MC 5136
Candidate
Amenda Chow, Applied Math, University of À¶Ý®ÊÓƵ
Title
Control of Hysteresis in theÌýLandau-Lifshitz Equation
Abstract
There are two main tools for determining the stability of nonlinear partial differential equations (PDEs): Lyapunov Theory and linearization. The former has the advantage of providingÌýstability results for nonlinear equations directly, while the latter considers the stability ofÌýlinear equations and then further justification is needed to show the linear stability impliesÌýstability of the nonlinear equation. Linearization has the advantage of investigating stabilityÌýon a simpler equation; however, the justification can be difficult to prove.
Both Lyapunov theory and linearization are applied to the Landau-Lifshitz equation, aÌýnonlinear PDE that describes the behaviour of magnetization inside a magnetic object. It isÌýknown that the Landau-Lifshitz equation has an infinite number of stable equilibrium points.ÌýWe present a control that forces the system from one equilibrium to another. This is provedÌýusing Lyapunov Theory. The linear Landau-Lifshitz equation is also investigated because itÌýprovides insight to the nonlinear equation. The linear model is shown to be well-posed andÌýits eigenvalue problem is solved. The resulting eigenvalues suggest an appropriate controlÌýfor the nonlinear Landau-Lifshitz equation. Mathematically, the control causes the initialÌýequilibrium to no longer be an equilibrium and the second point to be an asymptoticallyÌýstable equilibrium point. This implies the magnetization has moved to the second equilibriumÌýand hence the control objective is successfully achieved.
The existence of multiple stable equilibria is closely related to hysteresis. This is a phenomenon that is often characterized by a looping behaviour; however, the existence of aÌýloop is not sufficient to identify hysteretic systems. A precise definition is required, whichÌýis presented, and applied to the Landau-Lifshitz equation (both linear and nonlinear) toÌýestablish the presence of hysteresis.