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 where the effective field H e is given by H e = ∆u + h d (u) + H a . The demagnetizing field h d (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 H a (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])

GILLES CARBOU, STÉPHANE LABBÉ AND EMMANUEL TRÉLAT
where h δ (u) = ∂ 2 u ∂x 2 − u 2 e 2 − u 3 e 3 + δe 1 . Here, (e 1 , e 2 , e 3 ) is the canonical basis of IR 3 and the nanowire is the real axis IRe 1 . The magnetic field is written δ(t)e 1 , where the function δ(·) is our control. Setting h(u) = u xx − u 2 e 2 − u 3 e 3 , this yields When δ ≡ 0, stationary solutions do exist, of the form and are called Bloch walls. Their stability properties were studied in [7]. When δ(·) ≡ δ is constant, the solution writes where is the rotation of angle θ around the axis IRe 1 . It corresponds to a rotation plus translation of the above wall along the nanowire. Notice the invariance of (3) through translations x → x − σ and rotations R θ around the axis e 1 . This generates a three-parameters family of particular solutions defined by called travelling wall profiles. Controlling these walls (position plus speed) might be relevant for coding and transporting some information. This is our aim here to derive a controllability result, with an eye on possible applications such as rapid recording. In [9], control properties were proven with piecewise constant controls. However, practical applications require the control to be smooth. Recall that the control here is an external magnetic field applied along the nanowire. The main result of [9] strongly uses the fact that the control is a piecewise constant function and our aim is here to extend this result to the case of smooth controls, hence closer to practical issues. Theorem 1.1. There exist ε 0 > 0 and δ 0 > 0 such that, for all δ 1 , δ 2 ∈ IR satisfying |δ i | ≤ δ 0 , i = 1, 2, for all σ 1 , σ 2 ∈ IR, for every ε ∈ (0, ε 0 ), there exist T > 0 and a control function δ(·) ∈ C ∞ (IR, IR) such that, for every solution u of (3) associated with the control δ(·) and satisfying there exists a real number θ 2 such that Moreover, there exists real numbers θ 2 and σ 2 , with |θ 2 − θ 2 | + |σ 2 − σ 2 | ≤ ε, such that u(t, ·) − u δ2,θ 2 ,σ 2 (t, ·) H 2 −→ t→+∞ 0. In the proof of the main result, we shall choose control laws δ(·) so that where T > 0 is large, δ |[0,T ] is a smooth function such that tδ remains small, and the function δ is smooth overall IR. Notice that this control shares robustness properties in H 2 norm. The time T is required to be large enough. It follows from this result that the family of travelling wall profiles (6) is approximately controllable in H 2 norm, locally in δ and globally in σ, in time sufficiently large.

2.1.
Preliminaries. The following formulas, easy to establish, will be useful next: It is clear from Equation (2) that the solution u has a constant norm. Up to normalizing, assume this norm is equal to 1. Set v(t, x) = R −δ(t)t (u(t, x − δ(t)t)); then, v has a constant norm too, equal to 1. Using the above formulas, computations lead to Note that: Then, easy but lengthy computations, not reported here, show that v is solution of (11) if and only if r = r 1 r 2 satisfies hal-00313779, version 1 -27 Aug 2008

GILLES CARBOU, STÉPHANE LABBÉ AND EMMANUEL TRÉLAT
where and • H 2 (r) is the quadratic form on IR 2 defined by and uniformly with respect to the variable x ∈ IR. Then, we infer that there exists a constant C > 0 such that, if r 2 I R 2 = r 2 ≤ 1 2 and |δ| ≤ 1, then, for all p, q ∈ IR 2 , for all x, t, ε ∈ IR, From this a priori estimate, one might consider R(t, δ,δ, x, r, r x , r xx ) as a remainder term in Equation (12). The proof uses stability properties established for the linear operator A, so as to establish. We next follow the same lines as in [9].

2.2.
Change of coordinates. The operator L is a self-adjoint operator on L 2 (IR), of domain H 2 (IR), and L = − * with = ∂ x + th x Id (one has * = −∂ x + th x Id). It follows that L is nonpositive, and that ker L = ker is the one dimensional subspace of L 2 (IR) generated by 1 ch x . In particular, the operator L, restricted to the subspace E = (ker L) ⊥ , is negative. Moreover, combining the facts that L |(ker L) ⊥ is negative and that Spec J = {1 + i, 1 − i}, it follows that the operator A, restricted to the subspace E = (ker A) ⊥ , is negative.
In what follows, solutions r of (12) are written as the sum of an element of ker A and of an element of E. Since Equation (11) is invariant with respect to translations in x and rotations around the axis e 1 , for every Λ = (θ, σ) ∈ IR 2 , M Λ (x) = R θ M 0 (x − σ) is solution of (11). Define The mapping is a diffeomorphism from a neighborhood U of zero in IR 2 × E into a neighborhood V of zero in H 2 (IR). Indeed, if r = R Λ + W with W ∈ E, then, by definition, Conversely, if Λ ∈ IR 2 satisfies ((15)), then W = r − R Λ ∈ E. The mapping h : IR 2 −→ IR 2 , defined by h(Λ) = ( R Λ , a 1 L 2 , R Λ , a 2 L 2 ) is smooth and satisfies dh(0) = −2 Id, thus is a local diffeomorphism at (0, 0). It follows easily that Ψ is a local diffeomorphism at zero. Therefore, every solution r of (12), as long as it stays 1 in the neighborhood V, can be written as where W (t, ·) ∈ E and Λ(t) ∈ IR 2 , for every t ≥ 0, and (Λ(t), W (t, ·)) ∈ U. In these new coordinates 2 , Equation (12) leads to (see [7] for the details of computations) where R : are nonlinear mappings, for which there exist constants K > 0 and η > 0 such that for every W ∈ E, every δ ∈ IR, every t ≥ 0, and every Λ ∈ IR 2 satisfying Λ I R 2 ≤ η. Note that, since L is selfadjoint, it follows that AW ∈ E, for every W ∈ E, and thus (17) makes sense.

Asymptotic estimates. Denoting
Remark 2. It follows from Remark 1 that, on the subspace E = (ker A) ⊥ , V(W ) is a norm, which is equivalent to the norm W 2 (H 2 (I R 2 )) .
Consider a solution (W, Λ) of (17), such that W (0, ·) = W 0 (·) and Λ(0) = Λ 0 . Since L is selfadjoint, one has Concerning the first term of the right-hand side of (21), one computes AW, and, using Remark 1, there exists a constant C 1 > 0 such that Concerning the second term of the right-hand side of (21), one deduces from the Cauchy-Schwarz inequality, from Remark 1, and from the estimate (18), that where, to get the last line, we used the inequality here, ξ denotes some real number to be chosen later. One infers from (21), (22) and (23) that Fix > 0; then, under the a priori estimates and The existence of a constant C 2 > 0 follows from Remark 2. Therefore, choosing ξ > 0 large enough, there exist constants C 3 > 0 and C 4 > 0 such that, if holds, and if the control function δ(·) is chosen so that and for every t ≥ 0, then for every s ∈ [0, T ], and moreover, one deduces from (17), (19), and (27) that, if the a priori estimate (24) holds, then From the above a priori estimates, we infer that, if the quantity Λ(0) I R 2 + W (0, ·) (H 2 (I R)) 2 is small enough, and if the control function δ fits the conditions (25) and (26), then Λ(t) I R 2 remains small, for every t ≥ 0, and W (t, ·) (H 2 (I R)) 2 is exponentially decreasing to 0.
Finally we must choose a smooth control function such that u(t, x) is close to u δ1,θ1,σ1 (t, x) at initial time, and close to u δ2,θ2,σ2 (t, x) for large times. Hence, we can choose the function δ such that δ(t) = δ 1 for t ≤ 0. Then, with the reasoning above, we enforce v(t, x) to remain close to M 0 (x), that is, the solution u(t, x) follows the profile u δ(t),θ1,σ1 (t, x). At times t ≥ T , we require u(t, x) to be close to u δ2,θ2,σ2 (t, x) for some θ 2 ; one must have, for t ≥ T , −σ 1 + δ(t)t = −σ 2 + δ 2 t, and hence, To conclude, observe that it is possible to choose a function δ and a time T > 0 large enough, such that δ is smooth on IR and satisfies the above requirements and the estimates (25) and (26).

SMOOTH CONTROL OF NANOWIRES 9
The first part of the theorem, on the interval [0, T ], then follows from the above considerations.