Motion planing for an ensemble of Bloch equations towards the south pole with smooth bounded control

Abstract

One considers the control problem of an ensemble of Bloch equations (non-interacting half-spins) in a static magnetic field B0. The state M(t,⋅) belongs to the Sobolev space H1((ω∗,ω∗),S2) where the parameter ω∈(ω∗,ω∗) is the Larmor frequency. Previous works have constructed a Lyapunov based stabilizing feedback in a convenient H1-norm that assures local L∞-convergence of the initial state M0(ω) to the south pole, solving locally the approximate steering problem from M0 towards the south pole. However, the corresponding control law contains a comb of periodic π-Rabi pulses (Dirac impulses), corresponding to strongly unbounded control. The present work proposes smooth uniformly bounded time-varying controls for this local steering problem, where the Rabi pulses are replaced by adiabatic following smooth pulses. Furthermore, simulations show that this new strategy produces faster convergence, even for initial conditions “relatively far” from the south pole.