Control of harmonic map heat flow with an external field

TL;DR

This paper demonstrates how to control the harmonic map heat flow using an external magnetic field, enabling the steering of the system to any desired state within a short time under specific initial conditions.

Contribution

It introduces a novel control method for harmonic map heat flow via external magnetic fields, combining spatial and uniform controls and reducing the problem to a local controllability framework.

Findings

System can be steered to any ground state within short time

Control method applies to initial data in a hemisphere

Explicit solution of algebraic system enables control conversion

Abstract

We investigate the control problem of harmonic map heat flow by means of an external magnetic field. In contrast to the situation of a parabolic system with internal or boundary control, the magnetic field acts as the coefficients of the lower order terms of the equation. We show that for initial data whose image stays in a hemisphere, with one control acting on a subset of the domain plus a spatial-independent control acting on the whole domain, the state of the system can be steered to any ground state, i.e. any given unit vector, within any short time. To achieve this, in the first step a spatial independent control is applied to steer the solution into a small neighborhood of the peak of the hemisphere. Then under stereographic projection, the original system is reduced to an internal parabolic control system with initial data sufficiently close to $0$ such that the existing method for local controllability can be applied. The key process in this step is to give an explicit solution of an underdetermined algebraic system such that the affine type control can be converted into an internal control.