# Feedback control of charged ideal fluids

## TL;DR

This paper extends control theory to ideal charged fluids in magnetic fields, designing feedback controls that stabilize specific fluid flows using Lie-Poisson structures and energy-momentum methods.

## Contribution

It introduces a feedback control framework for charged ideal fluids with magnetic fields, ensuring closed-loop equations are Lie-Poisson and enabling stabilization of fluid flows.

## Key findings

- Feedback control depends on the Eulerian state.
- Closed loop equations are Lie-Poisson.
- System can be stabilized for long channels.

## Abstract

The theory of controlled mechanical systems of [6, 3, 4] is extended to the case of ideal incompressible fluids consisting of charged particles in the presence of an external magnetic field. The resulting control is of feedback type and depends on the Eulerian state of the controlled system. Moreover, the control is set up so that the corresponding closed loop equations are Lie-Poisson. This implies that the energy-momentum method of [1, 11] can be used to find a stabilizing control. As an example the case of planar parallel shear flow with an inflection point is treated. A state dependent feedback control is constructed which stabilizes the system for an arbitrarily long channel.