Control in the spaces of ensembles of points

We study the controlled dynamics of the ensembles of points of a Riemannian manifold $M$. Parameterized ensemble of points of $M$ is the image of a continuous map $gamma:Theta o M$, where $Theta$ is a compact set of parameters. The dynamics of ensembles is defined by the action $gamma( heta) mapsto P_t(gamma( heta))$ of the semigroup of diffeomorphisms $P_t:M o M, t in mathbb{R}$, generated by the controlled equation $dot{x}=f(x,u(t))$ on $M$. Therefore, any control system on $M$ defines a control system on (generally infinite-dimensional) space $mathcal{E}_Theta(M)$ of the ensembles of points. We wish to establish criteria of controllability for such control systems. As in our previous work [A. Agrachev, Y. Baryshnikov, and A. Sarychev, ESAIM Control Optim. Calc. Var., 22 (2016), pp. 921--938], 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 the genericity of the exact controllability property for them. We also find a 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 the Rashevsky--Chow theorem for finite- and infinite-dimensional manifolds.