Control of ensemble systems on special orthogonal groups

In this paper, we study the control of an ensemble of time-invariant bilinear systems defined on the special orthogonal group SO(n). This type of ensemble control systems appears in various application domains, such as the manipulation of quantum spin ensembles and motion planning for a population of robots. We establish an explicit algebraic necessary and sufficient condition to examine the controllability of systems on SO(n) by using the terminology from the theory of symmetric groups, which provides a transparent means to analyze the underlying Lie algebras. In addition, we show the equivalence between controllability and ensemble controllability of individual and ensemble systems, respectively, for systems evolving on SO(n).