Chin. Phys. B, 2014, Vol. 23(4): 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

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
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.
Keywords:  boundary optimal control      heat equation      variable domain      finite element method      control parameterization method
Received:  28 October 2013      Revised:  06 December 2013      Accepted manuscript online:
 PACS: 02.60.Lj (Ordinary and partial differential equations; boundary value problems) 02.30.Yy (Control theory)
Fund: 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).
Corresponding Authors:  Xu Chao     E-mail:  cxu@zju.edu.cn