  Recent years have witnessed a wave of research activities in systems science
toward the study of population systems. The driving force behind this shift was
geared by numerous emerging and ever-changing technologies in life and physical
sciences and engineering, from neuroscience, biology, and quantum physics to
robotics, where many control-enabled applications involve manipulating a large
ensemble of structurally identical dynamic units, or agents. Analyzing
fundamental properties of ensemble control systems in turn plays a foundational
and critical role in enabling and, further, advancing these applications, and
the analysis is largely beyond the capability of classical control techniques.
In this paper, we consider an ensemble of time-invariant linear systems
evolving on an infinite-dimensional space of continuous functions. We exploit
the notion of separating points and techniques of polynomial approximation to
develop necessary and sufficient ensemble controllability conditions. In
particular, we introduce an extended notion of controllability matrix, called
Ensemble Controllability Gramian. This means enables the characterization of
ensemble controllability through evaluating controllability of each individual
system in the ensemble. As a result, the work provides a unified framework with
a systematic procedure for analyzing control systems defined on an
infinite-dimensional space by a finite-dimensional approach.
