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

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