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 continuous map γ:Theta → M, where Theta is a compact set of parameters. The dynamics of ensembles is defined by the action γ(θ) mapsto Pt(γ(θ)) of the semigroup of diffeomorphisms Pt:M → M, t ∈ ℝ, generated by the controlled equation dotx=f(x,u(t)) on M. Therefore any control system on M defines a control system on (generally infinite-dimensional) space E_Theta(M) of the ensembles of points. We wish to establish criteria of controllability for such control systems. As in our previous work ([1]) 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.