Time-periodic feedback stabilization for an ensemble of half-spin systems

Karine Beauchard (1) , P.S. Pereira da Silva (2) , Pierre Rouchon (3)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 Laboratory of Automation and Control, Department of Telecommunications and Control Engineering
3 CAS - Centre Automatique et Systèmes

Feedback stabilization of an ensemble of non interacting half spins described by Bloch equations is considered. This system may be seen as a prototype for infinite dimensional systems with continuous spectrum. We propose an explicit feedback law that stabilizes asymptotically the system around a uniform state of spin +1/2 or -1/2. The closed-loop stability analysis is done locally around the equilibrium. The local convergence is shown to be a weak asymptotic convergence for the H1 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.

Mots clés: nonlinear systems; stabilization; quantum ensembles; Lyapunov stability; LaSalle invariance; infinite dimensional systems; continuous spectrum

Domaines: Automatique / Robotique

Karine Beauchard, P.S. Pereira da Silva, Pierre Rouchon. Time-periodic feedback stabilization for an ensemble of half-spin systems. 8th IFAC Symposium on Nonlinear Control Systems, Sep 2010, Bologna, Italy. ⟨hal-00577044⟩