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

• Main Authors: Maciel Neto, Ulisses Alves, Pereira Da Silva, Paulo Sergio, Rouchon, Pierre
• Other Authors: Amazônia Azul Tecnologias de Defesa S.A., Polytechnic School of the University of São Paulo (Brazil), Universidade de São Paulo (USP), Centre Automatique et Systèmes (CAS), MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), QUANTum Information Circuits (QUANTIC), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Sorbonne Université (SU)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), This study was financed in part by the Coordenacao de Aperfei¸coamento de Pessoal de Nıvel Superior -Brazil (CAPES) - Finance Code 001. The second author was partially supported by CNPq, Brazil, Project 305546/2016-3, and by FAPESP, Brazil, Project 18/17463-7. This project has received some funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 884762 ).
• Format: Report
• Language: English
• Published: HAL CCSD 2021
• Subjects: Lyapunov feedback stabilization; Adiabatic control; Quantum systems; Ensemble controllability; Nonlinear systems; Bloch equations; Control of PDEs; [SPI.AUTO]Engineering Sciences [physics]/Automatic; South Pole; South pole
• Online Access: https://hal-mines-paristech.archives-ouvertes.fr/hal-03325203; https://hal-mines-paristech.archives-ouvertes.fr/hal-03325203/document; https://hal-mines-paristech.archives-ouvertes.fr/hal-03325203/file/autosam.pdf

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 H 1 ((ω * , ω *), S 2) where the parameter ω ∈ (ω * , ω *) is the Larmor frequency. Previous works have constructed a Lyapunov based stabilizing feedback in a convenient H 1-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 propose 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.