# Stabilization for an ensemble of half-spin systems

Authors: Karine Beauchard, Paulo Sérgio Pereira da Silva, Pierre Rouchon, Automatica, Vol 48, no 1, pp. 68–76, January 2012, DOI: 10.1016/j.automatica.2011.09.050

Feedback stabilization of an ensemble of non interacting halfspins 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 H1 topology. This local convergence is shown to be a weak asymptotic convergence for the H1 topology and thus a strong convergence for the C0 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.

Download PDF

BibTeX:

@Article{,

author = {Paulo Sérgio Pereira da Silva Karine Beauchard, Pierre Rouchon},

title = {Stabilization for an ensemble of half-spin systems},

journal = {Automatica},

volume = {48},

number = {1},

pages = {68–76},

year = {2012},

}

Feedback stabilization of an ensemble of non interacting halfspins 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 H1 topology. This local convergence is shown to be a weak asymptotic convergence for the H1 topology and thus a strong convergence for the C0 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.

Download PDF

BibTeX:

@Article{,

author = {Paulo Sérgio Pereira da Silva Karine Beauchard, Pierre Rouchon},

title = {Stabilization for an ensemble of half-spin systems},

journal = {Automatica},

volume = {48},

number = {1},

pages = {68–76},

year = {2012},

}