  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.
