Ensemble control on lie groups

In this article, we study a class of control problems which involves controlling a continuum of dynamical systems with different dynamics by using the same control signal. We call such problems control of ensembles. We will specially focus on the systems evolving on SO(3). 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. From the mathematical control point of view, the challenge is to simultaneously steer a continuum of systems between points of interest with the same control signal. This raises some new and unexplored questions about ensemble controllability. We show that controllability of an ensemble can be understood by the study of the algebra of polynomials defined by the non-commuting vector fields that govern the system dynamics. A systematic study of these systems has immediate applications to broad areas of control of systems in quantum and nano domains, such as coherent spectroscopy and quantum information processing. The new mathematical structures appearing in such problems are excellent motivation for new developments in control theory.