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March 2018, 8(1): 35-56. doi: 10.3934/mcrf.2018002

## Optimal voltage control of non-stationary eddy current problems

 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

Dedicated to Prof. Dr. Eduardo Casas on the occasion of his 60th birthday

Received  March 2017 Revised  September 2017 Published  January 2018

Fund Project: 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.

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.

Citation: Fredi Tröltzsch, Alberto Valli. Optimal voltage control of non-stationary eddy current problems. Mathematical Control & Related Fields, 2018, 8 (1) : 35-56. doi: 10.3934/mcrf.2018002
##### References:
 [1] A. Alonso Rodríguez, E. Bertolazzi, R. Ghiloni and A. Valli, Construction of a finite element basis of the first de Rham cohomology group and numerical solution of 3D magnetostatic problems, SIAM J. Numer. Anal., 51 (2013), 2380-2402. [2] A. Alonso Rodríguez, E. Bertolazzi, R. Ghiloni and A. Valli, Finite element simulation of eddy current problems using magnetic scalar potentials, J. Comput. Phys., 294 (2015), 503-523. [3] A. Alonso Rodríguez and A. Valli, Eddy Current Approximation of Maxwell Equations, Springer-Verlag Italia, Milan, 2010. [4] L. Arnold and B. von Harrach, A unified variational formulation for the parabolic-elliptic eddy current equations, SIAM J. Appl. Math., 72 (2012), 558-576. [5] A. Bermudez, B. López Rodríguez, R. Rodríguez and P. Salgado, Numerical solution of transient eddy current problems with input current intensities as boundary data, IMA J. Numer. Anal., 32 (2012), 1001-1029. [6] V. Bommer and I. Yousept, Optimal control of the full time-dependent Maxwell equations, ESAIM Math. Model. Numer. Anal., 50 (2016), 237-261. [7] A. Bossavit, Most general 'non-local' boundary conditions for the Maxwell equations in a bounded region, COMPEL, 19 (2000), 239-245. [8] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Springer-Verlag, Berlin, 1992. [9] P. E. Druet, O. Klein, J. Sprekels, F. Tröltzsch and I. Yousept, Optimal control of three-dimensional state-constrained induction heating problems with nonlocal radiation effects, SIAM J. Control Optim., 49 (2011), 1707-1736. [10] R. Griesse and K. Kunisch, Optimal control for a stationary MHD system in velocity-current formulation, SIAM J. Control Optim., 45 (2006), 1822-1845. [11] M. Gunzburger and C. Trenchea, Analysis and discretization of an optimal control problem for the time-periodic MHD equations, J. Math. Anal. Appl., 308 (2005), 440-466. [12] M. Hinze, Control of weakly conductive fluids by near wall Lorentz forces, GAMM-Mitt., 30 (2007), 149-158. [13] D. Hömberg and J. Sokołowski, Optimal shape design of inductor coils for surface hardening, Numer. Funct. Anal. Optim., 42 (2003), 1087-1117. [14] D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition, Math. Comput. Modelling, 38 (2003), 1003-1028. [15] L. S. Hou and A. J. Meir, Boundary optimal control of MHD flows, Appl. Math. Optim., 32 (1995), 143-162. [16] L. S. Hou and S. S. Ravindran, Computations of boundary optimal control problems for an electrically conducting fluid, J. Comput. Phys., 128 (1996), 319-330. [17] M. Kolmbauer, The Multiharmonic Finite Element and Boundary Element Method for Simulation and Control of Eddy Current Problems, Ph.D thesis, Johannes Kepler University Linz, 2012. [18] M. Kolmbauer and U. Langer, A robust preconditioned MinRes solver for distributed time-periodic eddy current optimal control problems, SIAM J. Sci. Comput., 34 (2012), B785-B809. [19] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003. [20] S. Nicaise, S. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields, Comput. Methods Appl. Math., 14 (2014), 555-573. [21] S. Nicaise, S. Stingelin and F. Tröltzsch, Optimal control of magnetic fields in flow measurement, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 579-605. [22] S. Nicaise and F. Tröltzsch, Optimal control of some quasilinear Maxwell equations of parabolic type, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1375-1391. [23] S. S. Ravindran, Real-time computational algorithm for optimal control of an MHD flow system, SIAM J. Sci. Comput., 26 (2005), 1369-1388. [24] F. Tröltzsch and A. Valli, Modeling and control of low-frequency electromagnetic fields in multiply connected conductors, In System Modeling and Optimization (eds. L. Bociu, J.-A. Desideri, and A. Habbal), Springer, (2017), 505-516. [25] F. Tröltzsch and A. Valli, Optimal control of low-frequency electromagnetic fields in multiply connected conductors, Optimization, 65 (2016), 1651-1673. [26] I. Yousept, Optimal control of Maxwell's equations with regularized state constraints, Comput. Optim. Appl., 52 (2012), 559-581. [27] I. Yousept, Optimal bilinear control of eddy current equations with grad-div regularization, J. Numer. Math., 23 (2015), 81-98. [28] I. Yousept and F. Tröltzsch, PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages, ESAIM Math. Model. Numer. Anal., 46 (2012), 709-729.

show all references

##### References:
 [1] A. Alonso Rodríguez, E. Bertolazzi, R. Ghiloni and A. Valli, Construction of a finite element basis of the first de Rham cohomology group and numerical solution of 3D magnetostatic problems, SIAM J. Numer. Anal., 51 (2013), 2380-2402. [2] A. Alonso Rodríguez, E. Bertolazzi, R. Ghiloni and A. Valli, Finite element simulation of eddy current problems using magnetic scalar potentials, J. Comput. Phys., 294 (2015), 503-523. [3] A. Alonso Rodríguez and A. Valli, Eddy Current Approximation of Maxwell Equations, Springer-Verlag Italia, Milan, 2010. [4] L. Arnold and B. von Harrach, A unified variational formulation for the parabolic-elliptic eddy current equations, SIAM J. Appl. Math., 72 (2012), 558-576. [5] A. Bermudez, B. López Rodríguez, R. Rodríguez and P. Salgado, Numerical solution of transient eddy current problems with input current intensities as boundary data, IMA J. Numer. Anal., 32 (2012), 1001-1029. [6] V. Bommer and I. Yousept, Optimal control of the full time-dependent Maxwell equations, ESAIM Math. Model. Numer. Anal., 50 (2016), 237-261. [7] A. Bossavit, Most general 'non-local' boundary conditions for the Maxwell equations in a bounded region, COMPEL, 19 (2000), 239-245. [8] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Springer-Verlag, Berlin, 1992. [9] P. E. Druet, O. Klein, J. Sprekels, F. Tröltzsch and I. Yousept, Optimal control of three-dimensional state-constrained induction heating problems with nonlocal radiation effects, SIAM J. Control Optim., 49 (2011), 1707-1736. [10] R. Griesse and K. Kunisch, Optimal control for a stationary MHD system in velocity-current formulation, SIAM J. Control Optim., 45 (2006), 1822-1845. [11] M. Gunzburger and C. Trenchea, Analysis and discretization of an optimal control problem for the time-periodic MHD equations, J. Math. Anal. Appl., 308 (2005), 440-466. [12] M. Hinze, Control of weakly conductive fluids by near wall Lorentz forces, GAMM-Mitt., 30 (2007), 149-158. [13] D. Hömberg and J. Sokołowski, Optimal shape design of inductor coils for surface hardening, Numer. Funct. Anal. Optim., 42 (2003), 1087-1117. [14] D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition, Math. Comput. Modelling, 38 (2003), 1003-1028. [15] L. S. Hou and A. J. Meir, Boundary optimal control of MHD flows, Appl. Math. Optim., 32 (1995), 143-162. [16] L. S. Hou and S. S. Ravindran, Computations of boundary optimal control problems for an electrically conducting fluid, J. Comput. Phys., 128 (1996), 319-330. [17] M. Kolmbauer, The Multiharmonic Finite Element and Boundary Element Method for Simulation and Control of Eddy Current Problems, Ph.D thesis, Johannes Kepler University Linz, 2012. [18] M. Kolmbauer and U. Langer, A robust preconditioned MinRes solver for distributed time-periodic eddy current optimal control problems, SIAM J. Sci. Comput., 34 (2012), B785-B809. [19] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003. [20] S. Nicaise, S. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields, Comput. Methods Appl. Math., 14 (2014), 555-573. [21] S. Nicaise, S. Stingelin and F. Tröltzsch, Optimal control of magnetic fields in flow measurement, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 579-605. [22] S. Nicaise and F. Tröltzsch, Optimal control of some quasilinear Maxwell equations of parabolic type, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1375-1391. [23] S. S. Ravindran, Real-time computational algorithm for optimal control of an MHD flow system, SIAM J. Sci. Comput., 26 (2005), 1369-1388. [24] F. Tröltzsch and A. Valli, Modeling and control of low-frequency electromagnetic fields in multiply connected conductors, In System Modeling and Optimization (eds. L. Bociu, J.-A. Desideri, and A. Habbal), Springer, (2017), 505-516. [25] F. Tröltzsch and A. Valli, Optimal control of low-frequency electromagnetic fields in multiply connected conductors, Optimization, 65 (2016), 1651-1673. [26] I. Yousept, Optimal control of Maxwell's equations with regularized state constraints, Comput. Optim. Appl., 52 (2012), 559-581. [27] I. Yousept, Optimal bilinear control of eddy current equations with grad-div regularization, J. Numer. Math., 23 (2015), 81-98. [28] I. Yousept and F. Tröltzsch, PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages, ESAIM Math. Model. Numer. Anal., 46 (2012), 709-729.
The computational domain $\Omega$ with the conductor $\Omega_C$ and the electric ports $\Gamma_E$ and $\Gamma_J$ .
A first alternative geometrical configuration: a connected conductor $\Omega_C$ with five electric ports.
A second alternative geometrical configuration: a non-connected conductor $\Omega_C$ with four electric ports.
A third alternative geometrical configuration: a non-connected conductor $\Omega_C$ with two electric ports.
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