Asymptotic Stability of the Landau-Lifshitz Equation
Amenda Chow









Department of Mathematics and Statistics, York University, Canada
 amchow@yorku.ca
 
                            
Asymptotic stability, Equilibrium Points, Hysteresis, Lyapunov function, Nonlinear control systems, Partial differential equations            
            
          

The Landau–Lifshitz equation describes the behaviour of magnetic domains in ferromagnetic structures. Recently such structures have been found to be favourable for storing digital data. Stability of magnetic domains is important for this. Consequently, asymptotic stability of the equilibrium points in the Landau–Lifshitz equation are established.  A suitable Lyapunov function is presented. 






§ INTRODUCTION

The Landau–Lifshitz equation is a coupled set of nonlinear partial differential equations. One of its first appearances is in a 1935 paper <cit.>, in which this equation describes the behaviour of magnetic domains within a ferromagnetic structure. In recent applications, structures such as ferromagnetic nanowires have appeared in electronic devices used for storing digital information <cit.>.  In particular, data is encoded as a specific pattern of magnetic domains within a ferromagnetic nanowire. Consequently the Landau–Lifshitz equation continues to be widely explored, and its stability is of growing interest <cit.>. Stability of equilibrium points is also related to hysteresis <cit.>, and investigating stability lends insights into the hystereric behaviour that appears in the Landau–Lifshitz equation <cit.>. 

Stability results are often based on linearization <cit.>; however, in the proceeding discussion, asymptotic stability of the Landau-Lifshitz equation is established using Lyapunov theory. This is preferred because linearization leads only to an approximation. 


The difficulty with Lyapunov theory is often in the construction of an appropriate Lyapunov function.  Working in infinite-dimensions makes this more difficult; however, Lyapunov functions have been found for the Landau–Lifshitz equation <cit.>.  In both these works, a Lyapunov function establishes that the equilibrium points of the Landau-Lifshitz equation are stable. The work in <cit.> is extended here and asymptotic stability is shown. In particular, a nonlinear control is shown to steer the system to an asymptotically stable equilibrium point.  Control of the Landau–Lifshitz equation is crucial as this means the behaviour of the domain walls, which contains the encoded data, can be fully determined <cit.>. 

The control objective is to steer the system dynamics to any arbitrary equilibrium point, which requires asymptotic stability. This is presented in Theorem <ref>, which is the main result and can be found in Section <ref>.  A summary and future avenues are in the last section. To begin, a brief mathematical review of the Landau–Lifshitz equation is discussed next.




§ LANDAU-LIFSHITZ EQUATION


Consider a one dimensional ferromagnetic nanowire of length L>0. Let   𝐦(x,t)=(m_1(x,t),m_2(x,t),m_3(x,t)) be the magnetization of the ferromagnetic nanowire for some position x ∈ [0,L]  and time t ≥ 0 with initial magnetization 𝐦(x,0)=𝐦_0(x).  These dynamics are determined by  

    ∂𝐦/∂ t  = 𝐦×( 𝐦_xx+𝐮)-ν𝐦×(𝐦×(𝐦_xx+𝐮)) 
       𝐦_x(0,t)=  𝐦_x(L,t)=0, 
         || 𝐦(x,t)||_2 =1.

where × denotes the cross product and ||·||_2 is the Euclidean norm. Equation (<ref>) is the one–dimensional controlled Landau–Lifshitz equation <cit.>.  It satisfies the constraint in (<ref>), which means the magnitude of the magnetization is uniform at every point of the ferromagnet. The exchange energy is 𝐦_xx. Mathematically, 𝐦_xx denotes magnetization differentiated with respect to x twice. The parameter ν≥  0 is the damping parameter, which depends on the type of ferromagnet. The applied magnetic field, denoted 𝐮(t), acts as the control, and hence, when 𝐮(t)=0, equation (<ref>)  can be thought of as the uncontrolled Landau–Lifshitz equation. Neumann boundary conditions  (<ref>) are used here. 

Existence and uniqueness results  can be found in <cit.> and references therein.  Solutions to (<ref>) are defined on ℒ_2^3 = ℒ_2 ([0,L]; ℝ^3) with the usual inner–product and norm with domain

    D={   𝐦∈ℒ_2^3 :  𝐦_x ∈ℒ_2^3,  
       𝐦_xx∈ℒ_2^3,  𝐦_x(0)=𝐦_x(L) = 0}.

The notation ·_ℒ_2^3 is used for the norm.









§ ASYMPTOTIC STABILITY

For 𝐮(t)=0, the set of equilibrium points is

    E=   {𝐚=(a_1,a_2,a_3) : a_1,a_2,a_3
       ||𝐚||_2=1 },
 
which satisfies the boundary conditions in (<ref>) <cit.>.  It is clear E contains an infinite number of equilibria. A particular 𝐚∈ E is stable but not asymptotically stable <cit.>; however, the set E is asymptotically stable in the ℒ_2^3–norm <cit.>. 

Let 𝐫 be an arbitrary equilibrium point of E with r_1≠ 0 to ensure ||𝐫||_2=1, that is,  (<ref>), is  satisfied. Define the control in (<ref>) to be  


    𝐮=k(𝐫 -𝐦 )

where  k is a scalar constant. This is the same control used in <cit.>.  
For this control, 𝐫 is an equilibrium point of the controlled Landau-Lifshitz equation (<ref>). It is shown in Theorem <ref> that 𝐫 is locally asymptotically stable, which is the main result.   Simulations demonstrating asymptotic stability of the Landau-Lifshitz equation given the control in (<ref>) are shown in <cit.>. 


The following lemmas are needed in Theorem <ref>. Lemmas <ref> and <ref> appear in <cit.>, which can be obtained from the product rule and integration by parts, respectively. 

For 𝐦∈ℒ_2^3, the derivative of 𝐠=𝐦×𝐦_x is 𝐠_x=𝐦×𝐦_xx.



For 𝐦∈ℒ_2^3 satisfying (<ref>), 

    ∫_0^L (𝐦-𝐫)^T(𝐦×𝐦_xx)dx=0.




If 𝐫∈ E and 𝐦 satisfies (<ref>), then ||𝐦×𝐫||_2≤1.



Recall ||𝐦×𝐫||_2 = ||𝐦||_2||𝐫 ||_2sin(θ) where θ is the angle between 𝐦. Since 𝐫 and ||𝐦||_2=||𝐫||_2=1, then
||𝐦×𝐫||_2 = ||𝐦||_2||𝐫 ||_2sin(θ) ≤sin(θ)≤ 1.



For any 𝐫∈ E, there exists a range of positive values of k such that 𝐫 is a locally asymptotically stable equilibrium point of (<ref>)  in the ℒ_2^3–norm. 



Let B(𝐫,p)={𝐦∈ℒ_2^3: ||𝐦 -𝐫||_ℒ_2^3<p }⊂ D for some constant 0<p<2.  Note that since p<2, then  -𝐫∉ B(𝐫,p). For any 𝐦∈ B(𝐫,p), the Lyapunov function is 

    V(𝐦)=f(k)/2| | 𝐦-𝐫||_ℒ_2^3^2+1/2| |  𝐦_x||_ℒ_2^3^2

where f(k)>0 is a scalar function of k and |f(k)+k|≤1 for all k>0. Such functions exist. For example, f(k)=k for k∈(0,1/2].  

Taking the derivative of V,

    dV/dt   =∫_0^Lf(k)(𝐦 -𝐫)^T𝐦̇ dx+∫_0^L𝐦_x^T𝐦̇_xdx 
       =∫_0^Lf(k)(𝐦 -𝐫)^T𝐦̇ dx-∫_0^L𝐦_xx^T𝐦̇dx
       =∫_0^L(f(k)(𝐦 -𝐫)^T𝐦̇ -𝐦_xx^T𝐦̇)dx

where the dot notation means differentiation with respect to t. 

Letting 𝐡 = 𝐦 - 𝐫, the integrand in (<ref>) becomes

    f(k)𝐡^T𝐦̇ - 𝐦_xx^T𝐦̇

and equation (<ref>) becomes

    𝐦̇  = 𝐦×( 𝐦_xx-k𝐡)-ν𝐦×(𝐦×(𝐦_xx-k𝐡)).

It follows that 

    𝐡^T𝐦̇  =    𝐡^T[𝐦×( 𝐦_xx-k𝐡)-ν𝐦×(𝐦×(𝐦_xx-k𝐡))] 
    
    =   𝐡^T(𝐦×𝐦_xx) -k 𝐡^T(𝐦×𝐡) 
       -ν𝐡^T[𝐦×(𝐦×𝐦_xx)] +ν k  𝐡^T[𝐦×(𝐦×𝐡)] 
    
    =   𝐡^T(𝐦×𝐦_xx) -ν(𝐦×𝐦_xx)^T(𝐡×𝐦)
       +ν k  (𝐦×𝐡)^T(𝐡×𝐦)  
    
    =   𝐡^T(𝐦×𝐦_xx) -ν(𝐦×𝐦_xx)^T(𝐡×𝐦) 
       -ν k  ||𝐦×𝐡||_2^2

 and 

    𝐦_xx^T𝐦̇ =   𝐦_xx^T[  𝐦×( 𝐦_xx-k𝐡)-ν𝐦×(𝐦×(𝐦_xx-k𝐡)) ] 
    
     =   𝐦_xx^T(  𝐦×𝐦_xx) -k𝐦_xx^T(  𝐦×𝐡)
       -ν𝐦_xx^T[ 𝐦×(𝐦×𝐦_xx)]+ν k𝐦_xx^T[ 𝐦×(𝐦×𝐡)] 
    
     =   -k𝐦_xx^T(  𝐦×𝐡)-ν𝐦_xx^T[ 𝐦×(𝐦×𝐦_xx)] 
       +ν k𝐦_xx^T[ 𝐦×(𝐦×𝐡)] 
    
     =   -k𝐦_xx^T(𝐦×𝐡)-ν(𝐦×𝐦_xx)^T(  𝐦_xx×𝐦)
        +ν k(𝐦×𝐡)^T(𝐦_xx×𝐦) 
    
      =   -k𝐦_xx^T( 𝐦×𝐡)+ν||𝐦×𝐦_xx ||_2^2 
       +ν k(𝐦×𝐡)^T(𝐦_xx×𝐦).


Substituting (<ref>) and (<ref>) into equation (<ref>) leads to

    f(k)𝐡^T𝐦̇ - 𝐦_xx^T𝐦̇
    
    =    f(k)𝐡^T(𝐦×𝐦_xx) -ν f(k) (𝐦×𝐦_xx)^T(𝐡×𝐦) 
        -ν kf(k)  ||𝐦×𝐡||_2^2 +k𝐦_xx^T(  𝐦×𝐡)-ν||𝐦×𝐦_xx ||_2^2
       -ν k(𝐦×𝐡)^T(𝐦_xx×𝐦)
    
    =    (f(k)-k)𝐡^T(𝐦×𝐦_xx) -ν f(k) (𝐦×𝐦_xx)^T(𝐡×𝐦) 
        -ν kf(k)  ||𝐦×𝐡||_2^2 -ν||𝐦×𝐦_xx ||_2^2
       -ν k(𝐦×𝐡)^T(𝐦_xx×𝐦)
    
    =    (f(k)-k)𝐡^T(𝐦×𝐦_xx) 
       -ν (f(k)+k) (𝐦×𝐦_xx)^T(𝐡×𝐦)
       -ν kf(k)  ||𝐦×𝐡||_2^2 -ν||𝐦×𝐦_xx ||_2^2

Substituting this expression into equation (<ref>) leads to

    dV/dt  =   (f(k)-k) ∫_0^L𝐡^T(𝐦×𝐦_xx)dx
       -ν (f(k)+k)  ∫_0^L (𝐦×𝐦_xx)^T(𝐦×𝐡) dx
        - ν kf(k)∫_0^L  ||𝐦×𝐡||_2^2 dx-ν∫_0^L ||𝐦×𝐦_xx ||_2^2 dx.

From Lemma <ref>, the first integral equals zero since 𝐡 = 𝐦 -𝐫. It follows that 

    dV/dt  =   -ν (f(k)+k)  ∫_0^L (𝐦×𝐦_xx)^T(𝐦×𝐡) dx
        - ν kf(k)∫_0^L  ||𝐦×𝐡||_2^2 dx-ν∫_0^L ||𝐦×𝐦_xx ||_2^2 dx
    
    =    -ν (f(k)+k) ∫_0^L (𝐦×𝐦_xx)^T(𝐦×𝐡) dx 
       - ν kf(k) ||𝐦×𝐡||_ℒ_2^3^2-ν  ||𝐦×𝐦_xx ||_ℒ_2^3^2.

Applying the Cauchy-Schwarz Inequality to the integrand leads to

    dV/dt≤   ν |f(k)+k| ∫_0^L ||𝐦×𝐦_xx||_2||𝐦×𝐡||_2 dx 
       - ν kf(k) ||𝐦×𝐡||_ℒ_2^3^2-ν  ||𝐦×𝐦_xx ||_ℒ_2^3^2.

Since ||𝐦×𝐡||_2≤ 1 from  Lemma <ref>, then

    dV/dt≤   ν |f(k)+k| ∫_0^L ||𝐦×𝐦_xx||_2dx - ν kf(k) ||𝐦×𝐡||_ℒ_2^3^2
       -ν  ||𝐦×𝐦_xx ||_ℒ_2^3^2
    
    =   ν |f(k)+k|    ||𝐦×𝐦_xx||_ℒ_2^3^2 - ν kf(k) ||𝐦×𝐡||_ℒ_2^3^2
       -ν  ||𝐦×𝐦_xx ||_ℒ_2^3^2
    
    =   ν(|f(k)+k|-1)    ||𝐦×𝐦_xx||_ℒ_2^3^2 - ν kf(k) ||𝐦×𝐡||_ℒ_2^3^2.


Since |f(k)+k|≤1, then

    dV/dt   ≤ - ν kf(k) ||𝐦×𝐡||_ℒ_2^3^2.

Since k>0 and f(k)>0, then dV/dt≤ 0 with dV/dt =0 if and only if 𝐦×𝐡=0.  For example, suppose f(k)=k on k∈(0,1/2], which satisfies  |f(k)+k|≤1. Equation (<ref>) becomes
 
    dV/dt≤ - ν k^2 ||𝐦×𝐡||_ℒ_2^3^2

 which is clearly less than or equal to zero, and the value of k can be any number in the interval (0,1/2]. For instance, picking k=1/4, the Lyapunov function is
 
    V(𝐦)=1/8| | 𝐦-𝐫||_ℒ_2^3^2+1/2| |  𝐦_x||_ℒ_2^3^2.


Revisiting equation (<ref>), if 𝐦 =𝐫, then 𝐡=0 and hence dV/dt=0. On the other hand, dV/dt=0 implies 𝐦×𝐡=0, and hence 𝐫×𝐦=0. This is a system of algebraic equations,

    r_2m_3-r_3m_2   =0
    
    r_3m_1-r_1m_3   =0
    
    r_1m_2-r_2m_1   =0.

The solution is m_2=r_2/r_1m_1 and m_3=r_3/r_1m_1 for any m_1 and r_1≠0.  Given  ||𝐦||_2=1 and ||𝐫||_2=1, this leads to  m_1^2=r_1^2 and hence m_1=± r_1, which leads to m_2=± r_2 and m_3=± r_3; that is, 𝐦 = ±𝐫.  For V(𝐦) on B(𝐫,p), this implies 𝐦=𝐫 if dV/dt=0. Local  asymptotic stability  follows from Lyapunov's Theorem <cit.>.







§ DISCUSSIONS


Asymptotic stability of an arbitrary equilibrium point of the Landau–Lifshitz equation with Neumann boundary conditions is shown in Theorem <ref>. This is established using Lyapunov theory. The result in Theorem <ref> is an extension of the work presented in <cit.>.  

The control given in (<ref>) can be used to control  the hysteresis that often appears in magnetization dynamics including those described by the Landau–Lifshitz equation <cit.>. Figure <ref> depicts the input-output map of the Landau–Lifshitz equation in (<ref>). The output is the magnetization, 𝐦(x,t)=(m_1(x,t),m_2(x,t),m_2(x,t)), and the input is a periodic function denoted û(t) where ω is the frequency of this input. For each m_i with i=1,2,3, the input is the periodic function 0.01cos(ω t). For this periodic input, equation (<ref>) becomes

    ∂𝐦/∂ t  = 𝐦×( 𝐦_xx+𝐮)-ν𝐦×(𝐦×(𝐦_xx+𝐮)) +û(t).

 As the frequency of the (periodic) input approaches zero, loops appear in the input–output map for m_1(x,t), which indicates the presence of hysteresis <cit.>.  





Because hysteresis is characterized by multiple stable equilibrium points <cit.>, the ability to control the stability of equilibrium points implies the ability to control hysteresis. Such a control for the Landau–Lifshitz equation is given in (<ref>). This is a possible avenue for future exploration.



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