Higher-Order Mean-Field Control Barrier Functions
Abstract
Mean-field control barrier functions (MF-CBFs) enforce swarm safety through constraints on the agents' distribution. Previous formulations, including applications to stochastic coverage and shepherding, use first-order differential inequalities. These cannot directly enforce constraints whose barrier functions yield control-free first derivatives. Thus, we develop the theory of higher-order MF-CBFs, which ensures the positivity of the barrier functionals through higher-order differential inequalities, as in their finite-dimensional analogs. More specifically, we introduce the notion of relative degree for mean-field safety functionals and develop an analytic framework for computing their higher-order derivatives. Moreover, for practical examples of cross-correlation and self-correlation functionals, we show that repeated differentiation along mean-field dynamics is structure-preserving and reduces to kernel recursions. As an illustration, we apply higher-order MF-CBFs to double-integrator swarm tracking and avoidance problems, where position-only constraints lead to MF-CBFs of relative degree two.
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BibTeX
@misc{nurbekyan2026higherorder,
title = {{Higher-Order Mean-Field Control Barrier Functions}},
author = {Levon Nurbekyan and Samy Wu Fung},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33852},
eprint = {2609.33852},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33852},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Higher-Order Mean-Field Control Barrier Functions
AU - Levon Nurbekyan
AU - Samy Wu Fung
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33852
UR - https://arxiv.org/abs/2609.33852
AB - Mean-field control barrier functions (MF-CBFs) enforce swarm safety through constraints on the agents' distribution. Previous formulations, including applications to stochastic coverage and shepherding, use first-order differential inequalities. These cannot directly enforce constraints whose barrier functions yield control-free first derivatives. Thus, we develop the theory of higher-order MF-CBFs, which ensures the positivity of the barrier functionals through higher-order differential inequalities, as in their finite-dimensional analogs. More specifically, we introduce the notion of relative degree for mean-field safety functionals and develop an analytic framework for computing their higher-order derivatives. Moreover, for practical examples of cross-correlation and self-correlation functionals, we show that repeated differentiation along mean-field dynamics is structure-preserving and reduces to kernel recursions. As an illustration, we apply higher-order MF-CBFs to double-integrator swarm tracking and avoidance problems, where position-only constraints lead to MF-CBFs of relative degree two.
ER -