Stabilization for an ensemble of half-spin systems

doi:10.1016/j.automatica.2011.09.050

Automatica

Volume 48, Issue 1, January 2012, Pages 68-76

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Abstract

Feedback stabilization of an ensemble of non interacting half spins described by the Bloch equations is considered. This system may be seen as an interesting example for infinite dimensional systems with continuous spectra. We propose an explicit feedback law that stabilizes asymptotically the system around a uniform state of spin +1/2 or −1/2. The proof of the convergence is done locally around the equilibrium in the $H^{1}$ topology. This local convergence is shown to be a weak asymptotic convergence for the $H^{1}$ topology and thus a strong convergence for the $C^{0}$ topology. The proof relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. Numerical simulations illustrate the efficiency of these feedback laws, even for initial conditions far from the equilibrium.

Introduction

Most controllability results available for infinite dimensional

bilinear systems are related to systems with discrete spectra (see for instance, Beauchard and Coron, 2006, Beauchard and Laurent, 2010 for exact controllability results and Beauchard and Nersesyan, 2010, Chambrion et al., 2009, Nersesyan, 2009 for approximate controllability results). As far as we know, very few controllability studies consider systems admitting a continuous part in their spectra.

In Mirrahimi (2009) an approximate controllability result is given for a system with mixed discrete/continuous spectrum: the Schrödinger partial differential equation of a quantum particle in an $N$ -dimensional decaying potential is shown to be approximately controllable (in infinite time) to the ground bounded state when the initial state is a linear superposition of bounded states.

In Li and Khaneja, 2006, Li and Khaneja, 2009 a controllability notion, called ensemble controllability, is introduced and discussed for quantum systems described by a family of ordinary differential equations (Bloch equations) depending continuously on a finite number of scalar parameters and with a finite number of control inputs. Ensemble controllability means that it is possible to find open-loop controls that compensate for the dispersion in these scalar parameters: the goal is to simultaneously steer a continuum of systems between states of interest with the same control input. Such continuous family of ordinary differential systems sharing the same control inputs can be seen as an interesting example of infinite dimensional systems with purely continuous spectra.

The article Li and Khaneja (2009) highlights the role of Lie algebras and non-commutativity in the design of a compensating control sequence and consequently in the characterization of ensemble controllability. In Beauchard, Coron, and Rouchon (2010), this analysis is completed by functional analysis methods developed for infinite dimensional systems governed by partial differential equations (see, e.g., Coron, 2007, for samples of these methods). Several mathematical answers are given, with discrimination between approximate and exact controllability, and finite time and infinite time controllability, for the Bloch equation. In particular, it is proved that a priori bounded $L^{2}$ -controls are not sufficient to achieve exact controllability, but unbounded controls (containing, for example sums of Dirac masses) allow to recover controllability. For example, it is proved in Beauchard et al. (2010) that the Bloch equation is approximately controllable to the south pole of the Bloch Sphere (in the Sobolev space $H^{1}$ ) in finite time, with unbounded controls. The authors also propose explicit open loop (unbounded) controls for the local exact controllability to the north pole in infinite time.

The goal of this article is to investigate feedback stabilization of such specific infinite dimensional systems with continuous spectra. As in Mirrahimi (2009), the feedback design is based on a Lyapunov function closely related to the norm of the state space, a Banach space. The potential practical interest of such stabilization techniques consists in a simple algorithm providing an open-loop control steering from the initial state to the final state. This algorithm just consists in the numerical integration of the closed-loop system where the control values are recorded at each sample integration time.

Section 2 is devoted to the system model (continuum of Bloch equations), the control design and closed-loop simulations: the feedback law is the sum of a Dirac comb and a time-periodic feedback law based on a Lyapunov function; Proposition 1, proved in Appendix A, guarantees that the closed-loop initial value problem is always well defined; simulations illustrate the convergence rates observed for an initial state formed by a quarter of the equator on the Bloch sphere. In Section 3 we state and prove the main convergence result (Theorem 1): the closed-loop convergence toward the south pole is shown to be local and weak for the $H^{1}$ topology. The obstruction to global stabilization of our feedback scheme is also discussed: it is based on an explicit description of the Lasalle invariant set. Some concluding remarks are gathered in Section 4.

Section snippets

The studied model

We consider here an ensemble of non interacting half-spins in a static field ${(0, 0, B_{0})}^{T}$ in $R^{3}$ , subject to a transverse radio frequency field ${(\tilde{u} (t), \tilde{v} (t), 0)}^{T}$ in $R^{3}$ (the control input) (see Abragam, 1961, for an extended physical description). The ensemble of half-spins is described by the magnetization vector $M \in R^{3}$ depending on time $t$ but also on the Larmor frequency $ω = - γ B_{0}$ ( $γ$ is the gyromagnetic ratio). It obeys the Bloch equation: $\frac{\partial M}{\partial t} (t, ω) = (\tilde{u} (t) e_{1} + \tilde{v} (t) e_{2} + ω e_{3}) \times M (t, ω),$ where $- \infty < ω_{*} < ω^{*} < + \infty, ω \in (ω_{*}, ω$

Local stabilization

The main result of this paper shows that the control law (11) is a solution of the local stabilization problem stated at the end of the introduction.

Theorem 1

Consider system (6) with the feedback law (11). There exists $δ^{'} > 0$ such that, for every $M^{0} \in H^{1} ((ω_{*}, ω^{*}), S^{2})$ with ${‖ M^{0} + e_{3} ‖}_{H^{1}} \leq δ^{'}, M (t, .)$ converges weakly in $H^{1}$ to $- e_{3}$ when $t \to + \infty$ . In particular, as the injection of $H^{1}$ in $C^{0}$ is compact, $M (t, ω)$ converges to $- e_{3}$ when $t \to + \infty$ uniformly with respect to $ω \in (ω_{*}, ω^{*})$ (convergence in the sup norm of $C^{0}$ ).

The proof of

Conclusion

We have investigated here the stabilization of an infinite dimensional system admitting a continuous spectrum. We have designed a Lyapunov based feedback. Closed-loop simulations illustrate the asymptotic convergence toward the goal steady-state. We have provided a local and weak convergence result for the $H^{1}$ topology. Simulations indicate that the domain of attraction is far from being local and thus we can expect a large attraction domain for this feedback law. However, the stabilization is

Karine Beauchard was born in France in 1978. She graduated in Mathematics from the Ecole Normale Supérieure de Cachan. She received the Ph.D. degree in Mathematics in 2005 from the Université Paris Sud and the Habilitation degree in 2010 from the Ecole Normale Supérieure de Cachan. Since 2006, she has been a research associate CNRS at the laboratory CMLA (ENS Cachan). Since 2010, she is an associated professor at Ecole Polytechnique in Mathematics. Her research interests include analysis and

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Paulo Sérgio Pereira da Silva was born in Fortaleza (Brazil) in 1959. He received the degree in Engineering from the Instituto Militar de Engenharia, Rio de janeiro, in 1982. After some years (1983–1987) working as an engineer in projects for the development of the Brazilian air-traffic control system, he received an M.Sc. and a Doctor Degree, both in Electrical Engineering, respectively in 1988 and 1992. From September 1993 to February 1995, he stayed in France, holding an inspiring Post-Doc position in Laboratoire de Signaux et Systèmes (CNRS), under the supervision of Prof. Michel Fliess. He is an Associate Professor at the Laboratory of Automation and Control (LAC) of the Polytechnical School of University of São Paulo. His main research interests includes nonlinear system control theory and its applications, in particular, in quantum control.

Pierre Rouchon was born in 1960 in Saint-Etienne, France. Graduated from Ecole Polytechnique in 1983, he obtained his Ph.D. in Chemical Engineering at Mines ParisTech in 1990. In 2000, he obtained his “habilitation à diriger des recherches” in Mathematics at University Paris-Sud Orsay. From 1993 to 2005, he was an associated professor at Ecole Polytechnique in Applied Mathematics. From 1998 to 2002, he was the head of the Centre Automatique et Systèmes of Mines ParisTech. He is now a professor at Mines ParisTech. His fields of interest include nonlinear control and system theory with its applications. His contributions include differential flatness and its extension to infinite dimensional systems, nonlinear observers and symmetries, process control, motion planing and tracking for mechanical systems, feedback stabilization and estimation for electrical drives, internal combustion engines and quantum systems.

^☆: KB and PR were partially supported by the “Agence Nationale de la Recherche” (ANR), Projet Blanc C-QUID number BLAN-3-139579. PSPS was partially supported by CNPq Conselho Nacional de Desenvolvimento Cientifico e Tecnologico, and Fundação do Amparo a Pesquisa do Estado de São Paulo FAPESP, Brazil. The material in this paper was partially presented at the 8th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2010), September 1–3 2010, Bologna, Italy. This paper was recommended for publication in revised form by Associate Editor Gang Tao under the direction of Editor Miroslav Krstic.

¹: Tel.: +33 0 1 40 51 90 00; fax: +33 0 1 40 51 91 65.

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Stabilization for an ensemble of half-spin systems☆

Abstract

Introduction

Section snippets

The studied model

Local stabilization

Conclusion

Journal of Functional Analysis

Journal de Mathématiques Pures et Appliquées

Comptes Rendus Mathématiques

Annales de l’Institut Henri Poincaré Analyse Non Linéaire

Journal of Magnetic Resonance

Annales de l’Institut Henri Poincaré Analyse Non Linéaire

Automatica

Principles of nuclear magnetism

Feedback stabilization of distributed semilinear systems

Applied Mathematics & Optimization