| In this thesis, we study a class of control systems referred to as ensemble control. The control objective is to guide a collection of structurally identical systems, differing by only a parameter, to a desired target state using the same control signal. The notion of ensemble control comes from the study of complex spin dynamics in nuclear magnetic resonance spectroscopy and imaging. Research in ensemble control is directly applicable to dynamical systems with unstable parameters, as well as other areas of neuroscience and quantum control. The challenge of such control is to simultaneously steer a continuum of systems between points of interest with a single control signal. In this work, the theoretical derivation of the solution to such system and the necessary and sufficient conditions for controllability are derived and shown to be related to the singular system of the operator characterizing the system's dynamics. Singular value decomposition is utilized to obtain an accurate and numerically efficient method to approximate the singular system and to obtain the control input of the minimum norm. Several new methods are introduced using basis representations and the basic properties of inner products to compute the control input. |
