first-order optimality conditions are derived by Lagrange multiplier method. II. The state equation is discretized by the boundary stable finite element method, the control variables are discretized by the variational discrete method, the first-order optimality conditions of the discrete scheme are derived by the Lagrangian multiplier method, and the convex analysis is used. A posteriori error estimation of control variables and state variables is established by dual argument and other techniques. Three Based on the theoretical analysis, a numerical iterative algorithm for the state constrained optimal control problem is presented based on the primal duality strategy, and an adaptive algorithm is constructed based on the posteriori error estimator.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82;O232
【参考文献】
相关期刊论文 前2条
1 Michael Hinze;;VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS[J];Journal of Computational Mathematics;2009年Z1期
2 ;A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation[J];Numerical Mathematics:Theory,Methods and Applications;2008年03期
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