Optimal control for a phase field model of melting arising from inductive heating

Zonghong Xiong; Wei Wei; Ying Zhou; Yue Wang; Yumei Liao; Zonghong Xiong; Wei Wei; Ying Zhou; Yue Wang; Yumei Liao

doi:10.3934/math.2022007

AIMS Mathematics

2022, Volume 7, Issue 1: 121-142. doi: 10.3934/math.2022007

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Research article

Optimal control for a phase field model of melting arising from inductive heating

1.
Department of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China
2.
School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou, 550025, China
3.
School of Mathematics and Big Data, Guizhou Education University, Guiyang, Guizhou, 550018, China

Received: 02 June 2021 Accepted: 08 September 2021 Published: 08 October 2021
MSC : 35K55, 35Q60, 49J20, 49K20

Due to its unique performance of high efficiency, fast heating speed and low power consumption, induction heating is widely and commonly used in many applications. In this paper, we study an optimal control problem arising from a metal melting process by using a induction heating method. Metal melting phenomena can be modeled by phase field equations. The aim of optimization is to approximate a desired temperature evolution and melting process. The controlled system is obtained by coupling Maxwell's equations, heat equation and phase field equation. The control variable of the system is the external electric field on the local boundary. The existence and uniqueness of the solution of the controlled system are showed by using Galerkin's method and Leray-Schauder's fixed point theorem. By proving that the control-to-state operator $ P $ is weakly sequentially continuous and Fréchet differentiable, we establish an existence result of optimal control and derive the first-order necessary optimality conditions. This work improves the limitation of the previous control system which only contains heat equation and phase field equation.
- induction heating,
- optimal control,
- phase field equation,
- Maxwell's equations,
- existence,
- necessary conditions
Citation: Zonghong Xiong, Wei Wei, Ying Zhou, Yue Wang, Yumei Liao. Optimal control for a phase field model of melting arising from inductive heating[J]. AIMS Mathematics, 2022, 7(1): 121-142. doi: 10.3934/math.2022007

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Abstract

Due to its unique performance of high efficiency, fast heating speed and low power consumption, induction heating is widely and commonly used in many applications. In this paper, we study an optimal control problem arising from a metal melting process by using a induction heating method. Metal melting phenomena can be modeled by phase field equations. The aim of optimization is to approximate a desired temperature evolution and melting process. The controlled system is obtained by coupling Maxwell's equations, heat equation and phase field equation. The control variable of the system is the external electric field on the local boundary. The existence and uniqueness of the solution of the controlled system are showed by using Galerkin's method and Leray-Schauder's fixed point theorem. By proving that the control-to-state operator $ P $ is weakly sequentially continuous and Fréchet differentiable, we establish an existence result of optimal control and derive the first-order necessary optimality conditions. This work improves the limitation of the previous control system which only contains heat equation and phase field equation.

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