Control in the spaces of ensembles of points

We study the controlled dynamics of the {it ensembles of points} of a Riemannian manifold M. Parameterized ensemble of points of M is the image of a compact set of parameters under a continuous map. The dynamics of ensembles is defined by the action on the ensembles of the semigroup of diffeomorphisms on M, generated by a controlled system on M and a given control. Therefore any control system on M defines a control system on (generally infinite-dimensional) space of the ensembles of points. We wish to establish criteria of controllability for such control systems. As in our previous work we seek to adapt the Lie-algebraic approach of geometric control theory to the infinite-dimensional setting. We study the case of finite ensembles and prove genericity of exact controllability property for them. We also find sufficient approximate controllability criterion for continual ensembles and prove a result on motion planning in the space of flows on M. We discuss the relation of the obtained controllability criteria to various versions of Rashevsky-Chow theorem for finite- and infinite-dimensional manifolds.