Synthesis of optimal ensemble controls for linear systems using the singular value decomposition

An emerging and challenging area in mathematical control theory called Ensemble Control encompasses a class of problems that involves the guidance of an uncountably infinite collection of structurally identical dynamical systems, which are indexed by a parameter set, by applying the same open-loop control. The subject originates from the study of complex spin dynamics in Nuclear Magnetic Resonance (NMR) spectroscopy and imaging (MRI). A fundamental question concerns ensemble controllability, which determines the existence of controls that transfer the system between desired initial and target states. For ensembles of finite-dimensional time-varying linear systems, the necessary and sufficient controllability conditions and analytical optimal control laws have been shown to depend on the singular system of the operator characterizing the system dynamics. Because analytical solutions are available only in the simplest cases, there is a need to develop numerical methods for synthesizing these controls. We introduce a direct, accurate, and computationally efficient algorithm based on the singular value decomposition (SVD) that approximates ensemble controls of minimum norm for such systems. This method enables the application of ensemble control to engineering problems involving complex, time-varying, and high-dimensional linear dynamic systems.