SMOOTH CONTROL OF NANOWIRES BY MEANS OF A
MAGNETIC FIELD
Gilles Carbou
MAB, UMR 5466, CNRS, Université Bordeaux 1,
351 cours de la Libération, 33405 Talence cedex, France.
Stéphane Labbé
Université Joseph Fourier, Laboratoire Jean Kuntzmann, CNRS, UMR 5224,
51 rue des Mathématiques, B.P. 53, 38041 Grenoble Cedex 9, France.
Emmanuel Trélat
Université d’Orléans, Laboratoire MAPMO, CNRS, UMR 6628,
Fédération Denis Poisson, FR 2964,
Bat. Math., BP 6759, 45067 Orléans cedex 2, France.
Abstract. 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
∂u
= −u ∧ He − u ∧ (u ∧ He ),
∂t
where the effective field He is given by He = ∆u + hd (u) + Ha . The demagnetizing
field hd (u) is solution of the magnetostatic equations
(1)
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
= −u ∧ hδ (u) − u ∧ (u ∧ hδ (u)),
∂t
1991 Mathematics Subject Classification. Primary: 35B37, 93D15; Secondary: 93C20.
Key words and phrases. Landau-Lifschitz equation, control, stabilization.
The authors were partially supported by the ANR project SICOMAF (”SImulation et COntrôle
des MAtériaux Ferromagnétiques”).
1
2
GILLES CARBOU, STÉPHANE LABBÉ AND EMMANUEL TRÉLAT
2
where hδ (u) = ∂∂xu2 − u2 e2 − u3 e3 + δe1 . Here, (e1 , e2 , e3 ) is the canonical basis of IR3
and the nanowire is the real axis IRe1 . The magnetic field is written δ(t)e1 , where
the function δ(·) is our control. Setting h(u) = uxx − u2 e2 − u3 e3 , this yields
(3)
ut = −u ∧ h(u) − u ∧ (u ∧ h(u)) − δ(u ∧ e1 + u ∧ (u ∧ e1 )).
When δ ≡ 0, stationary solutions do exist, of the form


th x
(4)
M0 (x) =  0 
1
ch x
and are called Bloch walls. Their stability properties were studied in [7].
When δ(·) ≡ δ is constant, the solution writes
(5)
uδ (t, x) = Rδt M0 (x + δt),
where

1
0


Rθ = 
 0

0
cos θ
sin θ
0



− sin θ 


cos θ
is the rotation of angle θ around the axis IRe1 . It corresponds to a rotation plus
translation of the above wall along the nanowire.
Notice the invariance of (3) through translations x 7→ x − σ and rotations Rθ
around the axis e1 . This generates a three-parameters family of particular solutions
defined by
(6)
uδ,θ,σ (t, x) = MΛ uδ (t, x) = Rδt+θ M0 (x + δt − σ)
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 ε0 > 0 and δ0 > 0 such that, for all δ1 , δ2 ∈ IR satisfying
|δi | ≤ δ0 , i = 1, 2, for all σ1 , σ2 ∈ IR, for every ε ∈ (0, ε0 ), 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
(7)
∃θ1 ∈ IR | ku(0, ·) − uδ1 ,θ1 ,σ1 (0, ·)kH 2 ≤ ε,
there exists a real number θ2 such that
(8)
ku(T, ·) − uδ2 ,θ2 ,σ2 (T, ·)kH 2 ≤ ε.
Moreover, there exists real numbers θ20 and σ20 , with |θ20 − θ2 | + |σ20 − σ2 | ≤ ε, such
that
(9)
0
0
ku(t, ·) − uδ2 ,θ2 ,σ2 (t, ·)kH 2 −→ 0.
t→+∞
SMOOTH CONTROL OF NANOWIRES
3
In the proof of the main result, we shall choose control laws δ(·) so that
δ1
if t ≤ 0,
(10)
δ(t) =
2
δ2 + σ1 −σ
if t ≥ T,
t
where T > 0 is large, δ|[0,T ] is a smooth function such that tδ̇ 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. Proof of Theorem 1.1
Similarly as in [7, 8, 9], it is relevant to first reexpress the Landau-Lifschitz
equation in adapted coordinates.
2.1. Preliminaries. The following formulas, easy to establish, will be useful next:


0
0
0
d
Rθ = 0 − sin θ − cos θ = Rθ+ π2 − e1 eT1 = R π2 Rθ − e1 eT1 ;
• dθ
0 cos θ
− sin θ
• v ∧ e1 = −R π2 v + v1 e1 ;
• Rθ u ∧ Rθ v = Rθ (u ∧ v);
• a ∧ (b ∧ c) = b(a.c) − c(a.b);
• Rθ (IRe1 ) = IRe1 .
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
(11)
vt = −v ∧ h(v) − v ∧ (v ∧ h(v)) − δ(vx + v1 v − e1 ) − tδ̇(vx − v3 e2 + v2 e3 ),
where we recall that h(v) = vxx − v2 e2 − v3 e3 . Define


 
1
0


M1 (x) =  ch0 x  and M2 = 1 .
0
−th x
In the frame (M0 (x), M1 (x), M2 ), the solution v : IR+ × IR −→ S 2 ⊂ IR3 writes in
the form
p
v(t, x) = 1 − r1 (t, x)2 − r2 (t, x)2 M0 (x) + r1 (t, x)M1 (x) + r2 (t, x)M2 .
Note that:
1
1
sh x
1
M1 (x), M10 (x) = −
M0 , M000 (x) = − 2 M1 (x)− 2 M0 ;
ch x
ch x
ch x
ch x
1
1
• e1 = th x M0 +
M1 (x), e2 = M2 , e3 =
M0 − th x M1 (x);
ch x
ch x
2
• h(M0 ) = − 2 M0 ;
ch x
• M0 ∧ M1 = M2 , M0 ∧ M2 = −M1 , M1 ∧ M2 = M0 ;
Then, easy but lengthycomputations,
not reported here, show that v is solution of
r1
(11) if and only if r =
satisfies
r2
• M00 (x) =
(12)
rt = Ar + R(t, δ, δ̇, x, r, rx , rxx ),
4
GILLES CARBOU, STÉPHANE LABBÉ AND EMMANUEL TRÉLAT
where
` 0
R(t, δ, δ̇, x, r, rx , rxx ) = − δ
r + G(r)rxx + H1 (x, r)rx + H2 (r)(rx , rx )
0 `
(13)
+ P (x, r) − δB(x, r) − tδ̇C(x, r),
and
L L
with L = ∂xx + (1 − 2th 2 x)Id;
−L L
• ` = ∂x + th x Id;
• G(r) is the matrix defined by
• A=
r22
r1 r2

p
p

1 − krk2
G(r) = 
p

r12
− 1 − krk2 + 1
−p
2
1 − krk
1 − krk2

1 − krk2 − 1

;

r1 r2
+
p
−p
1 − krk2
• H1 (x, r) is the matrix defined by
 p
r2 1 − krk2 − r1 r22
2

H1 (x, r) = p
1 − krk2 ch x
r2 − r23
−r2 + r2 r12
p

;
1−
krk2 r2
+
r1 r22
• H2 (r) is the quadratic form on IR2 defined by

p
1 − krk2 r1 + r2
(1 − krk )X X + (r X) 
;
H2 (r)(X, X) =
p
(1 − krk2 )3/2
2
1 − krk r2 − r1
2
T
T
2
 1

P (x, r)
 , with
• P (x, r) = 
2
P (x, r)
p
1
sh x
1
P (x, r) =2r2 ( 1 − krk2 − 1) 2 − 2r1 r2 2 − 2r1 krk2 2
ch x
ch x
ch x
p
p
sh x
2
3
2
2
− 2r1 1 − krk
+ r1 + r2 (1 − 1 − krk ) + r1 r22 ,
ch 2 x
and
p
1
sh x
1
P 2 (x, r) = − 2r1 ( 1 − krk2 − 1) 2 + 2r12 2 − 2r2 krk2 2
ch x
ch x
ch x
p
sh x
− 2r1 r2 1 − krk2 2 + r2 krk2 ,
ch x
p
p
1
1 − krk2 − 1 + r12
• B(x, r) = (∂x +th x)r+
+th x ( 1 − krk2 −1)r,
r
r
ch x
p1 2
1 − krk2 1
0 −1
• C(x, r) = ∂x + th x
r+
.
1 0
−1
ch x
SMOOTH CONTROL OF NANOWIRES
5
It is clear that there holds
G(r) = O(krk2 ),
H1 (x, r) = O(krk),
H2 (r) = O(krk),
P (x, r) = O(krk2 ),
B(x, r) = O(krk + krx k),
C(x, r) = O(krk + krx k),
uniformly with respect to the variable x ∈ IR. Then, we infer that there exists a
constant C > 0 such that, if krk2IR2 = krk2 ≤ 21 and |δ| ≤ 1, then, for all p, q ∈ IR2 ,
for all x, t, ε ∈ IR,
kR(t, δ, ε, x, r,p, q)kIR2 ≤ C |δ|krkIR2 + |δ|kpkIR2 + t|ε| + t|ε|kpkIR2
(14)
+ krk2IR2 kqkIR2 + krkIR2 kpkIR2 + krkIR2 kpk2IR2 + krk2IR2 .
From this a priori estimate, one might consider R(t, δ, δ̇, x, r, rx , rxx ) 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 L2 (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 L2 (IR) generated by ch1 x . In particular, the operator L, restricted to
the subspace E = (ker L)⊥ , is negative.
Remark 1. On the subspace E:
• the norms k(−L)1/2 f kL2 (IR) and kf kH 1 (IR) are equivalent;
• the norms kLf kL2 (IR) and kf kH 2 (IR) are equivalent;
• the norms k(−L)3/2 f kL2 (IR) and kf kH 3 (IR) are equivalent.
Writing A = JL, with
1 1
J=
,
−1 1
it is clear that the kernel of A is ker A = ker L × ker L; it is the two dimensional
space of L2 (IR2 ) generated by
1 0
a1 (x) =
and a2 (x) = ch x .
1
0
ch x
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 e1 , for every Λ = (θ, σ) ∈ IR2 ,
MΛ (x) = Rθ M0 (x − σ) is solution of (11). Define
hMΛ (x), M1 (x)i
RΛ (x) =
,
hMΛ (x), M2 i
the coordinates of MΛ (x) in the mobile frame (M1 (x), M2 (x)).
6
GILLES CARBOU, STÉPHANE LABBÉ AND EMMANUEL TRÉLAT
The mapping
Ψ : IR2 × E −→ H 2 (IR)
(Λ, W ) −
7 → r(x) = RΛ (x) + W (x)
is a diffeomorphism from a neighborhood U of zero in IR2 × E into a neighborhood
V of zero in H 2 (IR). Indeed, if r = RΛ + W with W ∈ E, then, by definition,
(15)
hr, a1 iL2 = hRΛ , a1 iL2
and hr, a2 iL2 = hRΛ , a2 iL2 .
Conversely, if Λ ∈ IR2 satisfies ((15)), then W = r − RΛ ∈ E. The mapping
h : IR2 −→ IR2 , defined by h(Λ) = (hRΛ , a1 iL2 , hRΛ , a2 iL2 ) 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 stays1 in the neighborhood V,
can be written as
(16)
r(t, ·) = RΛ(t) (·) + W (t, ·),
where W (t, ·) ∈ E and Λ(t) ∈ IR2 , for every t ≥ 0, and (Λ(t), W (t, ·)) ∈ U. In these
new coordinates2, Equation (12) leads to (see [7] for the details of computations)
(17)
Wt (t, x) = AW (t, x) + R(t, δ, ε, Λ(t), x, W (t, x), Wx (t, x), Wxx (t, x)),
Λ0 (t) = M(Λ(t), W (t, ·), Wx (t, ·)),
2
2
2
where R : IR × IR × IR × IR2 × IR × H 2 (IR) × H 1 (IR) × L2 (IR) −→ E and
2
2
M : IR2 × H 1 (IR) × L2 (IR) −→ IR2 are nonlinear mappings, for which there
exist constants K > 0 and η > 0 such that
(18)
(19)
kR(t, δ, ε, Λ, ·, W, Wx , Wxx )k(H 1 (IR))2
≤ K kΛkIR2 + |δ| + t|ε| + kW k(H 2 (IR))2 kW k(H 3 (IR))2 + Kt|ε|,
|M(Λ, W, Wx )| ≤ K kΛkIR2 + kW k(H 1 (IR))2 kW k(H 1 (IR))2 ,
for every W ∈ E, every δ ∈ IR, every t ≥ 0, and every Λ ∈ IR2 satisfying kΛkIR2 ≤ η.
Note that, since L is selfadjoint, it follows that AW ∈ E, for every W ∈ E, and thus
(17) makes sense.
2
W1
2.3. Asymptotic estimates. Denoting W =
, define on H 2 (IR) × IR2
W2
the function
2
1
1
1 L 0
(20)
V(W ) = W
= kLW1 k2L2 (IR) + kLW2 k2L2 (IR) .
2
0 L
2
2
2
2
(L (IR))
p
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 kW k2(H 2 (IR2 )) .
1This a priori estimate will be a consequence of the stability property derived next.
2This decomposition is actually quite standard and has been used e.g. in [14] to establish
stability properties of static solutions of semilinear parabolic equations, and in [2, 19] to prove
stability of travelling waves.
SMOOTH CONTROL OF NANOWIRES
7
Consider a solution (W, Λ) of (17), such that W (0, ·) = W0 (·) and Λ(0) = Λ0 .
Since L is selfadjoint, one has
(21)
2 d
L W1
V(W (t, ·)) = AW,
L 2 W2
dt
(L2 (IR))2
(−L)1/2
0
(−L)3/2 W1
.
+
R(t,
δ,
ε,
Λ,
·,
W,
W
,
W
),
x
xx
0
(−L)1/2
(−L)3/2 W2
(L2 (IR))2
Concerning the first term of the right-hand side of (21), one computes
2 L W1
AW,
= −k(−L)3/2 W1 k(L2 (IR))2 − k(−L)3/2 W2 k(L2 (IR))2 ,
L2 W2
(L2 (IR2 ))2
and, using Remark 1, there exists a constant C1 > 0 such that
2 L W1
≤ −C1 kW k2(H 3 (IR))2 .
(22)
AW,
L2 W2
2
2
(L (IR))
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
(23)
(−L)1/2
0
0
(−L)3/2 W1
R(t,
δ,
ε,
Λ,
·,
W,
W
,
W
),
x
xx
(−L)1/2
(−L)3/2 W2
2
2
(L (IR))
≤ kR(t, δ, ε, Λ, ·, W, Wx , Wxx )k(H 1 (IR))2 kW k(H 3 (IR))2
≤ K kΛkIR2 + |δ| + t|ε| + kW k(H 2 (IR))2 kW k2(H 3 (IR))2 + t|ε|kW k(H 3 (IR))2
1
ξ2
≤ K kΛkIR2 + |δ| + t|ε| + kW k(H 2 (IR))2 + 2 kW k2(H 3 (IR))2 + t2 ε2 ,
2ξ
2
where, to get the last line, we used the inequality
t|ε|kW k(H 3 (IR))2 ≤
ξ2 2 2
1
t ε + 2 kW k2(H 3 (IR))2 ;
2
2ξ
here, ξ denotes some real number to be chosen later.
One infers from (21), (22) and (23) that
1
d
V(W ) ≤ −C1 +K kΛkIR2 + |δ| + t|δ̇| + kW k(H 2 (IR))2 + 2
kW k2(H 3 (IR))2
dt
2ξ
ξ2
+ t2 δ̇ 2 .
2
Fix > 0; then, under the a priori estimates
kΛ(t)kIR2 + |δ| + t|δ̇| + kW (t, ·)k(H 2 (IR))2 +
and
ξ2 2 2
t δ̇ ≤ ,
2
1
C1
≤
2
2ξ
2K
8
GILLES CARBOU, STÉPHANE LABBÉ AND EMMANUEL TRÉLAT
there holds
d
C1
V(W (t, ·)) ≤ − kW (t, ·)k2(H 3 (IR))2 + dt
2
C1
≤ − kW (t, ·)k2(H 2 (IR))2 + 2
≤ −C2 V(W (t, ·)) + .
The existence of a constant C2 > 0 follows from Remark 2. Therefore, choosing ξ > 0 large enough, there exist constants C3 > 0 and C4 > 0 such that, if
C1
kW (0, ·)k(H 2 (IR))2 ≤ 6K
, if the a priori estimate
max kΛ(s)kIR2 ≤
(24)
0≤s≤t
C1
6K
holds, and if the control function δ(·) is chosen so that
|δ(t)| + t|δ̇(t)| ≤
(25)
C1
6K
and
t2 δ̇(t)2 ≤ 2/ξ 2
(26)
for every t ≥ 0, then
(27)
kW (s, ·)k(H 2 (IR))2 ≤ C3 e−C4 s kW (0, ·)k(H 2 (IR))2 + C3 ,
for every s ∈ [0, T ], and moreover, one deduces from (17), (19), and (27) that, if
the a priori estimate (24) holds, then
Z t
C1 C3
kΛ(t)kIR2 ≤ kΛ(0)kIR2 +
kW (0, ·)k(H 2 (IR))2
e−C4 s ds
4
0
Z t
(28)
e−2C4 s ds
+ KC32 kW (0, ·)k2(H 2 (IR))2
0
≤ kΛ(0)kIR2
C2
C1 C3
kW (0, ·)k(H 2 (IR))2 + K 3 kW (0, ·)k2(H 2 (IR))2 .
+
4C4
2C4
From the above a priori estimates, we infer that, if the quantity kΛ(0)kIR2 +
kW (0, ·)k(H 2 (IR))2 is small enough, and if the control function δ fits the conditions
(25) and (26), then kΛ(t)kIR2 remains small, for every t ≥ 0, and kW (t, ·)k(H 2 (IR))2
is exponentially decreasing to 0.
Finally we must choose a smooth control function such that u(t, x) is close to
uδ1 ,θ1 ,σ1 (t, x) at initial time, and close to uδ2 ,θ2 ,σ2 (t, x) for large times. Hence, we
can choose the function δ such that δ(t) = δ1 for t ≤ 0. Then, with the reasoning
above, we enforce v(t, x) to remain close to M0 (x), that is, the solution u(t, x)
follows the profile uδ(t),θ1 ,σ1 (t, x). At times t ≥ T , we require u(t, x) to be close to
uδ2 ,θ2 ,σ2 (t, x) for some θ2 ; one must have, for t ≥ T ,
−σ1 + δ(t)t = −σ2 + δ2 t,
and hence,
σ1 − σ2
.
t
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).
δ(t) = δ2 +
SMOOTH CONTROL OF NANOWIRES
9
The first part of the theorem, on the interval [0, T ], then follows from the above
considerations.
For the second part, we use a stronger version of the estimate (27), namely,
kW (s, ·)k(H 2 (IR))2 ≤ C3 e−C4 s kW (0, ·)k(H 2 (IR))2 + ξ 2 t2 δ̇(t)2 .
Since t2 δ̇(t)2 is integrable, it follows from the above estimate, and from (17) and
(19), that kΛ0 (t)kIR2 is integrable on [0, +∞). Hence, Λ(t) has a limit in IR2 , denoted
Λ∞ = (θ∞ , σ∞ ), as t tends to +∞. The theorem follows with θ20 = θ2 + θ∞ and
σ20 = σ2 + σ∞ .
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