抛物型最优控制问题的全离散Crank-Nicolson有限元法

张馨丹; 赵建平; 侯延仁

doi:10.13568/j.cnki.651094.651316.2023.05.18.0001

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抛物型最优控制问题的全离散Crank-Nicolson有限元法

更新时间：2026-01-21

- 抛物型最优控制问题的全离散Crank-Nicolson有限元法
- Journal of Xinjiang University (Natural Science Edition in Chinese and English) Vol. 41, Issue 2, Pages: 196-205(2024)
- 作者机构：
  
  1. 新疆大学数学与系统科学学院
  2. 西安交通大学数学与统计学院
- 作者简介：
- 基金信息：
- DOI：10.13568/j.cnki.651094.651316.2023.05.18.0001
  CLC： O241.82;O232
- Published：2024
- 稿件说明：
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[1]张馨丹,赵建平,侯延仁.抛物型最优控制问题的全离散Crank-Nicolson有限元法[J].新疆大学学报(自然科学版)(中英文),2024,41(02):196-205+245.
[1]张馨丹,赵建平,侯延仁.抛物型最优控制问题的全离散Crank-Nicolson有限元法[J].新疆大学学报(自然科学版)(中英文),2024,41(02):196-205+245. DOI： 10.13568/j.cnki.651094.651316.2023.05.18.0001.

DOI：10.13568/j.cnki.651094.651316.2023.05.18.0001.

摘要

针对具有积分控制约束的抛物型最优控制问题

提出了一种基于Crank-Nicolson格式的全离散有限元法.使用分段线性有限元对状态进行空间离散

采用Crank-Nicolson格式进行时间离散

对控制变量采用分段线性近似

从而得到离散的最优性系统.证明了状态变量、伴随状态变量和控制变量的误差估计

并通过数值算例验证理论结果.

Abstract

Aiming at parabolic optimal control problem with integral control constraints

a fully discrete finite element method based on Crank-Nicolson scheme is proposed. The state is discretized by piecewise linear finite elements for the space discretization

Crank-Nicolson scheme for time discretization

and the control variables are approximated by piecewise linear function approximation

so as to obtain a discrete optimal system. The error estimates of state variables

adjoint state variables and control variables are proved

and the theoretical results are verified by numerical examples.

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references

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