H 1 -Control of an ensemble of half-spin systems replacing Rabi pulses by adiabatic following

Ulisses Alves Maciel Neto (1) , Paulo Sergio Pereira da Silva (1) , Karine Beauchard (2) , Pierre Rouchon (3)

1 Polythecnical School
2 ENS Rennes - École normale supérieure - Rennes
3 CAS - Centre Automatique et Systèmes

This work considers the control of an ensemble of non-interacting half-spin systems (Bloch equations) in a vertical static field B0 subject to a pair of controlled radio-frequency inputs (u1(t),u2(t)) acting on the hor- izontal plane. The state M(t,ω) ∈ S2 belongs to the Bloch sphere S2, and it is indexed by the Larmor fre- quency ω ∈ (ω∗ , ω ∗ ). Previous works have constructed a local stabilizing feedback based on a Lyapunov func- tional which is essentially a convenient H 1 -norm of a Sobolev space H 1 ((ω∗ , ω ∗ ), S2 ) (see Beachard, Pereira da Silva and Rouchon [3]-[4]). This feedback assures local L∞ -convergence of the initial state M0 (ω ) to −e3 . 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 ∥M0(ω) + e3∥H1 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 ∥M(lT,·)+ e3∥L∞ ≤ ε. Simulations has shown that this new strat- egy 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 ∥M0(ω)+e3∥H1 is “relatively big”.

Domaines: Automatique / Robotique

Ulisses Alves Maciel Neto, Paulo Sergio Pereira da Silva, Karine Beauchard, Pierre Rouchon. H 1 -Control of an ensemble of half-spin systems replacing Rabi pulses by adiabatic following. 58th Conference on Decision and Control - CDC 2019, Dec 2019, Nice, France. ⟨hal-02431019⟩