Stabilization for an ensemble of half-spin systems

TL;DR

This paper develops an explicit feedback control law to stabilize an ensemble of non-interacting half-spin systems described by Bloch equations, demonstrating local convergence and effectiveness through simulations.

Contribution

It introduces a novel explicit feedback stabilization method for infinite-dimensional spin systems with proven local convergence.

Findings

Feedback law achieves asymptotic stabilization around uniform spin states.

Convergence proof adapts LaSalle invariance principle to infinite dimensions.

Numerical simulations confirm effectiveness even from far initial conditions.

Abstract

Feedback stabilization of an ensemble of non interacting half spins described by Bloch equations is considered. This system may be seen as a prototype for infinite dimensional systems with continuous spectrum. We propose an explicit feedback law that stabilizes asymptotically the system around a uniform state of spin +1/2 or -1/2. The proof of the convergence is done locally around the equilibrium in the H^1 topology. This local convergence is shown to be a weak asymptotic convergence for the H^1 topology and thus a strong convergence for the C^0 topology. The proof relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. Numerical simulations illustrate the efficiency of these feedback laws, even for initial conditions far from the equilibrium