Singular extremals of optimal control problems with $L^1$ cost
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912739716235264 |
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| author | Agrachev, Andrei Beschastnyi, Ivan Motta, Michele |
| author_facet | Agrachev, Andrei Beschastnyi, Ivan Motta, Michele |
| contents | We study the optimal control problem for a control-affine system, where we want to minimize the $L^1$ norm of the control. First, we show how Pontryagin Maximum Principle (PMP) applies to this problem and we divide the extremal trajectories into two categories: regular and singular extremals. Then, we obtain a strong generalized Legendre-Clebsch condition for singular extremals and we show that this condition together with the absence of conjugate points is sufficient to ensure local strong optimality. We provide also some geometric examples where we apply our results. Finally, we prove that generalized Legendre-Clebsch condition is necessary for optimality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21527 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular extremals of optimal control problems with $L^1$ cost Agrachev, Andrei Beschastnyi, Ivan Motta, Michele Optimization and Control Dynamical Systems 49K15 (Primary), 93C10 (Secondary) We study the optimal control problem for a control-affine system, where we want to minimize the $L^1$ norm of the control. First, we show how Pontryagin Maximum Principle (PMP) applies to this problem and we divide the extremal trajectories into two categories: regular and singular extremals. Then, we obtain a strong generalized Legendre-Clebsch condition for singular extremals and we show that this condition together with the absence of conjugate points is sufficient to ensure local strong optimality. We provide also some geometric examples where we apply our results. Finally, we prove that generalized Legendre-Clebsch condition is necessary for optimality. |
| title | Singular extremals of optimal control problems with $L^1$ cost |
| topic | Optimization and Control Dynamical Systems 49K15 (Primary), 93C10 (Secondary) |
| url | https://arxiv.org/abs/2511.21527 |