Collective steering in finite time: controllability on $\text{GL}^+(n,\mathbb{R})$
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915039252840448 |
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| author | Abdelgalil, Mahmoud Georgiou, Tryphon T. |
| author_facet | Abdelgalil, Mahmoud Georgiou, Tryphon T. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_18766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Collective steering in finite time: controllability on $\text{GL}^+(n,\mathbb{R})$ Abdelgalil, Mahmoud Georgiou, Tryphon T. Optimization and Control Systems and Control 93-xx, 49-xx, 53-xx, 22-xx 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. |
| title | Collective steering in finite time: controllability on $\text{GL}^+(n,\mathbb{R})$ |
| topic | Optimization and Control Systems and Control 93-xx, 49-xx, 53-xx, 22-xx |
| url | https://arxiv.org/abs/2411.18766 |