{
  "abstract": "We consider the problem of computing Taylor approximations to the optimal feedback law and value function for infinite-horizon finite-dimensional nonlinear optimal control problem---also known as Al'brekht's method. The scalability of existing Al'brekht-based approaches to computing approximate solutions to the Hamilton-Jacobi-Bellman equation associated with the optimal control problem has been severely limited by computational resources. The computational time issue has been partially mitigated by recent works leveraging tensor-structure to develop efficient numerical methods. However, RAM requirements still have remained a major hindrance to scaling beyond state dimensions of a few hundred or low thousands. We propose an acceleration to compute the expensive higher-order terms in a reduced space, mitigating the curse of dimensionality. A fast and scalable \\textsc{Matlab} implementation is provided open-source for general purpose use, with the ability to accommodate arbitrary polynomial nonlinearities in the control-affine state equation, as well as polynomial state-dependence in the cost function, and the ability to handle systems in generalized form with a non-identity mass matrix, such as finite element models. Combined with the exploitation of sparsity and modern low-rank Riccati solvers, this permits computing approximations to polynomial feedback laws based on Al'brekht's method in hundreds of thousands of states.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15942",
  "author_total": 2,
  "authors": [
    {
      "name": "Nicholas A. Corbin"
    },
    {
      "name": "Boris Kramer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f6ch550eg4gqp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15942",
  "fulltext_url": null,
  "id": "f6ch550eg4gqp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15942",
  "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.DS"
    ],
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.15942",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f6ch550eg4gqp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15942",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15942"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Nonlinear Feedback Control in High Dimensions using the Polynomial-Polynomial Regulator",
  "title_text": "Nonlinear Feedback Control in High Dimensions using the Polynomial-Polynomial Regulator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/f6ch550eg4gqp",
  "venue": "arXiv",
  "versions": []
}