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

Mahmoud Abdelgalil; Tryphon T. Georgiou

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arXiv · 2411.18766

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

Mahmoud Abdelgalil·Tryphon T. Georgiou

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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.

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BibTeX RIS

Mahmoud Abdelgalil, Tryphon T. Georgiou. 2024-11-27. Collective steering in finite time: controllability on $\text{GL}^+(n,\mathbb{R})$. https://arxiv.org/abs/2411.18766

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