Control and stabilization of steady-states in a finite-length ferromagnetic nanowire
ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 2, pp. 301-323.

We consider a finite-length ferromagnetic nanowire, in which the evolution of the magnetization vector is governed by the Landau–Lifshitz equation. We first compute all steady-states of this equation, and prove that they share a quantization property in terms of a certain energy. We study their local stability properties. Then we address the problem of controlling and stabilizing steady-states by means of an external magnetic field induced by a solenoid rolling around the nanowire. We prove that, for a generic placement of the solenoid, any steady-state can be locally exponentially stabilized with a feedback control. Moreover we design this feedback control in an explicit way by considering a finite-dimensional linear control system resulting from a spectral analysis. Finally, we prove that we can steer approximately the system from any steady-state to any other one, provided that they have the same energy level.

Received:
DOI: 10.1051/cocv/2014028
Classification: 58F15, 58F17, 53C35
Keywords: Landau–Lifshitz equation, nanowire, control, Kalman condition, feedback stabilization
Privat, Yannick 1; Trélat, Emmanuel 2

1 CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
2 Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, 75005 Paris, France.
@article{COCV_2015__21_2_301_0,
     author = {Privat, Yannick and Tr\'elat, Emmanuel},
     title = {Control and stabilization of steady-states in a finite-length ferromagnetic nanowire},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {301--323},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {2},
     year = {2015},
     doi = {10.1051/cocv/2014028},
     mrnumber = {3348399},
     zbl = {1351.78007},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2014028/}
}
TY  - JOUR
AU  - Privat, Yannick
AU  - Trélat, Emmanuel
TI  - Control and stabilization of steady-states in a finite-length ferromagnetic nanowire
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2015
SP  - 301
EP  - 323
VL  - 21
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2014028/
DO  - 10.1051/cocv/2014028
LA  - en
ID  - COCV_2015__21_2_301_0
ER  - 
%0 Journal Article
%A Privat, Yannick
%A Trélat, Emmanuel
%T Control and stabilization of steady-states in a finite-length ferromagnetic nanowire
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2015
%P 301-323
%V 21
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2014028/
%R 10.1051/cocv/2014028
%G en
%F COCV_2015__21_2_301_0
Privat, Yannick; Trélat, Emmanuel. Control and stabilization of steady-states in a finite-length ferromagnetic nanowire. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 2, pp. 301-323. doi : 10.1051/cocv/2014028. http://archive.numdam.org/articles/10.1051/cocv/2014028/

D.A. Allwood et al., Submicrometer ferromagnetic NOT gate and shift register. Science 296 (2002) 2003–2006. | DOI

F. Alouges and A. Soyeur, On global weak solutions for Landau–Lifshitz equations: existence and nonuniqueness. Nonlinear Anal. 18 (1992) 1070–1084. | DOI | MR | Zbl

D. Atkinson et al., Magnetic domain-wall dynamics in a submicrometre ferromagnetic structure. Nature Mater. 2 (2003) 85–87. | DOI

G.S.D. Beach, C. Nistor, C. Knutson, M. Tsoi and J.L. Erskine, Dynamics of field-driven domain-wall propagation in ferromagnetic nanowires. Nature Mater. 4 (2005) 741–744. | DOI

F. Brown, Micromagnetics. Wiley, New York (1963).

G. Carbou and S. Labbé, Stability for walls in ferromagnetic nanowire. Discrete Contin. Dyn. Syst. Ser. B 6 (2006) 273–290. | MR | Zbl

G. Carbou and P. Fabrie, Regular solutions for Landau–Lifshitz equation in R 3 . Commun. Appl. Anal. 5 (2001) 17–30. | MR | Zbl

G. Carbou and S. Labbé, Stabilization of walls for nanowires of finite length. ESAIM: COCV 18 (2012) 1–21. | Numdam | MR | Zbl

G. Carbou, S. Labbé and E. Trélat, Control of travelling walls in a ferromagnetic nanowire. Discrete Contin. Dyn. Syst. Ser. S 1 (2008) 51–59. | MR | Zbl

G. Carbou, S. Labbé and E. Trélat, Smooth control of nanowires by means of a magnetic field. Commun. Pure Appl. Anal. 8 (2009) 871–879. | DOI | MR | Zbl

J.-M. Coron and E. Trélat, Global steady-state controllability of 1-D semilinear heat equations. SIAM J. Control Optim. 43 (2004) 549–569. | DOI | MR | Zbl

J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1-D semilinear wave equations. Commun. Contemp. Math. 8 (2006) 535–567. | DOI | MR | Zbl

A. De Simone, H. Knüpfer and F. Otto, 2-d stability of the Néel wall. Calc. Var. Partial Differ. Equ. 27 (2006) 233–253. | DOI | MR | Zbl

Y. Egorov and V. Kondratiev, On spectral theory of elliptic operators. Birkhäuser (1996). | MR | Zbl

J. Grollier et al., Switching a spin valve back and forth by current-induced domain wall motion. Appl. Phys. Lett. 83 (2003) 509–511. | DOI

A. Hubert and R. Schäfer, Magnetic domains: the analysis of magnetic microstructures. Springer-Verlag (2000).

R. Ignat and B. Merlet, Lower bound for the energy of Bloch Walls in micromagnetics. Arch. Ration. Mech. Anal. 199 (2011) 369–406. | DOI | MR | Zbl

H.K. Khalil, Nonlinear Systems. Macmillan Publishing Company, New York (1992). | MR | Zbl

S. Labbé, Y. Privat and E. Trélat, Stability properties of steady-states for a network of ferromagnetic nanowires. J. Differ. Equ. 253 (2012) 1709–1728. | DOI | MR | Zbl

L. Landau and E. Lifshitz, Electrodynamics of continuous media, Course of theoretical Physics. Vol. 8. Translated from the russian by J.B. Sykes and J.S. Bell. Pergamon Press, Oxford-London-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass (1960). | MR | Zbl

C. Melcher, Global solvability of the Cauchy problem for the Landau–Lifshitz-Gilbert equation in higher dimensions. Indiana University Math. J. 61 (2013) 1175–1200. | DOI | MR | Zbl

C. Melcher and M. Ptashnyk, Landau–Lifshitz-Slonczewski equations: global weak and classical solutions. SIAM J. Math. Anal. 45 (2013) 407–429. | DOI | MR | Zbl

T. Ono et al., Propagation of a domain wall in a submicrometer magnetic wire. Science 284 (1999) 468–470. | DOI

S. Parkin et al., Magnetic domain-wall racetrack memory. Science 320 (2008) 190–194. | DOI

D. Sanchez, Behaviour of the Landau–Lifshitz equation in a periodic thin layer. Asymptot. Anal. 41 (2005) 41–69. | MR | Zbl

E. Trélat, Contrôle optimal (French) [Optimal control], Théorie & applications [Theory and applications]. Math. Concrètes [Concrete Mathematics]. Vuibert, Paris (2005). | MR | Zbl

M. Tsoi, R.E. Fontana and S.S.P. Parkin, Magnetic domain wall motion triggered by an electric current. Appl. Phys. Lett. 83 (2003) 2617–2619. | DOI

A. Visintin, On Landau–Lifshitz equations for ferromagnetism. Japan J. Appl. Math. 2 (1985) 69–84. | DOI | MR | Zbl

A. Zettl, Sturm–Liouville theory. Vol. 121 of Math. Surveys & Monographs. AMS, Providence (2005). | MR | Zbl

Cited by Sources: