Control of Inhomogeneous Ensembles

In this thesis, we study a class of control problems which involves controlling a large number of dynamical systems with different values of parameters governing the system dynamics by using the same control signal. We call such problems control of inhomogeneous 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 with dispersion in natural frequencies and the strengths of the applied radio frequency (rf) field. A systematic study of these systems has immediate applications to broad areas of the control of systems in quantum and nano domains, such as coherent spectroscopy and quantum information processing. 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 signal. This raises the intriguing question about ensemble controllability. We show that controllability of an ensemble can be understood by the study of the algebra of polynomials defined by the noncommuting vector fields governing the system dynamics. In practical magnetic resonance applications, this work leads to the design of a compensating pulse sequence. The new mathematical structures arising from such problems are excellent motivation for new developments in control theory.

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