{
  "abstract": "This note establishes a geometric foundation for trajectory-manifold representations of deterministic nonlinear systems in a behavioral setting motivated by data-enabled predictive control. For a discrete-time system $x_{k+1}=f(x_k,u_k)$ with measured state and a $C^r$ transition map, $r\\geq 1$, we consider the terminal-state-augmented finite-horizon behavior consisting of all admissible state-input trajectories over a prediction horizon $N$. We prove that this behavior is a $C^r$ embedded submanifold of the ambient trajectory space with intrinsic dimension $n+Nm$, where $n$ and $m$ are the state and input dimensions. Moreover, the rollout map from the admissible initial-state and input coordinates $(x_0,\\mathbf u)$ is a $C^r$ diffeomorphism onto the behavior manifold, providing explicit global smooth coordinates. This yields a canonical exact encoder--decoder representation and implies that any exact differentiable latent representation of the full behavior must have latent dimension at least $n+Nm$. The geometric result does not require controllability, stabilizability, or invertibility of the dynamics. Corresponding results are given for zero-order-hold sampled continuous-time systems and fixed-step numerical transition maps. These results provide the deterministic geometric foundation for subsequent data-driven approximation and predictive-control development.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19079",
  "author_total": 1,
  "authors": [
    {
      "name": "Arda Bayer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a6gn9xrr78m7t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19079",
  "fulltext_url": null,
  "id": "a6gn9xrr78m7t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19079",
  "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",
      "cs.SY"
    ],
    [
      "arxiv",
      "eess.SY"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2207|Advanced Control Systems Optimization"
    ]
  ],
  "openalex": "W7213513538",
  "pdf_url": "https://arxiv.org/pdf/2609.19079",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a6gn9xrr78m7t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.19079",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19079"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Trajectory Manifolds for Nonlinear Data-Enabled Predictive Control",
  "title_text": "Trajectory Manifolds for Nonlinear Data-Enabled Predictive Control",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/a6gn9xrr78m7t",
  "venue": "arXiv",
  "versions": []
}