Control in the spaces of ensembles of points

Abstract

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 $\gamma :\Theta \to M$, where $\Theta$ is a compact set of parameters. The dynamics of ensembles is defined by the action $\gamma (\theta )\mapsto P_t(\gamma (\theta ))$ of the semigroup of diffeomorphisms $P_t:M\to M,t\in \mathbb{R}$, 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 ([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.

24 pages