Issue |
ESAIM: COCV
Volume 31, 2025
|
|
---|---|---|
Article Number | 48 | |
Number of page(s) | 32 | |
DOI | https://doi.org/10.1051/cocv/2025034 | |
Published online | 04 June 2025 |
A converse robust-safety theorem for differential inclusions
1
CNRS, Gipsa-lab, Grenoble INP, Université Grenoble Alpes,
Grenoble, France
2
Department of Electrical and Computer Engineering, University of Colorado Bulder,
Colorado, USA
* Corresponding author: mohamed.maghenem@gipsa-lab.fr
Received:
28
November
2023
Accepted:
21
March
2025
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 on converse Lyapunov theorems, 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.
Mathematics Subject Classification: 93A10-26B05
Key words: Robust safety / Differential inclusions / Barrier functions / Converse theorem
© The authors. Published by EDP Sciences, SMAI 2025
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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