Collective steering in finite time: controllability on $\text{GL}^+(n,\mathbb{R})$

TL;DR

This paper investigates the controllability of multiple particles with linear dynamics on the space of matrices with positive determinant, demonstrating feasibility of finite-time rearrangement and discussing the limitations of feedback control laws.

Contribution

It establishes the feasibility of finite-time steering on 1+(n,2R) and analyzes the non-existence of universal, continuous feedback laws for such tasks.

Findings

Any rearrangement is always feasible in finite time.

Optimal feedback control policies may not exist.

No universal continuous feedback law can achieve all rearrangements.

Abstract

We consider the problem of steering a collection of n particles that obey identical n-dimensional linear dynamics via a common state feedback law towards a rearrangement of their positions, cast as a controllability problem for a dynamical system evolving on the space of matrices with positive determinant. We show that such a task is always feasible and, moreover, that it can be achieved arbitrarily fast. We also show that an optimal feedback control policy to achieve a similar feat, may not exist. Furthermore, we show that there is no universal formula for a linear feedback control law to achieve a rearrangement, optimal or not, that is everywhere continuous with respect to the specifications. We conclude with partial results on the broader question of controllability of dynamics on orientation-preserving diffeomorphisms.