{
  "abstract": "Gradient flows of twice continuously differentiable strongly convex and convex objectives are, respectively, incrementally exponentially stable and non-expansive in the Euclidean metric. In this work, we investigate to what extent these properties extend to nonconvex objectives satisfying generalized Polyak-Łojasiewicz inequalities. For objectives with a unique minimizer, we establish conditions under which the gradient flow is contracting in a state-dependent Riemannian metric and incrementally exponentially stable. For objectives with nonunique minimizers, we generalize the non-expansivity of convex objectives to incremental stability for more general nonconvex objectives. We focus on incremental stability due to its implication to convergence to equilibrium points; when paired with a Lyapunov function it implies pointwise convergence to a minimizer and continuity of the endpoint map. Along the way, we derive new structural results for incrementally stable nonlinear systems and extend the analysis to Heavy-Ball and Dynamical Inertial Newton flows.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40897",
  "author_total": 4,
  "authors": [
    {
      "name": "Andreas Oliveira"
    },
    {
      "name": "Arthur C. B. de Oliveira"
    },
    {
      "name": "Mario Sznaier"
    },
    {
      "name": "Eduardo Sontag"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8xb30n164h973.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40897",
  "fulltext_url": null,
  "id": "8xb30n164h973",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40897",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.SY"
    ],
    [
      "arxiv",
      "eess.SY"
    ],
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40897",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8xb30n164h973.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40897",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40897"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Incremental Stability of Gradient Flows under Generalized Polyak-Łojasiewicz Inequalities and Its Implications for General Systems",
  "title_text": "Incremental Stability of Gradient Flows under Generalized Polyak-Łojasiewicz Inequalities and Its Implications for General Systems",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/8xb30n164h973",
  "venue": "arXiv",
  "versions": []
}