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.

DOI: 10.1051/cocv/2014028

Keywords: Landau–Lifshitz equation, nanowire, control, Kalman condition, feedback stabilization

^{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://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/

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