Control of the
Landau-Lifshitz Equation
Amenda Chow
Department of Applied Mathematics University of Waterloo [email protected]
What is control theory?
Definition: Control theory is the introduction of an input into a dynamical system to steer the system to a desired objective.
Objective: control stability.
How is stability defined?
How is the control objective achieved?
Defining Stability
˜
z stable z asymptotically stable˜
Feedback Control
Dynamical System -output, y(t)
The error is e(t) = r(t) − y(t). Objective is for e(t) → 0 as t → ∞.
Feedback Control
i -
- Controller -Dynamical System -
6 -
reference input, r(t) error, e(t) control, u(t)
output, y(t)
The error is e(t) = r(t) − y(t).
Objective is for e(t) → 0 as t → ∞.
Control of the Landau–Lifshitz Equation
Relevance and Motivation
• Describes magnetization in a ferromagnetic object, e.g. nanowire
• Ferromagnets are used to store data, found in hard disks, credit cards, etc.
• Each set of data stored is uniquely assigned to a specific stable equilibrium of the ferromagnet.
• Magnetic states can be changed by an applied magnetic field (control)
• Unfortunately, more than one stable equilibrium is possible for a particular input of the applied magnetic field.
• Controlling the stable states allows for exact storage of data
Nonlinear PDE Example: Landau–Lifshitz Equation
∂m
∂t = m × mxx− νm × (m × mxx) , m(x, 0) = m0(x) Three coupled nonlinear PDEs
mx(0, t) = 0 and mx(L, t) = 0
m(x, t) =
m1(x, t) m2(x, t) m3(x, t)
, mi(x, t) ∈ L2[0, L]
ν ≥ 0 is the damping parameter
Nanowire
Magnetization
Solution to Landau-Lifshitz Equation
Initial condition: m0(x) = (sin(2πx), cos(2πx), 0) Settles to a stable equilibirium, a = (0, −0.6, 0).
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m1 m2
m3
Landau–Lifshitz Equation: Equilibria and its Stability
Any a = (a1, a2, a3) is an equilibrium where ai is a constant.
The set of equilibrium points is E = {(a1, a2, a3) : a1, a2, a3 constants}.
E is asymptotically stable.
Proof uses Lyapunov’s Theorem, not linearization.
Control of the Landau–Lifshitz Equation
Suppose system is initially at a stable equilibrium, a.
Let r be another stable equilibrium with r 6= a.
Objective: Force system to move from a to r.
How: Use feedback control Control causes: a is no longer an equilibrium
r is an equilibrium r is asymptotically stable
Control of the Landau–Lifshitz Equation
- d - k - Landau–Lifshitz -
-6
r e(t) u(t)
m(t)
∂m
∂t = m × mxx− νm × (m × mxx) + bu(t) e(t) = r − m
u(t) = ke(t)
where k and b are nonzero real constants.
Objective: control, u, forces e → 0 which implies m → r
Controlled Landau–Lifshitz Equation
The controlled Landau–Lifshitz equation is
∂m
∂t = m × mxx− νm × (m × mxx) + bk (r − m)
a is no longer an equilibrium
X
r is an equilibrium
X
Is r asymptotically stable?
Yes, if k ≥ 8νL4 b .
Proof uses Lyapunov’s Theorem, not linearization.
Controlled Landau–Lifshitz Equation
The controlled Landau–Lifshitz equation is
∂m
∂t = m × mxx− νm × (m × mxx) + bk (r − m)
a is no longer an equilibrium
X
r is an equilibrium
X
Is r asymptotically stable?
Yes, if k ≥ 8νL4 b .
Proof uses Lyapunov’s Theorem, not linearization.
Controlled Landau–Lifshitz Equation
The controlled Landau–Lifshitz equation is
∂m
∂t = m × mxx− νm × (m × mxx) + bk (r − m)
a is no longer an equilibrium
X
r is an equilibrium
X
Is r asymptotically stable?
Yes, if k ≥ 8νL4 b .
Proof uses Lyapunov’s Theorem, not linearization.
Controlled Landau–Lifshitz Equation
The controlled Landau–Lifshitz equation is
∂m
∂t = m × mxx− νm × (m × mxx) + bk (r − m)
a is no longer an equilibrium
X
r is an equilibrium
X
Is r asymptotically stable?
Yes, if k ≥ 8νL4 b .
Proof uses Lyapunov’s Theorem, not linearization.
Controlled Landau–Lifshitz Equation
The controlled Landau–Lifshitz equation is
∂m
∂t = m × mxx− νm × (m × mxx) + bk (r − m)
a is no longer an equilibrium
X
r is an equilibrium
X
Is r asymptotically stable?
Yes, if k ≥ 8νL4 b .
Proof uses Lyapunov’s Theorem, not linearization.
Control Results for the Landau–Lifshitz Equation
r is globally asymptotically stable
• control from an arbitrary initial magnetization to an arbitrary stable equilibrium
• control from one arbitrary stable equilibrium to another arbitrary stable equilibrium
• analysis does not require linearization
Control from one Stable Equilibrium to Another Stable Equilibrium
Initial condition: m0(x) = (sin(2πx), cos(2πx), 0) Control from a = (0, −0.6, 0) to r = (0, 0, 1)
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m m3
Control from an Initial Magnetization to a Stable Equilibrium
Initial condition: m0(x) = (sin(2πx), cos(2πx), 0) Control to r = (0, 0, 1)
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m m3
Control to Any Stable Equilibrium (r globally asymptotically stable)
Initial condition: m0(x) = (sin(2πx), cos(2πx), 0) Control to r = (0, 0, 12)
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m m3
Future Research
• control from an arbitrary magnetization to another arbitrary magnetization
• relax requirements
k ≥ 8νL4 b
on control parameters
• analysis of stability in nonlinear infinite–dimensional systems
References
A. Chow, K.A. Morris, “Control of Hysteresis in the Landau–Lifshitz Equation”, Automatica, submitted
A. Chow, K.A. Morris, “Hysteresis in the Linearized Landau–Lifshitz Equation”, in Proceedings of the American Control Conference, 2014.
R. al Jamal, A. Chow and K.A. Morris, “Linearized Stability Analysis of Nonlinear Partial Differential Equations”, In Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, 2014.
