Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch Equations

TL;DR

This paper investigates the controllability of the Bloch equation for spin ensembles, revealing limitations in exact controllability but demonstrating approximate controllability and explicit control strategies for asymptotic state transfer.

Contribution

It provides new mathematical insights into the controllability of infinite-dimensional systems with continuous spectra, specifically addressing the Bloch equation with dispersion.

Findings

Not exactly controllable in finite time with bounded controls

Approximately controllable in finite time with unbounded controls

Explicit controls for asymptotic exact controllability to uniform states

Abstract

We study the controllability of the Bloch equation, for an ensemble of non interacting half-spins, in a static magnetic field, with dispersion in the Larmor frequency. This system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum, whose controllability is not well understood. We provide several mathematical answers, with discrimination between approximate and exact controllability, and between finite time or infinite time controllability: this system is not exactly controllable in finite time $T$ with bounded controls in $L^2(0,T)$, but it is approximately controllable in $L^\infty$ in finite time with unbounded controls in $L^{\infty}_{loc}([0,+\infty))$. Moreover, we propose explicit controls realizing the asymptotic exact controllability to a uniform state of spin +1/2 or -1/2.