# Control in the spaces of ensembles of points

## TL;DR

This paper investigates controllability of ensemble dynamics on Riemannian manifolds, establishing criteria for exact and approximate controllability in finite and infinite-dimensional settings using geometric control theory.

## Contribution

It extends Lie-algebraic controllability methods to infinite-dimensional ensemble spaces and provides new criteria for controllability and motion planning.

## Key findings

- Proves generic exact controllability for finite ensembles.
- Provides sufficient approximate controllability criteria for continual ensembles.
- Discusses relations to Rashevsky-Chow theorem in infinite-dimensional contexts.

## 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.