# Asymptotic ensemble stabilizability of the Bloch equation

## TL;DR

This paper develops control strategies to asymptotically stabilize an ensemble of non-interacting spins in the down position, addressing both finite and countably infinite cases with bounded, continuous controls.

## Contribution

It introduces novel feedback control methods for stabilizing spin ensembles, including a sequence of controls for infinite ensembles, with convergence guarantees.

## Key findings

- Finite ensemble stabilization with feedback control
- Sequence of controls for countably infinite ensembles
- Controls are uniformly bounded and continuous

## Abstract

In this paper we are concerned with the stabilizability to an equilibrium point of an ensemble of non interacting half-spins. We assume that the spins are immersed in a static magnetic field, with dispersion in the Larmor frequency, and are controlled by a time varying transverse field. Our goal is to steer the whole ensemble to the uniform "down" position. Two cases are addressed: for a finite ensemble of spins, we provide a control function (in feedback form) that asymptotically stabilizes the ensemble in the "down" position, generically with respect to the initial condition. For an ensemble containing a countable number of spins, we construct a sequence of control functions such that the sequence of the corresponding solutions pointwise converges, asymptotically in time, to the target state, generically with respect to the initial conditions. The control functions proposed are uniformly bounded and continuous.