# American Institute of Mathematical Sciences

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

## Control of a network of magnetic ellipsoidal samples

 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

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.
Citation: Shruti Agarwal, Gilles Carbou, Stéphane Labbé, Christophe Prieur. Control of a network of magnetic ellipsoidal samples. Mathematical Control & 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. 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. doi: 10.1016/0362-546X(92)90196-L. [3] S. W. Anwane, "Fundamentals of Electromagnetic Fields,", Infinity Science Press, (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. doi: 10.1137/070683064. [5] William F. Brown, "Micromagnetics,", Wiley, (1963). [6] Gilles Carbou, Stability of static walls for a three-dimensional model of ferromagnetic material,, J. Math. Pures Appl., 93 (2010), 183. 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. [8] Gilles Carbou and Pierre Fabrie, Regular solutions for Landau-Lifschitz equation in $\R^3$,, Commun. Appl. Anal., 5 (2001), 17. [9] Gilles Carbou, Stéphane Labbé and Emmanuel Trélat, Control of travelling walls in a ferromagnetic nanowire,, Discrete Contin. Dyn. Syst., 1 (2008), 51. [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. 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. doi: 10.1006/jmaa.1997.5467. [12] D. J. Griffiths, "Introduction to Electrodynamics,", 3rd edition, (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. 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. 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, VIII (1969). [17] J. A. Osborn, Demagnetizing factors of the general ellipsoid,, Phys. Rev., 67 (1945), 351. doi: 10.1103/PhysRev.67.351. [18] Augusto Visintin, On Landau Lifschitz equation for ferromagnetism,, Japan Journal of Applied Mathematics, 1 (1985), 69. doi: 10.1007/BF03167039.

show all references

##### References:
 [1] François Alouges and Karine Beauchard, Magnetization switching on small ferromagnetic ellipsoidal samples,, ESAIM Control Optim. Calc. Var., 15 (2009), 676. 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. doi: 10.1016/0362-546X(92)90196-L. [3] S. W. Anwane, "Fundamentals of Electromagnetic Fields,", Infinity Science Press, (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. doi: 10.1137/070683064. [5] William F. Brown, "Micromagnetics,", Wiley, (1963). [6] Gilles Carbou, Stability of static walls for a three-dimensional model of ferromagnetic material,, J. Math. Pures Appl., 93 (2010), 183. 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. [8] Gilles Carbou and Pierre Fabrie, Regular solutions for Landau-Lifschitz equation in $\R^3$,, Commun. Appl. Anal., 5 (2001), 17. [9] Gilles Carbou, Stéphane Labbé and Emmanuel Trélat, Control of travelling walls in a ferromagnetic nanowire,, Discrete Contin. Dyn. Syst., 1 (2008), 51. [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. 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. doi: 10.1006/jmaa.1997.5467. [12] D. J. Griffiths, "Introduction to Electrodynamics,", 3rd edition, (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. 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. 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, VIII (1969). [17] J. A. Osborn, Demagnetizing factors of the general ellipsoid,, Phys. Rev., 67 (1945), 351. doi: 10.1103/PhysRev.67.351. [18] Augusto Visintin, On Landau Lifschitz equation for ferromagnetism,, Japan Journal of Applied Mathematics, 1 (1985), 69. doi: 10.1007/BF03167039.
 [1] Gaël Bonithon. Landau-Lifschitz-Gilbert equation with applied eletric current. Conference Publications, 2007, 2007 (Special) : 138-144. doi: 10.3934/proc.2007.2007.138 [2] Gilles Carbou, Stéphane Labbé, Emmanuel Trélat. Control of travelling walls in a ferromagnetic nanowire. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 51-59. doi: 10.3934/dcdss.2008.1.51 [3] Thierry Goudon, Frédéric Lagoutière, Léon M. Tine. The Lifschitz-Slyozov equation with space-diffusion of monomers. Kinetic & Related Models, 2012, 5 (2) : 325-355. doi: 10.3934/krm.2012.5.325 [4] Evelyne Miot, Mario Pulvirenti, Chiara Saffirio. On the Kac model for the Landau equation. Kinetic & Related Models, 2011, 4 (1) : 333-344. doi: 10.3934/krm.2011.4.333 [5] Claude Stolz. On estimation of internal state by an optimal control approach for elastoplastic material. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 599-611. doi: 10.3934/dcdss.2016014 [6] D. Blömker, S. Maier-Paape, G. Schneider. The stochastic Landau equation as an amplitude equation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 527-541. doi: 10.3934/dcdsb.2001.1.527 [7] Huicong Li. Effective boundary conditions of the heat equation on a body coated by functionally graded material. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1415-1430. doi: 10.3934/dcds.2016.36.1415 [8] Kay Kirkpatrick. Rigorous derivation of the Landau equation in the weak coupling limit. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1895-1916. doi: 10.3934/cpaa.2009.8.1895 [9] Gilles Carbou, Stéphane Labbé. Stability for static walls in ferromagnetic nanowires. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 273-290. doi: 10.3934/dcdsb.2006.6.273 [10] Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Adhesive flexible material structures. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 553-574. doi: 10.3934/dcdsb.2012.17.553 [11] Yoshinori Morimoto, Karel Pravda-Starov, Chao-Jiang Xu. A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation. Kinetic & Related Models, 2013, 6 (4) : 715-727. doi: 10.3934/krm.2013.6.715 [12] Kleber Carrapatoso. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinetic & Related Models, 2016, 9 (1) : 1-49. doi: 10.3934/krm.2016.9.1 [13] N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647 [14] Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601 [15] Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173 [16] Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871 [17] Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665 [18] Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359 [19] Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311 [20] Jungho Park. Bifurcation and stability of the generalized complex Ginzburg--Landau equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1237-1253. doi: 10.3934/cpaa.2008.7.1237

2017 Impact Factor: 0.542