Ensemble Control of Bloch Equations

In this article, we study a class of control problems which involves controlling a continuum of dynamical systems with different values of parameters characterizing the system dynamics by using the same control signal. We call such problems control of ensembles. The motivation for looking into these problems comes from the manipulation of an ensemble of nuclear spins in nuclear magnetic resonance (NMR) spectroscopy and imaging (MRI) with dispersions in natural frequencies and the strengths of the applied radio frequency (rf) field. From the standpoint of mathematical control theory, the challenge is to simultaneously steer a continuum of systems between points of interest with the same control input. This raises some new and unexplored questions about controllability of such systems. We show that controllability of an ensemble can be understood by the study of the algebra of polynomials defined by the noncommuting vector fields that govern the system dynamics. A systematic study of these systems has immediate applications to broad areas of control of ensembles of quantum systems as arising in coherent spectroscopy and quantum information processing. The new mathematical structures appearing in such problems are excellent motivation for new developments in control theory.