抛物型最优控制问题的全离散Crank-Nicolson有限元法  

Fully Discrete Finite Element Method Based on the Crank-Nicolson Scheme for Parabolic Optimal Control Problem

作　　者：张馨丹 赵建平[1] 侯延仁 ZHANG Xindan;ZHAO Jianping;HOU Yanren(School of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830017,China;School of Mathematics and Statistics,Xi’an Jiaotong University,Xi’an Shaanxi 710049,China)

机构地区：[1]新疆大学数学与系统科学学院,新疆乌鲁木齐830017 [2]西安交通大学数学与统计学院,陕西西安710049

出　　处：《新疆大学学报（自然科学版中英文）》2024年第2期196-205,245,共11页Journal of Xinjiang University(Natural Science Edition in Chinese and English)

基　　金：国家自然科学基金“沙丘长时间移动演变行为的动力学模型及其数值模拟研究”(61962056);新疆维吾尔自治区自然科学基金“带界面的抛物型最优控制问题高效数值算法研究”(2022D01C409)。

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.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.

关 键 词：抛物型最优控制问题 CRANK-NICOLSON格式 最优性系统 

分 类 号：O241.82[理学—计算数学]