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

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 H 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