A Converse Robust-Safety Theorem for Differential Inclusions
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913455462678528 |
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| author | Maghenem, Mohamed Ghanbarpour, Masoumeh |
| author_facet | Maghenem, Mohamed Ghanbarpour, Masoumeh |
| contents | This paper establishes the equivalence between robust safety and the existence of a barrier function certificate for differential inclusions. More precisely, for a robustly-safe differential inclusion, a barrier function is constructed as the time-to-impact function with respect to a specifically-constructed reachable set. Using techniques from set-valued and nonsmooth analysis, we show that such a function, although being possibly discontinuous, certifies robust safety by verifying a condition involving the system's solutions. Furthermore, we refine this construction, using smoothing techniques from the literature of converse Lyapunov theory, to provide a smooth barrier certificate that certifies robust safety by verifying a condition involving only the barrier function and the system's dynamics. In comparison with existing converse robust-safety theorems, our results are more general as they allow the safety region to be unbounded, the dynamics to be a general continuous set-valued map, and the solutions to be non-unique. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_11364 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Converse Robust-Safety Theorem for Differential Inclusions Maghenem, Mohamed Ghanbarpour, Masoumeh Optimization and Control This paper establishes the equivalence between robust safety and the existence of a barrier function certificate for differential inclusions. More precisely, for a robustly-safe differential inclusion, a barrier function is constructed as the time-to-impact function with respect to a specifically-constructed reachable set. Using techniques from set-valued and nonsmooth analysis, we show that such a function, although being possibly discontinuous, certifies robust safety by verifying a condition involving the system's solutions. Furthermore, we refine this construction, using smoothing techniques from the literature of converse Lyapunov theory, to provide a smooth barrier certificate that certifies robust safety by verifying a condition involving only the barrier function and the system's dynamics. In comparison with existing converse robust-safety theorems, our results are more general as they allow the safety region to be unbounded, the dynamics to be a general continuous set-valued map, and the solutions to be non-unique. |
| title | A Converse Robust-Safety Theorem for Differential Inclusions |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2208.11364 |