Control of a network of magnetic ellipsoidal samples

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June 2011, 1(2): 129-147. doi: 10.3934/mcrf.2011.1.129

Control of a network of magnetic ellipsoidal samples

Shruti Agarwal ^1, , Gilles Carbou ^2, , Stéphane Labbé ^3, and Christophe Prieur ^4,

1.	Indian Institute of Technology Madras, Department of Mathematics, Chennai - 600 036, India
2.	IMB, Université Bordeaux, 351 cours la Libération, 33405 Talence, France
3.	Laboratoire Jean Kuntzmann, Université de Grenoble, Tour IRMA, 51 rue des Mathématiques, BP 53, 38041 Grenoble Cedex 9, France
4.	Department of Automatic Control, Gipsa-lab, 961 rue de la Houille Blanche, BP 46, 38402 Grenoble Cedex, France

Received October 2010 Revised February 2011 Published June 2011

Abstract
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In this work, we present a mathematical study of stability and controllability of one-dimensional network of ferromagnetic particles. The control is the magnetic field generated by a dipole whose position and whose amplitude can be selected. The evolution of the magnetic field in the network of particles is described by the Landau-Lifschitz equation. First, we model a network of ellipsoidal shape ferromagnetic particles. Then, we prove the stability of relevant configurations and discuss the controllability by the means of the external magnetic field induced by the magnetic dipole. Finally some numerical results illustrate the stability and the controllability results.

Keywords: Landau-Lifschitz equation, Control., Ferromagnetic material.

Mathematics Subject Classification: Primary: 93B05, 35Q93; Secondary: 93C1.

Citation: Shruti Agarwal, Gilles Carbou, Stéphane Labbé, Christophe Prieur. Control of a network of magnetic ellipsoidal samples. Mathematical Control and Related Fields, 2011, 1 (2) : 129-147. doi: 10.3934/mcrf.2011.1.129

References:

[1]	François Alouges and Karine Beauchard, Magnetization switching on small ferromagnetic ellipsoidal samples, ESAIM Control Optim. Calc. Var., 15 (2009), 676-711. doi: 10.1051/cocv:2008047.
[2]	François Alouges and Alain Soyeur, On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal., 18 (1992), 1071-1084. doi: 10.1016/0362-546X(92)90196-L.
[3]	S. W. Anwane, "Fundamentals of Electromagnetic Fields," Infinity Science Press, Hingham, Masschusetts, 2007.
[4]	L'ubomír Baňas, Sören Bartels and Andreas Prohl, A convergent implicit finite element discretization of the Maxwell-Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 46 (2008), 1399-1422. doi: 10.1137/070683064.
[5]	William F. Brown, "Micromagnetics," Wiley, New York, 1963.
[6]	Gilles Carbou, Stability of static walls for a three-dimensional model of ferromagnetic material, J. Math. Pures Appl., 93 (2010), 183-203. doi: 10.1016/j.matpur.2009.10.004.
[7]	Gilles Carbou and Pierre Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differential Integral Equations, 14 (2001), 213-229.
[8]	Gilles Carbou and Pierre Fabrie, Regular solutions for Landau-Lifschitz equation in $\R^3$, Commun. Appl. Anal., 5 (2001), 17-30.
[9]	Gilles Carbou, Stéphane Labbé and Emmanuel Trélat, Control of travelling walls in a ferromagnetic nanowire, Discrete Contin. Dyn. Syst., 1 (2008), 51-59.
[10]	Shijin Ding, Boling Guo, Junyu Lin and Ming Zeng, Global existence of weak solutions for Landau-Lifshitz-Maxwell equations, Discrete Contin. Dyn. Syst., 17 (2007), 867-890. doi: 10.3934/dcds.2007.17.867.
[11]	Boling Guo and Fengqiu Su, Global weak solution for the Landau-Lifshitz-Maxwell equation in three space dimensions, J. Math. Anal. Appl., 211 (1997), 326-346. doi: 10.1006/jmaa.1997.5467.
[12]	D. J. Griffiths, "Introduction to Electrodynamics," 3rd edition, PearsonBenjamin Cummings, San Francisco, CA, 2008.
[13]	Stéphane Labbé, "Simulation Numérique du Comportement Hyperfréquence des Matériaux Ferromagnétiques," Editions Universitaires Européennes, 2010.
[14]	Stéphane Labbé, Fast computation for large magnetostatic systems adapted for micromagnetism, SIAM J. Sci. Comp., 26 (2005), 2160-2175. doi: 10.1137/030601053.
[15]	Stéphane Labbé and Pierre-Yves Bertin, Microwave polarisability of ferrite particles with non-uniform magnetization, Journal of Magnetism and Magnetic Materials, 206 (1999), 93-105. doi: 10.1016/S0304-8853(99)00537-5.
[16]	L. Landau and E. Lifschitz, "Electrodynamique des Milieux Continus, Cours de Physique Théorique," (French) [Electrodynamic of Continuous Media, Theoretical Physics Course], VIII, Mir, Moscou, 1969.
[17]	J. A. Osborn, Demagnetizing factors of the general ellipsoid, Phys. Rev., 67 (1945), 351-357. doi: 10.1103/PhysRev.67.351.
[18]	Augusto Visintin, On Landau Lifschitz equation for ferromagnetism, Japan Journal of Applied Mathematics, 1 (1985), 69-84. doi: 10.1007/BF03167039.

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References:

[1]	François Alouges and Karine Beauchard, Magnetization switching on small ferromagnetic ellipsoidal samples, ESAIM Control Optim. Calc. Var., 15 (2009), 676-711. doi: 10.1051/cocv:2008047.
[2]	François Alouges and Alain Soyeur, On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal., 18 (1992), 1071-1084. doi: 10.1016/0362-546X(92)90196-L.
[3]	S. W. Anwane, "Fundamentals of Electromagnetic Fields," Infinity Science Press, Hingham, Masschusetts, 2007.
[4]	L'ubomír Baňas, Sören Bartels and Andreas Prohl, A convergent implicit finite element discretization of the Maxwell-Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 46 (2008), 1399-1422. doi: 10.1137/070683064.
[5]	William F. Brown, "Micromagnetics," Wiley, New York, 1963.
[6]	Gilles Carbou, Stability of static walls for a three-dimensional model of ferromagnetic material, J. Math. Pures Appl., 93 (2010), 183-203. doi: 10.1016/j.matpur.2009.10.004.
[7]	Gilles Carbou and Pierre Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differential Integral Equations, 14 (2001), 213-229.
[8]	Gilles Carbou and Pierre Fabrie, Regular solutions for Landau-Lifschitz equation in $\R^3$, Commun. Appl. Anal., 5 (2001), 17-30.
[9]	Gilles Carbou, Stéphane Labbé and Emmanuel Trélat, Control of travelling walls in a ferromagnetic nanowire, Discrete Contin. Dyn. Syst., 1 (2008), 51-59.
[10]	Shijin Ding, Boling Guo, Junyu Lin and Ming Zeng, Global existence of weak solutions for Landau-Lifshitz-Maxwell equations, Discrete Contin. Dyn. Syst., 17 (2007), 867-890. doi: 10.3934/dcds.2007.17.867.
[11]	Boling Guo and Fengqiu Su, Global weak solution for the Landau-Lifshitz-Maxwell equation in three space dimensions, J. Math. Anal. Appl., 211 (1997), 326-346. doi: 10.1006/jmaa.1997.5467.
[12]	D. J. Griffiths, "Introduction to Electrodynamics," 3rd edition, PearsonBenjamin Cummings, San Francisco, CA, 2008.
[13]	Stéphane Labbé, "Simulation Numérique du Comportement Hyperfréquence des Matériaux Ferromagnétiques," Editions Universitaires Européennes, 2010.
[14]	Stéphane Labbé, Fast computation for large magnetostatic systems adapted for micromagnetism, SIAM J. Sci. Comp., 26 (2005), 2160-2175. doi: 10.1137/030601053.
[15]	Stéphane Labbé and Pierre-Yves Bertin, Microwave polarisability of ferrite particles with non-uniform magnetization, Journal of Magnetism and Magnetic Materials, 206 (1999), 93-105. doi: 10.1016/S0304-8853(99)00537-5.
[16]	L. Landau and E. Lifschitz, "Electrodynamique des Milieux Continus, Cours de Physique Théorique," (French) [Electrodynamic of Continuous Media, Theoretical Physics Course], VIII, Mir, Moscou, 1969.
[17]	J. A. Osborn, Demagnetizing factors of the general ellipsoid, Phys. Rev., 67 (1945), 351-357. doi: 10.1103/PhysRev.67.351.
[18]	Augusto Visintin, On Landau Lifschitz equation for ferromagnetism, Japan Journal of Applied Mathematics, 1 (1985), 69-84. doi: 10.1007/BF03167039.

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