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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[1]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001

[2]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[3]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559

[4]

Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101

[5]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47

[6]

William G. Litvinov. Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions. Journal of Industrial & Management Optimization, 2011, 7 (2) : 291-315. doi: 10.3934/jimo.2011.7.291

[7]

Kei Matsuura, Mitsuharu Otani. Exponential attractors for a quasilinear parabolic equation. Conference Publications, 2007, 2007 (Special) : 713-720. doi: 10.3934/proc.2007.2007.713

[8]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076

[9]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011

[10]

Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128

[11]

Eduardo Casas, Konstantinos Chrysafinos. Analysis and optimal control of some quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 607-623. doi: 10.3934/mcrf.2018025

[12]

Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51

[13]

Mehdi Badra, Kaushik Bal, Jacques Giacomoni. Existence results to a quasilinear and singular parabolic equation. Conference Publications, 2011, 2011 (Special) : 117-125. doi: 10.3934/proc.2011.2011.117

[14]

Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485

[15]

Andrei V. Dmitruk, Nikolai P. Osmolovskii. Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323

[16]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003

[17]

Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control & Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019

[18]

Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations & Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020

[19]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[20]

M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (11)
  • HTML views (87)
  • Cited by (0)

Other articles
by authors

[Back to Top]