Stabilization of an arbitrary profile for an ensemble of half-spin systems

Karine Beauchard (1) , Paulo Sergio Pereira da Silva (2) , Pierre Rouchon (3)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 Escola Politecnica
3 CAS - Centre Automatique et Systèmes

We consider the feedback stabilization of a variable profile for an ensemble of non interacting half-spins described by the Bloch equations. We propose an explicit feedback law that stabilizes asymptotically the system around a given arbitrary target profile. The convergence proof is done when the target profile is entirely in the south hemisphere or in the north hemisphere of the Bloch sphere. The convergence holds for initial conditions in a H1 neighbourhood of this target profile. This convergence is shown for the weak 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 target profile.

Mots clés: Nonlinear systems; Lyapunov stabilization; LaSalle invariance; Quantum systems; Bloch equations; Ensemble controllability; Infinite dimensional system

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Karine Beauchard, Paulo Sergio Pereira da Silva, Pierre Rouchon. Stabilization of an arbitrary profile for an ensemble of half-spin systems. Automatica, 2013, 49 (7), pp.2133-2137. ⟨10.1016/j.automatica.2013.03.011⟩. ⟨hal-00831611v2⟩