抛物型最优控制问题的三次B样条有限元方法

doi:10.6040/j.issn.1671-9352.0.2022.280

摘要/Abstract

摘要： 对一类四阶非线性抛物方程最优控制问题提出一种三次B样条有限元方法。状态变量和对偶状态变量用具有更好光滑性的分片三次B样条连续函数进行逼近,控制变量由分片常数函数进行逼近。这样得到的状态变量和对偶状态变量的数值解二阶连续可微。建立最优性系统的全离散格式,并用迭代法进行求解。最后建立数值算例,验证方法的有效性。

关键词: 三次B样条有限元方法, 最优控制问题, 四阶非线性抛物方程, 最优性系统, 迭代法

Abstract: A cubic B-spline finite element method is proposed for optimal control problems governed by a class of fourth-order nonlinear parabolic equations. The state and co-state variables are discretized by piecewise cubic B-spline continuous functions which have better smoothness and the control variable is approximated by piecewise constant functions. The numerical solutions of the state and co-state variables thus obtained are second-order continuously differentiable. A fully discrete scheme of the optimality system is established and solved by an iterative method. Finally, some numerical examples are presented to verify the effectivity of the proposed method.

Key words: cubic B-spline finite element method, optimal control problem, fourth-order nonlinear parabolic equation, optimality system, iterative method

中图分类号:

O241.82

杜芳芳,孙同军. 抛物型最优控制问题的三次B样条有限元方法[J]. 《山东大学学报(理学版)》, 2023, 58(4): 40-48.

DU Fang-fang, SUN Tong-jun. Cubic B-spline finite element method for parabolic optimal control problems[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(4): 40-48.

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