Local Exact Controllability of Landau-Lifshitz-Gilbert Equation

Abstract

We prove a local exact controllability result for controlled Landau--Lifshitz--Gilbert equations on ${\mathbb{T}}^2$: if the initial energy is sufficiently small, then for any terminal time $T>0$, there is a localised external magnetic field such that the system can be steered exactly to the terminal value of any nearby uncontrolled trajectory. We first transform the equation to a quasilinear parabolic system on ${\mathbb{R}}^2$ by a suitable stereographic chart. Then the Carleman estimate is established for the linearised system through a decomposition adapted to the self-adjoint and skew-adjoint structure of the conjugated adjoint operator. This yields observability and $L^{\infty }$-null controllability for the linearised system. The nonlinear projected equation is then recovered by a Kakutani fixed-point argument. We also obtain a semi-global controllability result under a hemisphere condition.