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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.
Privat, Yannick 1 ; Trélat, Emmanuel 2
@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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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, Tome 21 (2015) no. 2, pp. 301-323. doi: 10.1051/cocv/2014028
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