Smooth Control of Nanowires by Means of a Magnetic Field

We address the problem of control of the magnetic moment in a ferromagnetic nanowire by means of a magnetic field. Based on theoretical results for the 1D Landau-Lifschitz equation, we show a robust controllability result. 1. Model and control result The magnetic moment u of a ferromagnetic material is usually modeled as a unitary vector field, solution of the Landau-Lifschitz equation (1) ∂u ∂t = −u ∧He − u ∧ (u ∧He), where the effective field He is given by He = ∆u+ hd(u) +Ha. The demagnetizing field hd(u) is solution of the magnetostatic equations div B = div (H + u) = 0 and curl H = 0, where B is the magnetic induction. The applied field is denoted by Ha (see [3, 12, 17, 22] for more details on the modelization). Existence results have been established for the Landau-Lifschitz equation in [4, 5, 13, 21], numerical aspects have been investigated in [11, 15, 16], and asymptotic properties have been proved in [1, 6, 10, 18, 20]; control issues were addressed in [9]. Here we restrict ourselves to a one dimensional model, i.e., we consider a ferromagnetic nanowire, submitted to an external magnetic field applied along the axis of the wire and which is our control. The model is then written as (see [20]) (2) ∂u ∂t = −u ∧ hδ(u)− u ∧ (u ∧ hδ(u)), 1991 Mathematics Subject Classification. Primary: 35B37, 93D15; Secondary: 93C20.