A new Lagrangian approach to optimal control of second-order systems
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910861563527168 |
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| author | Konopik, Michael Leyendecker, Sigrid Maslovskaya, Sofya Ober-Blöbaum, Sina de Almagro, Rodrigo T. Sato Martín |
| author_facet | Konopik, Michael Leyendecker, Sigrid Maslovskaya, Sofya Ober-Blöbaum, Sina de Almagro, Rodrigo T. Sato Martín |
| contents | In this work, we propose and study a new approach to formulate the optimal control problem of second-order differential equations, with a particular interest in those derived from force-controlled Lagrangian systems. The formulation results in a new hyperregular control Langrangian and, thus, a new control Hamiltonian whose equations of motion provide necessary optimality conditions. We compare this approach to Pontryagin's maximum principle (PMP) in this setting, providing geometric insight into their relation. This leads us to define an extended Tulczyjew's triple with controls. Moreover, we study the relationship between Noether symmetries of this new formulation and those of the PMP. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new Lagrangian approach to optimal control of second-order systems Konopik, Michael Leyendecker, Sigrid Maslovskaya, Sofya Ober-Blöbaum, Sina de Almagro, Rodrigo T. Sato Martín Optimization and Control Mathematical Physics Differential Geometry Dynamical Systems 49K15, 34H05, 53Zxx, 53D05, 70G65, 70Hxx, 70H15, 70H50, 70H33, 70Q05 In this work, we propose and study a new approach to formulate the optimal control problem of second-order differential equations, with a particular interest in those derived from force-controlled Lagrangian systems. The formulation results in a new hyperregular control Langrangian and, thus, a new control Hamiltonian whose equations of motion provide necessary optimality conditions. We compare this approach to Pontryagin's maximum principle (PMP) in this setting, providing geometric insight into their relation. This leads us to define an extended Tulczyjew's triple with controls. Moreover, we study the relationship between Noether symmetries of this new formulation and those of the PMP. |
| title | A new Lagrangian approach to optimal control of second-order systems |
| topic | Optimization and Control Mathematical Physics Differential Geometry Dynamical Systems 49K15, 34H05, 53Zxx, 53D05, 70G65, 70Hxx, 70H15, 70H50, 70H33, 70Q05 |
| url | https://arxiv.org/abs/2503.04466 |