    • Previous Article
Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls
• MCRF Home
• This Issue
• Next Article
Second order optimality conditions for optimal control of quasilinear parabolic equations
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:
  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. Google Scholar  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. Google Scholar  A. Alonso Rodríguez and A. Valli, Eddy Current Approximation of Maxwell Equations, Springer-Verlag Italia, Milan, 2010. Google Scholar  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. Google Scholar  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. Google Scholar  V. Bommer and I. Yousept, Optimal control of the full time-dependent Maxwell equations, ESAIM Math. Model. Numer. Anal., 50 (2016), 237-261. Google Scholar  A. Bossavit, Most general 'non-local' boundary conditions for the Maxwell equations in a bounded region, COMPEL, 19 (2000), 239-245. Google Scholar  R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Springer-Verlag, Berlin, 1992. Google Scholar  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. Google Scholar  R. Griesse and K. Kunisch, Optimal control for a stationary MHD system in velocity-current formulation, SIAM J. Control Optim., 45 (2006), 1822-1845. Google Scholar  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. Google Scholar  M. Hinze, Control of weakly conductive fluids by near wall Lorentz forces, GAMM-Mitt., 30 (2007), 149-158. Google Scholar  D. Hömberg and J. Sokołowski, Optimal shape design of inductor coils for surface hardening, Numer. Funct. Anal. Optim., 42 (2003), 1087-1117. Google Scholar  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. Google Scholar  L. S. Hou and A. J. Meir, Boundary optimal control of MHD flows, Appl. Math. Optim., 32 (1995), 143-162. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003. Google Scholar  S. Nicaise, S. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields, Comput. Methods Appl. Math., 14 (2014), 555-573. Google Scholar  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. Google Scholar  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. Google Scholar  S. S. Ravindran, Real-time computational algorithm for optimal control of an MHD flow system, SIAM J. Sci. Comput., 26 (2005), 1369-1388. Google Scholar  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. Google Scholar  F. Tröltzsch and A. Valli, Optimal control of low-frequency electromagnetic fields in multiply connected conductors, Optimization, 65 (2016), 1651-1673. Google Scholar  I. Yousept, Optimal control of Maxwell's equations with regularized state constraints, Comput. Optim. Appl., 52 (2012), 559-581. Google Scholar  I. Yousept, Optimal bilinear control of eddy current equations with grad-div regularization, J. Numer. Math., 23 (2015), 81-98. Google Scholar  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. Google Scholar

show all references

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

##### References:
  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. Google Scholar  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. Google Scholar  A. Alonso Rodríguez and A. Valli, Eddy Current Approximation of Maxwell Equations, Springer-Verlag Italia, Milan, 2010. Google Scholar  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. Google Scholar  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. Google Scholar  V. Bommer and I. Yousept, Optimal control of the full time-dependent Maxwell equations, ESAIM Math. Model. Numer. Anal., 50 (2016), 237-261. Google Scholar  A. Bossavit, Most general 'non-local' boundary conditions for the Maxwell equations in a bounded region, COMPEL, 19 (2000), 239-245. Google Scholar  R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Springer-Verlag, Berlin, 1992. Google Scholar  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. Google Scholar  R. Griesse and K. Kunisch, Optimal control for a stationary MHD system in velocity-current formulation, SIAM J. Control Optim., 45 (2006), 1822-1845. Google Scholar  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. Google Scholar  M. Hinze, Control of weakly conductive fluids by near wall Lorentz forces, GAMM-Mitt., 30 (2007), 149-158. Google Scholar  D. Hömberg and J. Sokołowski, Optimal shape design of inductor coils for surface hardening, Numer. Funct. Anal. Optim., 42 (2003), 1087-1117. Google Scholar  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. Google Scholar  L. S. Hou and A. J. Meir, Boundary optimal control of MHD flows, Appl. Math. Optim., 32 (1995), 143-162. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003. Google Scholar  S. Nicaise, S. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields, Comput. Methods Appl. Math., 14 (2014), 555-573. Google Scholar  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. Google Scholar  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. Google Scholar  S. S. Ravindran, Real-time computational algorithm for optimal control of an MHD flow system, SIAM J. Sci. Comput., 26 (2005), 1369-1388. Google Scholar  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. Google Scholar  F. Tröltzsch and A. Valli, Optimal control of low-frequency electromagnetic fields in multiply connected conductors, Optimization, 65 (2016), 1651-1673. Google Scholar  I. Yousept, Optimal control of Maxwell's equations with regularized state constraints, Comput. Optim. Appl., 52 (2012), 559-581. Google Scholar  I. Yousept, Optimal bilinear control of eddy current equations with grad-div regularization, J. Numer. Math., 23 (2015), 81-98. Google Scholar  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. Google Scholar 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.
  Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639  Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222  Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 865-900. doi: 10.3934/dcdss.2018053  Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183  Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014  Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927  Yuji Harata, Yoshihisa Banno, Kouichi Taji. Parametric excitation based bipedal walking: Control method and optimization. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 171-190. doi: 10.3934/naco.2011.1.171  Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077  Javier A. Almonacid, Gabriel N. Gatica, Ricardo Oyarzúa, Ricardo Ruiz-Baier. A new mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent viscosity. Networks & Heterogeneous Media, 2020, 15 (2) : 215-245. doi: 10.3934/nhm.2020010  Boumedièene Chentouf, Sabeur Mansouri. Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021090  Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1845-1865. doi: 10.3934/dcdss.2020109  Michael Herty, Veronika Sachers. Adjoint calculus for optimization of gas networks. Networks & Heterogeneous Media, 2007, 2 (4) : 733-750. doi: 10.3934/nhm.2007.2.733  Heung Wing Joseph Lee, Chi Kin Chan, Karho Yau, Kar Hung Wong, Colin Myburgh. Control parametrization and finite element method for controlling multi-species reactive transport in a circular pool. Journal of Industrial & Management Optimization, 2013, 9 (3) : 505-524. doi: 10.3934/jimo.2013.9.505  Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041  Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240  Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141  Morteza Fotouhi, Mohsen Yousefnezhad. Homogenization of a locally periodic time-dependent domain. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1669-1695. doi: 10.3934/cpaa.2020061  G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1  Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969  Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339

2020 Impact Factor: 1.284