【摘要】：由對流擴散方程所描述的最優(yōu)控制問題被廣泛應用于很多領域,如:大氣污染控制問題,流體控制問題等,研究此類最優(yōu)控制問題的數(shù)值模擬具有重要的理論意義與應用價值.本文主要研究了求解狀態(tài)受限的對流擴散最優(yōu)控制問題的邊界穩(wěn)定有限元法,建立了殘量型后驗誤差估計.主要工作如下:一.采用Moreau-Yosida正則化和Lavrentiev正則化方法,建立狀態(tài)受限對流擴散最優(yōu)控制問題的正則化問題,利用拉格朗日乘子法推導連續(xù)的一階最優(yōu)性條件.二.采用邊界穩(wěn)定有限元法離散正則化后的狀態(tài)方程,采用變分離散法離散控制變量,利用拉格朗日乘子法推導離散格式的一階最優(yōu)性條件,運用凸分析、對偶論證等技術建立控制變量和狀態(tài)變量的后驗誤差估計.三.根據(jù)理論分析,基于原始對偶策略給出求解狀態(tài)受限最優(yōu)控制問題的數(shù)值迭代算法,并在此基礎上基于后驗誤差估計子構造自適應算法.
[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.
【學位授予單位】：山東師范大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.82;O232

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本文編號：2182774