Control and stabilization of steady-states in a finite-length ferromagnetic nanowire

We consider a finite-length ferromagnetic nanowire, in which the evolution of the magnetization vector is governed by the Landau-Lishitz 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.