H1-Control of an Ensemble of Half-Spin Systems Replacing Rabi Pulses by Adiabatic Following

This work considers the control of an ensemble of non-interacting half-spin systems (Bloch equations) in a vertical static field B
₀
subject to a pair of controlled radiofrequency inputs (u
₁
(t), u
₂
(t)) acting on the horizontal plane. The state M(t, ω) ∈ S
²
belongs to the Bloch sphere S
²
, and it is indexed by the Larmor frequency ω ∈ (ω
_*
, ω*). Previous works have constructed a local stabilizing feedback based on a Lyapunov functional which is essentially a convenient H
¹
-norm of a Sobolev space H
¹
((ω
_*
, ω*), S
²
) (see Beauchard, Pereira da Silva and Rouchon [3]-[4]). This feedback assures local L°°convergence of the initial state M
₀
(ω) to -e
₃
. However, the control law of that paper is a sum of a (infinite dimensional) state feedback with a T-periodic comb of π-Rabi pulses (Dirac impulses). The present work shows that one may replace this comb of Dirac pulses by adiabatic pulses. It is shown in the paper that, if ||M
₀
(ω) + e
₃
||H
₁
is small enough, our control strategy assures, for any ε > 0, the existence of a pair (T, ℓ) where T ∈ ℝ and ℓ ∈ N such that ||(ℓT,·) + e
₃
||
_L∞
≤ ε. Simulations has shown that this new strategy produces faster convergence than the one that is based on the comb of Rabi pulses. The new method seems to work well even for initial conditions such that ||M
₀
(ω)+e
₃
||H
₁
is “relatively big”.