  Controlling a large population, in the limit, a continuum, of structurally
identical dynamical systems with parametric variations is a pervasive task in
diverse applications in science and engineering. However, the severely
underactuated nature and the inability to avail comprehensive state feedback
information of such ensemble systems raise significant challenges in analysis
and design of ensemble systems. In this paper, we propose a moment-based
ensemble control framework, which incorporates and expands the method of
moments in probability theory to control theory. In particular, we establish an
equivalence between ensemble systems and their moment systems in terms of
control and their controllability properties by extending the Hausdorff moment
problem from the perspectives of differential geometry and dynamical systems.
The developments enable the design of moment-feedback control laws for closing
the loop in ensemble systems using the aggregated type of measurements. The
feasibility of this closed-loop control design procedure is validated both
mathematically and numerically.
