Control of the Landau–Lifshitz Equation

A.N. Chow and K.A. Morris

Department of Applied Mathematics, University of Waterloo, Canadaa29chow@uwaterloo.ca, kmorris@uwaterloo.ca

Abstract

The Landau–Lifshitz equation describes the dynamics of magnetization inside a ferromagnet. This equation is nonlinear andhas an infinite number of stable equilibria. It is desirable to control the system from one equilibrium to another. A controlthat moves the system from an arbitrary initial state, including an equilibrium point, to a specified equilibrium is presented.It is proven that the second point is an asymptotically stable equilibrium of the controlled system. The results are illustratedwith some simulations.

Key words: Asymptotic stability, Equilibrium, Lyapunov function, Nonlinear control systems, Partial differential equations

1 Introduction

The Landau–Lifshitz equation is a partial differentialequation (PDE), which describes the magnetic be-haviour within ferromagnetic structures. This equationwas originally developed to model the behaviour of do-main walls, which separate magnetic regions within aferromagnet [1]. Ferromagnets are often found in mem-ory storage devices such as hard disks, credit cards ortape recordings. Each set of data stored in a memorydevice is uniquely assigned to a specific stable magneticstate of the ferromagnet, and hence it is desirable tocontrol magnetization between different stable equilib-ria. This is difficult due to the presence of hysteresis inthe Landau–Lifshitz equation. Hysteresis indicates thepresence of multiple equilibria [2,3]. Because of this, aparticular control can lead to different magnetizations;that is, the particular path of magnetization dependson the initial state of the system and looping in theinput-output map is typical [2,3].

There is now an extensive body of results on control andstabilization of linear PDE’s; see for instance the books[4–7] and the review paper [8]. Stability results for theLandau-Lifshitz equation are often based on lineariza-tion [9–12]. In these works, the spectral properties ofthe linear operator are determined. In [13], sufficient as-sumptions are made that simplify a general form of theLandau-Lifshitz equation into an ordinary differentialequation; and based on this, the magnetization dynam-ics are shown to be stable.

The magnetic state of a ferromagnet can be changed by

an applied magnetic field, which is viewed as the con-trol. From this physically meaningful perspective, thecontrol enters the Landau-Lifshitz equation nonlinearly.In [14], the Landau–Lifshitz equation is linearized andshown to have an unstable equilibrium; and to stabi-lize this equilibrium a control that is the average of themagnetization in one direction and zero in the other twodirections is used. In [15,16], solutions to the Landau–Lifshitz equation are shown to be arbitrarily close to do-main walls given a constant control. Experiments andnumerical simulations demonstrating the control of do-mains walls in a nanowire are presented in [17,18].

In the next section, the uncontrolled Landau–Lifshitzequation is described. It is known to have multiple stableequilibria [19, Theorem 6.1.1]. In section 3, a control,acting as the applied magnetic field, is introduced intothe Landau–Lifshitz equation nonlinearly. The controlobjective is to steer the system dynamics between stableequilibrium points. Results demonstrate the controlledLandau–Lifshitz equation is stable, and the linearizedcontrolled Landau–Lifshitz equation is asymptoticallystable. In Section 4, simulations for the full equation arepresented.

2 Landau-Lifshitz Equation

Consider the magnetization

m(x, t) = (m1(x, t),m2(x, t),m3(x, t)),

at position x ∈ [0, L] and time t ≥ 0 in a long thin ferro-magnetic material of length L > 0. If only the exchange

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energy term is considered, the magnetization is modelledby the one–dimensional (uncontrolled) Landau–Lifshitzequation [20],[19, Chapter 6]

∂m

∂t= m×mxx − νm× (m×mxx) (1a)

m(x, 0) = m0(x) (1b)

where × denotes the cross product and ν ≥ 0 is thedamping parameter, which depends on the type of fer-romagnet. The term mxx denotes magnetization differ-entiated with respect to x twice. The Landau–Lifshitzequation sometimes includes a parameter called the gy-romagnetic ratio multiplying m ×mxx. The gyromag-netic ratio has been set to 1 for simplicity. For more onthe damping parameter and gyromagnetic ratio, see [21].

The Landau–Lifshitz equation is a coupled set of threenonlinear PDEs. It is assumed that there is no magneticflux at the boundaries and so Neumann boundary con-ditions are appropriate:

mx(0, t) = mx(L, t) = 0. (1c)

Existence and uniqueness of solutions to (1) with differ-ent degrees of regularity has been shown [22,23].

Theorem 1. [19, Lemma 6.3.1] If ||m0(x)||2 = 1, thesolution, m, to (1a) satisfies

||m(x, t)||2 = 1 (2)

where || · ||2 is the Euclidean norm.

The following statement is a more restrictive version ofthe theorem stated in [22].

Theorem 2. [22, Thm. 1.3,1.4]. If m0 ∈ H2(0, L),m0,x(0) = m0,x(L) = 0 and ‖m0‖2 = 1, then there ex-ists a time T ∗ > 0 and an unique solution m of (1)such that for all T < T ∗, m ∈ C([0, T ;H2(0, L)) ∩L2(0, L;H3(0, L)).

With more general initial conditions, solutions to (1)are defined on L3

2 = L2([0, L];R3) with the usual inner–product and norm. The notation ‖ · ‖L3

2is used for the

norm. Define the operator

f(m) = m×mxx − νm× (m×mxx) , (3)

and its domain

D = {m ∈ L32 : mx ∈ L3

2, mxx ∈ L32,

mx(0) = mx(L) = 0}. (4)

Theorem 3. [24, Theorem 4.7] The operator f(m) withdomain D generates a nonlinear contraction semigroupon L3

2.

Ferromagnets are magnetized to saturation [25, Sec-tion 4.1]; that is ||m0(x)||2 = Ms where Ms is the mag-netization saturation. In much of the literature, Ms isset to 1; see for example, [19, Section 6.3.1], [22,23,26].This convention is used here. Physically, this means thatat each point, x, the magnitude of m0(x) equals themagnetization saturation. The initial condition m0(x)is furthermore assumed to be real–valued, and hencem(x, t) for t > 0 is real–valued.

The set of equilibrium points of (1) is [19, Theorem 6.1.1]

E ={a = (a1, a2, a3) : a1, a2, a3 constants and aTa = 1}.(5)

Theorem 4. [24, Theorem 4.11] The equilibrium set in(5) is asymptotically stable in the L3

2–norm.

3 Controller Design

In current applications, the control enters as an ap-plied magnetic field [9,10,14–16]. More precisely, a con-trol, u(t), is introduced into the Landau-Lifshitz equa-tion (1a) as follows

∂m

∂t=m× (mxx + u)− νm× (m× (mxx + u))

=m×mxx − νm× (m×mxx)

+ m× u− νm× (m× u) , (6)

m(x, 0) = m0(x).

As for the uncontrolled system, the boundary condi-tions are mx(0, t) = mx(L, t) = 0. Equation (6) is theLandau-Lifshitz equation with a nonlinear control. Itsexistence and uniqueness results can be found in [22,Thm. 1.1,1.2] and is similar to Theorem 2.

As for the uncontrolled equation, since

1

2

∂||m(x, t)||2∂t

=mT ∂m

∂t=mT(m×mxx − νm× (m×mxx)

+ m× u− νm× (m× u)) = 0,

this implies ||m||2 = c, where c is a constant. The con-vention is to take c = 1. It follows that any equilibriumpoint is trivially stable in the L2-norm.

The goal is to choose a control so that the systemgoverned by the Landau–Lifshitz equation moves froman arbitrary initial condition, possibly an equilibriumpoint, to a specified equilibrium point r. The control

2

needs to be chosen so that r becomes a stable equilib-rium point of the controlled system. It can be shownthat zero is an eigenvalue of the linearized uncontrolledLandau–Lifshitz equation [24, Chapter 4.3.2]. For finite-dimensional linear systems, simple proportional controlof a system with a zero eigenvalue yields asymptotictracking of a specified state and this motivates choosingthe control

u = k(r−m) (7)

where r ∈ E is an equilibrium point of the uncontrolledequation (1) and k is a positive constant control param-eter.

Theorem 5. For any r ∈ E and any positive constantk with control defined in (7), r is a locally stable equi-librium point of (6) in the H1-norm. That is, for anyinitial condition m0(x) ∈ D, where D is defined in (4),the H1-norm of the error m− r does not increase.

PROOF. Let B(r, p) = {m ∈ L32 : ||m− r||L3

2< p} ⊂

D for some constant 0 < p < 2. Note that since p < 2,then −r /∈ B(r, p). For any m ∈ B(r, p), consider theH1-norm of the error

V (m) = k ||m− r||2L32

+ ||mx||2L32.

Taking the derivative of V ,

dV

dt=

∫ L

0

k(m− r)Tmdx+

∫ L

0

mTx mxdx

=

∫ L

0

k(m− r)Tmdx−∫ L

0

mTxxmdx

=

∫ L

0

(k(m− r)Tm−mT

xxm)dx. (8)

Let h = m− r, then the integrand becomes

khTm−mTxxm (9)

and equation (6) becomes

m = m× (mxx − kh)− νm× (m× (mxx − kh))

where the dot represents differentiation with respect tot. It follows that

hTm =hT (m×mxx)− ν (m×mxx)T

(h×m)

− νk||m× h||22 (10)

and

mTxxm =− kmT

xx (m× h) + ν||m×mxx||22+ νk (m× h)

T(mxx ×m) . (11)

Substituting (10) and (11) into equation (9) leads to

khTm−mTxxm =2νk (m×mxx)

T(m× h)

− νk2||m× h||22 − ν||m×mxx||22=− ν||m×mxx − km× h||22

Substituting this expression into equation (8) leads to

dV

dt=− ν||m× (mxx + u)||2L3

2≤ 0.

Thus, the H1-norm of the error does not increase.

For any equilibrium point r ∈ E of (6) and m ∈ D, letm = r+v where v is any admissible perturbation; thatis, v ∈ D and ‖r + v‖2 = 1. The linearization of (6) atr with control defined in (7) is

∂v

∂t=r× vxx − νr× (r× vxx)

+ kv × r− kνr× (v × r) (12)

v(0) = v0.

The following lemmas are needed in the proof of Theo-rem 8. The result in Lemma 6 is a consequence of theproduct rule.

Lemma 6. For m ∈ L32, the derivative of g = m×mx

is gx = m×mxx.

Lemma 7. For m ∈ L32 satisfying (1c),∫ L

0

(m− r)T(m×mxx)dx = 0.

PROOF. Integrating by parts, and applying Lemma 6and the boundary conditions (1c) implies that∫ L

0

(m− r)T(m×mxx)dx = −∫ L

0

mTx (m×mx)dx.

From properties of cross products, mTx (m × mx) =

mT(mx ×mx) = 0, and hence the integral is zero.

Theorem 8. Let r ∈ E be an equilibrium point of (12).For any positive constant k, r is a locally asymptoticallystable equilibrium of (12) in the L3

2-norm.

PROOF. For an admissible v with ||v− r||2 ≤ 2, con-sider the Lyapunov candidate

V (v) =1

2||v − r||2L3

2.

3

It is clear that V ≥ 0 for all v ∈ D and furthermore,V (v) = 0 only when v = r. Therefore, V (v) > 0 for allv ∈ D\{r}.

Taking the derivative of V (v) implies

dV

dt=

∫ L

0

(v − r)Tvdx

and substituting in (12) leads to

dV

dt=

∫ L

0

(v − r)T(r× vxx)dx

− ν∫ L

0

(v − r)T (r× (r× vxx)) dx

+ k

∫ L

0

(v − r)T(v × r)dx

− kν∫ L

0

(v − r)T (r× (v × r)) dx.

The first integral is zero, and to show this the proofis similar to the one in Lemma 7. Applying integratingby parts, Lemma 6 and the boundary conditions (1c)implies that

∫ L

0

(v − r)T(r× vxx)dx = −∫ L

0

vTx (v × vx)dx,

which is equal to zero from properties of cross products.The second integral can be written as

∫ L

0

(v − r)T (r× (r× vxx)) dx

=

∫ L

0

(r× vxx)T

((v − r)× r) dx

=

∫ L

0

(r× vxx)T

(v × r) dx,

then integrating by parts, and applying Lemma 6 andthe boundary conditions leads to

∫ L

0

(r× vxx)T

(v × r) dx = −∫ L

0

(vx × r)T

(r× vx) .

Therefore,

∫ L

0

(v − r)T (r× (r× vxx)) dx = ||vx × r||2L32.

Letting h = v − r, it follows that

dV

dt=− ν||vx × r||2L3

2+ k

∫ L

0

(v − r)T(v × r)dx

− kν∫ L

0

(v − r)T (r× (v × r)) dx

=− ν||vx × r||2L32

+ k

∫ L

0

hT(h× r)dx

− kν∫ L

0

hT (r× (h× r)) dx.

The first integral is zero since hT(h×r) = rT(h×h) = 0,and the last integral can be simplified using the fact that

hT (r× (h× r)) = (h× r)T

(h× r) .

Therefore,

dV

dt= −ν||vx × r||2L3

2− kν||h× r||2L3

2.

For k > 0,

dV

dt= −ν

(||vx × r||2L3

2+ k||h× r||2L3

2

)≤ 0

and furthermore, dV /dt = 0 if and only if vx × r = 0and h× r = v× r = 0. This is true only if v = αr whereα is any scalar. Since ||r + v||2 = 1 and ||v − r||2 ≤ 2,then α = 0.

It follows that r is a locally asymptotically stable equi-librium point of (12).

4 Example

Simulations illustrating the stabilization of the Landau-Lifshitz equation were done using a Galerkin approxi-mation with 12 linear spline elements. For the followingsimulations, the parameters are ν = 0.02 and L = 1with initial condition m0(x) = (sin(2πx), cos(2πx), 0).Figure 1 illustrates that the solution to the uncontrolledLandau–Lifshitz equation settles to r0 = (0,−0.6, 0).

Stabilization of the Landau–Lifshitz equation with non-linear control (6) is illustrated in Figure 2 with the sec-ond equilibrium point chosen to be r1 = (− 1√

2, 0, 1√

2).

The control parameter is k = 10. It is clear from thefigure that the system converges to the specified equi-librium point, r1. The control can also be applied afterthe dynamics have settled to r0 as shown in Figure 3.In Figure 4, the dynamics settle (without a control) tor0, then the control is applied in succession twice, whichforces the system from r0 to r1, and then finally tor2 = (0, 1, 0).

4

5 Conclusion

The Landau-Lifshitz equation is a nonlinear system ofPDEs with multiple equilibrium points. In applications,the control enters through an applied field and the con-trol enters nonlinearly. The presence of a 0 eigenvaluein the linearized equation suggested that a simple feed-back proportional control could be used to steer the sys-tem to an arbitrary equilibrium point. It was shown thatthe controlled Landau-Lifshitz equation with a nonlinearcontrol has a stable equilibrium point and the lineariza-tion has an asymptotically stable equilibrium point.

Simulations indicate that proportional control also sta-bilizes the fully nonlinear model. This suggests the non-linear equation has an asymptotically stable equilibriumpoint. Proving this remains an open research problem.This would be a significant contribution as the Landau–Lifshitz equation is not quasi-linear, which means lin-earization is not guaranteed to predict stability of thenonlinear equation [27,28].

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5

Fig. 1. Magnetization in the uncontrolled Landau–Lifshitzequation moves from initial condition m0(x), to the equilib-rium r0 = (0,−0.6, 0).

Fig. 2. Magnetization in the Landau–Lifshitz equation withnonlinear control moves from the initial condition m0(x)to the specified equilibrium r1 = (− 1√

2, 0, 1√

2) with control

parameter k = 10.

Fig. 3. Magnetization in the Landau–Lifshitz equationmoves from the initial condition m0(x) to the equilibriumr0 = (0,−0.6, 0) without a control. The control, u withk = 10 is then applied to the equation nonlinearly and steersthe dynamics to the specified equilibrium r1 = (− 1√

2, 0, 1√

2).

Fig. 4. Magnetization in the Landau–Lifshitz equationmoves from the initial condition m0(x) to the equilibriumr0 = (0,−0.6, 0) without a control. The control u withk = 10 is then applied to the equation nonlinearly and steersthe dynamics to the specified equilibrium r1 = (− 1√

2, 0, 1√

2),

and then to another equilibrium, r2 = (0, 1, 0).

6