{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33852",
  "author_total": 2,
  "authors": [
    {
      "name": "Levon Nurbekyan"
    },
    {
      "name": "Samy Wu Fung"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ewcggxaymvkm3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33852",
  "fulltext_url": null,
  "id": "ewcggxaymvkm3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33852",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1705|Distributed Control Multi-Agent Systems"
    ]
  ],
  "openalex": "W7214708845",
  "pdf_url": "https://arxiv.org/pdf/2609.33852",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ewcggxaymvkm3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33852",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33852"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Higher-Order Mean-Field Control Barrier Functions",
  "title_text": "Higher-Order Mean-Field Control Barrier Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/ewcggxaymvkm3",
  "venue": "arXiv",
  "versions": []
}