  In this paper, we study the control of a class of time-invariant linear
ensemble systems whose natural dynamics are linear in the system parameter.
This class of ensemble control systems arises from practical engineering and
physical applications, such as transport of quantum particles and control of
uncertain harmonic systems. We establish explicit algebraic criterions to
examine controllability of such ensemble systems. Our derivation is based on
the notion of polynomial approximation, where the elements of the reachable set
of the ensemble system are represented in polynomials of the system parameter
and used to approximate the desired state of interest. In addition, we
highlight the role of the spectra of the system matrices play in the
determination of ensemble controllability. Finally, illustrative examples and
numerical simulations for optimal control of this class of linear ensemble
systems are presented to demonstrate the theoretical results.
