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Ensemble control, an emerging research field focusing on the study of largepopulations of dynamical systems, has demonstrated great potential in numerousscientific and practical applications. Striking examples include pulse designfor exciting spin ensembles in quantum physics, neurostimulation for relievingneurological disorder symptoms, and path planning for steering robot swarms.However, the control targets in such applications are generally large-scalecomplex and severely underactuated ensemble systems, research into whichstretches the capability of techniques in classical control and dynamicalsystems theory to the very limit. This paper then devotes to advancing ourknowledge about controllability of linear ensemble systems by integrating toolsin modern algebra into the technique of separating points developed in ourrecent work. In particular, we give an algebraic interpretation of the dynamicsof linear systems in terms of actions of polynomials on vector spaces, and thisleads to the development of the functional canonical form of matrix-valuedfunctions, which can also be viewed as the generalization of the rationalcanonical form of matrices in linear algebra. Then, leveraging the technique ofseparating points, we achieve a necessary and sufficient characterization ofuniform ensemble controllability for time-invariant linear ensemble systems asthe ensemble controllability canonical form, in which the system and controlmatrices are in the functional canonical and block diagonal form, respectively.This work successfully launches a new research scheme by adopting and tailoringfinite-dimensional methods to tackle control problems involvinginfinite-dimensional ensemble systems, and lays a solid foundation for a moreinclusive ensemble control theory targeting a much broader spectrum of controland learning problems in both scientific research and practice.

Controllability Canonical Forms of Linear Ensemble Systems