The $Λ$-Set and Its Role in Local Controllability and Necessary Conditions for Free-Time Optimal Control
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912749002424320 |
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| author | Rashid, Mohammad H. M. |
| author_facet | Rashid, Mohammad H. M. |
| contents | This paper establishes a unified framework connecting local controllability, necessary conditions for optimality, and attainability in free-time optimal control problems. The central object of our investigation is the $Λ$-set, which governs the relationship between original control systems and their convexifications. Our main results demonstrate that emptiness of the $Λ$-set implies local controllability and guarantees the existence of minimizing sequences achieving the target in reduced time. We derive strengthened necessary conditions for time-optimal control and provide explicit constructive procedures for approximating generalized controls by ordinary trajectories. These results resolve longstanding questions about relaxation phenomena while extending classical theory to address modern challenges in non-convex optimization, establishing foundations for higher-order analysis in free-time problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04651 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $Λ$-Set and Its Role in Local Controllability and Necessary Conditions for Free-Time Optimal Control Rashid, Mohammad H. M. Optimization and Control 49K15, 93B05, 49J30, 93C15 This paper establishes a unified framework connecting local controllability, necessary conditions for optimality, and attainability in free-time optimal control problems. The central object of our investigation is the $Λ$-set, which governs the relationship between original control systems and their convexifications. Our main results demonstrate that emptiness of the $Λ$-set implies local controllability and guarantees the existence of minimizing sequences achieving the target in reduced time. We derive strengthened necessary conditions for time-optimal control and provide explicit constructive procedures for approximating generalized controls by ordinary trajectories. These results resolve longstanding questions about relaxation phenomena while extending classical theory to address modern challenges in non-convex optimization, establishing foundations for higher-order analysis in free-time problems. |
| title | The $Λ$-Set and Its Role in Local Controllability and Necessary Conditions for Free-Time Optimal Control |
| topic | Optimization and Control 49K15, 93B05, 49J30, 93C15 |
| url | https://arxiv.org/abs/2512.04651 |