{
  "abstract": "Prescribed-time controllers drive the state to the origin at a time $T$ chosen in advance, with gains that grow without bound as $t$ approaches $T$. This paper shows the growth is a necessity and computes the exact price of avoiding it. The starting point is a simple estimate: any trajectory with $\\|\\dot z(t)\\|\\le L(t)\\|z(t)\\|$ satisfies $\\|z(t)\\|\\ge\\|z(0)\\|e^{-\\int_0^t L}$. From this: a gain profile $L$ supports exact prescribed-time convergence, over the unrestricted class of closed-loop vector fields admitting it, if and only if its integral over $[0,T)$ is infinite; so every design must either use an envelope of infinite integral or violate the linear growth bound at the origin, as finite-time and fixed-time designs do. Under a finite gain budget $\\mathcal B=\\int_0^T L$ the best worst-case contraction over the class is exactly $e^{-\\mathcal B}$, a floor valid for every plant, and a finite-budget trajectory converges to a definite point away from the origin. A comparison lemma then carries the theory beyond linear growth: for envelopes $\\|\\dot z\\|\\le L(t)\\,ω(\\|z\\|)$, first arrival at the origin requires budget at least $\\int_0^{\\|z_0\\|}du/ω(u)$; arrival is impossible whenever this integral diverges, and when it is finite it is the exact threshold, attained inside the class. The classical settling expressions of finite-time control reappear as hitting-time floors, and the known fixed-time condition is the two-ended convergence of the same integral. For linear time-varying loops the results refine to logarithmic-norm budgets, corollaries of a known two-sided estimate, with the terminal error of gain freezing described exactly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17484",
  "author_total": 4,
  "authors": [
    {
      "name": "Osama F. Abdel Aal"
    },
    {
      "name": "Jiale Chen"
    },
    {
      "name": "Jairo Viola"
    },
    {
      "name": "YangQuan Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/davf9d1zf4vpt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17484",
  "fulltext_url": null,
  "id": "davf9d1zf4vpt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17484",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2207|Adaptive Control of Nonlinear Systems"
    ]
  ],
  "openalex": "W7221827318",
  "pdf_url": "https://arxiv.org/pdf/2610.17484",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/davf9d1zf4vpt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17484",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17484"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Fundamental Limits of Prescribed-Time Regulation: Gain Budgets, Accuracy Frontiers, and Arrival Costs",
  "title_text": "Fundamental Limits of Prescribed-Time Regulation: Gain Budgets, Accuracy Frontiers, and Arrival Costs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/davf9d1zf4vpt",
  "venue": "arXiv",
  "versions": []
}