Second-Order $Λ$-Sets and Extensions to Non-Smooth, Hybrid, and Stochastic Optimal Control
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912756170489856 |
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| author | Rashid, Mohammad H. M |
| author_facet | Rashid, Mohammad H. M |
| contents | This paper develops a comprehensive extension of the $Λ$-set framework for optimal control, introducing second-order $Λ$-sets and generalizing the theory to non-smooth, hybrid, and stochastic hybrid systems. We first establish second-order necessary conditions that incorporate curvature information of the reachable set, providing refined optimality criteria that bridge classical second-variation methods with the geometric $Λ$-set approach. The framework is then extended to Filippov systems with discontinuous dynamics and to hybrid dynamical systems with state-dependent switching, yielding new necessary conditions for optimality in these settings. Furthermore, we introduce stochastic $Λ$-sets for systems subject to both continuous diffusion and discrete random switching, connecting the framework to Peng's stochastic maximum principle. Throughout the paper, detailed examples -- including nonholonomic systems, mechanical systems with friction, and stochastic temperature control -- illustrate the theoretical developments and demonstrate the practical applicability of the extended $Λ$-set theory. The results unify and generalize existing maximum principles, offering a powerful geometric tool for analyzing optimal control problems across a broad spectrum of system classes, from classical smooth systems to modern stochastic hybrid systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_09126 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Second-Order $Λ$-Sets and Extensions to Non-Smooth, Hybrid, and Stochastic Optimal Control Rashid, Mohammad H. M Optimization and Control 49K15, 49K45, 93C30, 93C65, 93E20 This paper develops a comprehensive extension of the $Λ$-set framework for optimal control, introducing second-order $Λ$-sets and generalizing the theory to non-smooth, hybrid, and stochastic hybrid systems. We first establish second-order necessary conditions that incorporate curvature information of the reachable set, providing refined optimality criteria that bridge classical second-variation methods with the geometric $Λ$-set approach. The framework is then extended to Filippov systems with discontinuous dynamics and to hybrid dynamical systems with state-dependent switching, yielding new necessary conditions for optimality in these settings. Furthermore, we introduce stochastic $Λ$-sets for systems subject to both continuous diffusion and discrete random switching, connecting the framework to Peng's stochastic maximum principle. Throughout the paper, detailed examples -- including nonholonomic systems, mechanical systems with friction, and stochastic temperature control -- illustrate the theoretical developments and demonstrate the practical applicability of the extended $Λ$-set theory. The results unify and generalize existing maximum principles, offering a powerful geometric tool for analyzing optimal control problems across a broad spectrum of system classes, from classical smooth systems to modern stochastic hybrid systems. |
| title | Second-Order $Λ$-Sets and Extensions to Non-Smooth, Hybrid, and Stochastic Optimal Control |
| topic | Optimization and Control 49K15, 49K45, 93C30, 93C65, 93E20 |
| url | https://arxiv.org/abs/2512.09126 |