Optimal voltage control of non-stationary eddy current problems

Fredi Tröltzsch; Alberto Valli

doi:10.3934/mcrf.2018002

Article Contents

2018, Volume 8, Issue 1: 35-56. Doi: 10.3934/mcrf.2018002

This issue Previous Article Second order optimality conditions for optimal control of quasilinear parabolic equations Next Article Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls

Optimal voltage control of non-stationary eddy current problems

Fredi Tröltzsch^1, , and
Alberto Valli^2,

1.
Institut für Mathematik, Technische Universität Berlin, D-10623 Berlin, Germany,
2.
Dipartimento di Matematica, Università di Trento, 38123 Trento, Italy

* Corresponding author: Fredi Tröltzsch
* Corresponding author: Fredi Tröltzsch

Received: February 28, 2017

Revised: August 31, 2017

Published: January 2018

The first author was supported by Einstein Center for Mathematics Berlin (ECMath), project D-SE9. The second author is pleased to thank the Institute of Mathematics of the Technische Universität Berlin, the Research Center Matheon and the Einstein Center for Mathematics Berlin (ECMath) for their kind hospitality.

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Abstract

A mathematical model is set up that can be useful for controlled voltage excitation in time-dependent electromagnetism.The well-posedness of the model is proved and an associated optimal control problem is investigated. Here, the controlfunction is a transient voltage and the aim of the control is the best approximation of desired electric and magnetic fields insuitable $L^2$ -norms.Special emphasis is laid on an adjoint calculus for first-order necessary optimality conditions.Moreover, a peculiar attention is devoted to propose a formulation for which the computational complexity of the finite element solution method is substantially reduced.

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Mathematics Subject Classification: Primary: 35Q60, 49K20; Secondary: 65M60.

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Figure 1. The computational domain $\Omega$ with the conductor $\Omega_C$ and the electric ports $\Gamma_E$ and $\Gamma_J$ .

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Figure 2. A first alternative geometrical configuration: a connected conductor $\Omega_C$ with five electric ports.

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Figure 3. A second alternative geometrical configuration: a non-connected conductor $\Omega_C$ with four electric ports.

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Figure 4. A third alternative geometrical configuration: a non-connected conductor $\Omega_C$ with two electric ports.

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