  In this article, we investigate the problem of simultaneously steering an
uncountable family of finite dimensional time-varying linear systems. We call
this class of control problems Ensemble Control, a notion coming from the study
of spin dynamics in Nuclear Magnetic Resonance (NMR) spectroscopy and imaging
(MRI). This subject involves controlling a continuum of parameterized dynamical
systems with the same open-loop control input. From a viewpoint of mathematical
control theory, this class of problems is challenging because it requires
steering a continuum of dynamical systems between points of interest in an
infinite dimensional state space by use of the same control function. The
existence of such a control raises fundamental questions of ensemble
controllability. We derive the necessary and sufficient controllability
conditions and an accompanying analytical optimal control law for ensemble
control of time-varying linear systems. We show that ensemble controllability
is in connection with singular values of the operator characterizing the system
dynamics. In addition, we study the problem of optimal ensemble control of
harmonic oscillators to demonstrate our main results. We show that the optimal
solutions are pertinent to the study of time-frequency limited signals and
prolate spheroidal wave functions. A systematic study of ensemble control
systems has immediate applications to systems with parameter uncertainties as
well as to broad areas of quantum control systems as arising in coherent
spectroscopy and quantum information processing. The new mathematical
structures appearing in such problems are an excellent motivation for new
developments in control theory.
