An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

doi:10.1088/1674-1056/23/4/040201

中国物理B ›› 2014, Vol. 23 ›› Issue (4): 40201-040201.doi: 10.1088/1674-1056/23/4/040201

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An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

于欣^a, 任志刚^b, 许超^b

a Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China;
b The State Key Laboratory of Industrial Control Technology and Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou 310027, China

收稿日期:2013-10-28 修回日期:2013-12-06 出版日期:2014-04-15 发布日期:2014-04-15
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 61374096 and 61104048) and the Natural Science Foundation of Zhejiang Province of China (Grant No. Y6110751).

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

Yu Xin (于欣)^a, Ren Zhi-Gang (任志刚)^b, Xu Chao (许超)^b

a Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China;
b The State Key Laboratory of Industrial Control Technology and Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou 310027, China

Received:2013-10-28 Revised:2013-12-06 Online:2014-04-15 Published:2014-04-15
Contact: Xu Chao E-mail:cxu@zju.edu.cn
About author:02.60.Lj; 02.30.Yy
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 61374096 and 61104048) and the Natural Science Foundation of Zhejiang Province of China (Grant No. Y6110751).

摘要/Abstract

摘要： In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of the boundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming (SQP) algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.

关键词: boundary optimal control, heat equation, variable domain, finite element method, control parameterization method

Abstract: In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of the boundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming (SQP) algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.

Key words: boundary optimal control, heat equation, variable domain, finite element method, control parameterization method

中图分类号: (Ordinary and partial differential equations; boundary value problems)

02.60.Lj

02.30.Yy (Control theory)

于欣, 任志刚, 许超. An approximation for the boundary optimal control problem of a heat equation defined in a variable domain[J]. 中国物理B, 2014, 23(4): 40201-040201.

Yu Xin (于欣), Ren Zhi-Gang (任志刚), Xu Chao (许超). An approximation for the boundary optimal control problem of a heat equation defined in a variable domain[J]. Chin. Phys. B, 2014, 23(4): 40201-040201.

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An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

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