December  2017, 10(6): 1375-1391. doi: 10.3934/dcdss.2017073

Optimal control of some quasilinear Maxwell equations of parabolic type

1. 

LAMAV and FR CNRS 2956, Université de Valenciennes et du Hainaut Cambrésis, Institut des Sciences et Techniques of Valenciennes, F-59313 -Valenciennes Cedex 9, France

2. 

Technische Universität Berlin, Institut für Mathematik, Str. des 17. Juni 136, Sekr. MA 4-5, D-10623 Berlin, Germany

Dedicated to the 60th birthday of Tomáš Roubíček on

Received  May 2016 Revised  July 2016 Published  June 2017

Fund Project: The second author was supported by Einstein Center for Mathematics Berlin (ECMath), project D-SE 9.

An optimal control problem is studied for a quasilinear Maxwell equation of nondegenerate parabolic type. Well-posedness of the quasilinear state equation, existence of an optimal control, and weak Gâteaux-differentiability of the control-to-state mapping are proved. Based on these results, first-order necessary optimality conditions and an associated adjoint calculus are derived.

Citation: Serge Nicaise, Fredi Tröltzsch. Optimal control of some quasilinear Maxwell equations of parabolic type. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1375-1391. doi: 10.3934/dcdss.2017073
References:
[1]

C. AmroucheC. BernardiM. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823-864.  doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.  Google Scholar

[2]

F. BachingerU. Langer and J. Schöberl, Numerical analysis of nonlinear multiharmonic eddy current problems, Numer. Math., 100 (2005), 593-616.  doi: 10.1007/s00211-005-0597-2.  Google Scholar

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G. Bärwolff and M. Hinze, Optimization of semiconductor melts, ZAMM Z. Angew. Math. Mech., 86 (2006), 423-437.  doi: 10.1002/zamm.200410247.  Google Scholar

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V. Bommer and I. Yousept, Optimal control of the full time-dependent Maxwell equations, ESAIM Math. Model. Numer. Anal., 50 (2016), 237-261.  doi: 10.1051/m2an/2015041.  Google Scholar

[5]

P.E. DruetO. KleinJ. SprekelsF. 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.  doi: 10.1137/090760544.  Google Scholar

[6]

R. Griesse and K. Kunisch, Optimal control for a stationary MHD system in velocity-current formulation, SIAM J. Control Optim., 45 (2006), 1822-1845.  doi: 10.1137/050624236.  Google Scholar

[7]

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.  doi: 10.1016/j.jmaa.2004.11.022.  Google Scholar

[8]

M. Hinze, Control of weakly conductive fluids by near wall Lorentz forces, GAMM-Mitt., 30 (2007), 149-158.  doi: 10.1002/gamm.200790004.  Google Scholar

[9]

D. Hömberg and J. Sokolowski, Optimal shape design of inductor coils for surface hardening, Numer. Funct. Anal. Optim., 42 (2003), 1087-1117.  doi: 10.1137/S0363012900375822.  Google Scholar

[10]

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.  doi: 10.1137/110842533.  Google Scholar

[11]

S. NicaiseS. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields, Computational Methods in Applied Mathematics, 14 (2014), 555-573.  doi: 10.1515/cmam-2014-0022.  Google Scholar

[12]

S. NicaiseS. Stingelin and F. Tröltzsch, Optimal control of magnetic fields in flow measurement, Discrete and Continuous Dynamical Systems-S, 8 (2015), 579-605.   Google Scholar

[13]

S. Nicaise and F. Tröltzsch, A coupled Maxwell integrodifferential model for magnetization processes, Mathematische Nachrichten, 287 (2014), 432-452.  doi: 10.1002/mana.201200206.  Google Scholar

[14]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, volume 153 of International Series of Numerical Mathematics, Birkhäuser/Springer Basel AG, Basel, second edition, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar

[15]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, volume 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[16]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, volume 112. American Math. Society, Providence, 2010. Google Scholar

[17]

I. Yousept, Optimal control of Maxwell's equations with regularized state constraints, Comput. Optim. Appl., 52 (2012), 559-581.  doi: 10.1007/s10589-011-9422-2.  Google Scholar

[18]

I. Yousept and F. Tröltzsch, PDE-constrained optimization of time-dependent 3d electromagnetic induction heating by alternating voltages, ESAIM M2AN, 46 (2012), 709-729.  doi: 10.1051/m2an/2011052.  Google Scholar

[19]

I. Yousept, Optimal control of quasilinear H(curl)-elliptic partial differential equations in magnetostatic field problems, SIAM J. Control Optim., 51 (2013), 3624-3651.  doi: 10.1137/120904299.  Google Scholar

show all references

References:
[1]

C. AmroucheC. BernardiM. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823-864.  doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.  Google Scholar

[2]

F. BachingerU. Langer and J. Schöberl, Numerical analysis of nonlinear multiharmonic eddy current problems, Numer. Math., 100 (2005), 593-616.  doi: 10.1007/s00211-005-0597-2.  Google Scholar

[3]

G. Bärwolff and M. Hinze, Optimization of semiconductor melts, ZAMM Z. Angew. Math. Mech., 86 (2006), 423-437.  doi: 10.1002/zamm.200410247.  Google Scholar

[4]

V. Bommer and I. Yousept, Optimal control of the full time-dependent Maxwell equations, ESAIM Math. Model. Numer. Anal., 50 (2016), 237-261.  doi: 10.1051/m2an/2015041.  Google Scholar

[5]

P.E. DruetO. KleinJ. SprekelsF. 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.  doi: 10.1137/090760544.  Google Scholar

[6]

R. Griesse and K. Kunisch, Optimal control for a stationary MHD system in velocity-current formulation, SIAM J. Control Optim., 45 (2006), 1822-1845.  doi: 10.1137/050624236.  Google Scholar

[7]

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.  doi: 10.1016/j.jmaa.2004.11.022.  Google Scholar

[8]

M. Hinze, Control of weakly conductive fluids by near wall Lorentz forces, GAMM-Mitt., 30 (2007), 149-158.  doi: 10.1002/gamm.200790004.  Google Scholar

[9]

D. Hömberg and J. Sokolowski, Optimal shape design of inductor coils for surface hardening, Numer. Funct. Anal. Optim., 42 (2003), 1087-1117.  doi: 10.1137/S0363012900375822.  Google Scholar

[10]

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.  doi: 10.1137/110842533.  Google Scholar

[11]

S. NicaiseS. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields, Computational Methods in Applied Mathematics, 14 (2014), 555-573.  doi: 10.1515/cmam-2014-0022.  Google Scholar

[12]

S. NicaiseS. Stingelin and F. Tröltzsch, Optimal control of magnetic fields in flow measurement, Discrete and Continuous Dynamical Systems-S, 8 (2015), 579-605.   Google Scholar

[13]

S. Nicaise and F. Tröltzsch, A coupled Maxwell integrodifferential model for magnetization processes, Mathematische Nachrichten, 287 (2014), 432-452.  doi: 10.1002/mana.201200206.  Google Scholar

[14]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, volume 153 of International Series of Numerical Mathematics, Birkhäuser/Springer Basel AG, Basel, second edition, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar

[15]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, volume 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[16]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, volume 112. American Math. Society, Providence, 2010. Google Scholar

[17]

I. Yousept, Optimal control of Maxwell's equations with regularized state constraints, Comput. Optim. Appl., 52 (2012), 559-581.  doi: 10.1007/s10589-011-9422-2.  Google Scholar

[18]

I. Yousept and F. Tröltzsch, PDE-constrained optimization of time-dependent 3d electromagnetic induction heating by alternating voltages, ESAIM M2AN, 46 (2012), 709-729.  doi: 10.1051/m2an/2011052.  Google Scholar

[19]

I. Yousept, Optimal control of quasilinear H(curl)-elliptic partial differential equations in magnetostatic field problems, SIAM J. Control Optim., 51 (2013), 3624-3651.  doi: 10.1137/120904299.  Google Scholar

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