# Boundary control of time-harmonic eddy current equations

@article{Antil2022BoundaryCO, title={Boundary control of time-harmonic eddy current equations}, author={Harbir Antil and Hugo D'iaz}, journal={ArXiv}, year={2022}, volume={abs/2209.15129} }

. Motivated by various applications, this article develops the notion of boundary control for Maxwell’s equations in the frequency domain. Surface curl is shown to be the appropriate regularization in order for the optimal control problem to be well-posed. Since, all underlying variables are assumed to be complex valued, the standard results on diﬀerentiability do not directly apply. Instead, we extend the notion of Wirtinger derivatives to complexiﬁed Hilbert spaces. Optimality conditions are…

## References

SHOWING 1-10 OF 51 REFERENCES

### Finite Element Analysis of an Optimal Control Problem in the Coefficients of Time-Harmonic Eddy Current Equations

- MathematicsJ. Optim. Theory Appl.
- 2012

The main findings are convergence results of the finite element discretization of the optimal control problem, including sensitivity analysis, regularity results, existence of an optimal control, and optimality conditions.

### Optimal bilinear control of eddy current equations with grad–div regularization

- MathematicsJ. Num. Math.
- 2015

An optimal bilinear control problem governed by time-harmonic eddy current equations is considered to estimate the electric conductivity of a 3D bounded isotropic domain and an optimal control approach based on grad-div regularization and divergence penalization is introduced.

### Optimal control of quasilinear H(curl)-elliptic partial differential equations in magnetostatic field problems

- Mathematics
- 2013

This paper examines the mathematical and numerical analysis for optimal control problems governed by quasilinear H(curl)-elliptic partial differential equations. We consider a mathematical model…

### Numerical treatment of realistic boundary conditions for the eddy current problem in an electrode via Lagrange multipliers

- MathematicsMath. Comput.
- 2005

This paper deals with the finite element solution of the eddy current problem in a bounded conducting domain, crossed by an electric current and subject to boundary conditions appropriate from a…

### Optimal control of the full time-dependent maxwell equations

- Mathematics
- 2016

This paper analyzes the optimal control of the full time-dependent Maxwell equations. Our goal is to find an optimal current density and its time-dependent amplitude which steer the electric and…

### Optimal control of an eddy current problem with a dipole source

- MathematicsJournal of Mathematical Analysis and Applications
- 2020

### An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations

- Mathematics, Computer ScienceMath. Comput.
- 1999

An iteration-by-subdomain procedure is proven to converge, showing that the preconditioner implicitly defined by the iterative procedure is optimal, and proving a regularity theorem for Dirichlet and Neumann harmonic fields.

### Incompressible Finite Elements via Hybridization. Part II: The Stokes System in Three Space Dimensions

- Computer ScienceSIAM J. Numer. Anal.
- 2005

A method that gives exactly incompressible velocity approximations to Stokes flow in three space dimensions is introduced, which generalizes to allow for varying polynomial degrees on different mesh elements and to incorporate certain non-standard but physically relevant boundary conditions.

### Optimal control of Maxwell’s equations with regularized state constraints

- MathematicsComput. Optim. Appl.
- 2012

This paper regularizes the optimal control problem of Maxwell’s equations in the presence of pointwise state constraints by penalizing the pointwiseState constraints by establishing an a priori error estimate for the discretization error in the $\boldsymbol{H}(\bold{curl})$-norm.

### Numerical analysis of electric field formulations of the eddy current model

- MathematicsNumerische Mathematik
- 2005

This paper deals with finite element methods for the numerical solution of the eddy current problem in a bounded conducting domain crossed by an electric current, subjected to boundary conditions…