{
  "abstract": "Duality and dual representations are essential tools for systems theory and controller synthesis. For any finite-dimensional causal linear time-invariant (CLTI) system, there exists a dual CLTI system with identical $H_\\infty$ and $H_2$ norm. These relations are found by simply taking the adjoint of the transfer function. For nonlinear, time-varying or infinite-dimensional systems, however, such a transfer function may not exist or be difficult to obtain. For duality in the $H_\\infty$ norm, this obstacle is overcome by using the equivalent time-domain, induced $L_2$-norm characterization. For $H_2$ norms there is no such equivalent characterization and hence proxy norms such as induced energy to peak and impulse to energy norms are used. These proxy norms bound the $H_2$ norm in the CLTI case. However, neither of these $H_2$ proxy norms equal the same $H_2$ proxy of the canonical dual representation. In this paper, we overcome this by proposing a dual representation whose induced energy to peak proxy equals the impulse to energy proxy of the primal and vice-versa. This relation is developed in the context of Partial Integral Equation (PIE) representations. The result is then used to establish primal-dual operator inequalities for both proxy norms similar in form to the primal-dual LMIs used to characterize the $H_2$ norm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21355",
  "author_total": 5,
  "authors": [
    {
      "name": "Danilo Braghini"
    },
    {
      "name": "Tristan Thomas"
    },
    {
      "name": "Sachin Shivakumar"
    },
    {
      "name": "Javad M. Velni"
    },
    {
      "name": "Matthew M. Peet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bkbmg7frtqk1g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21355",
  "fulltext_url": null,
  "id": "bkbmg7frtqk1g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21355",
  "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",
      "2207|Stability and Controllability of Differential Equations"
    ]
  ],
  "openalex": "W7222610034",
  "pdf_url": "https://arxiv.org/pdf/2610.21355",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bkbmg7frtqk1g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21355",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21355"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Dual Representations and Proxy Characterization of $H_2$ Norm for Partial Integral Equations",
  "title_text": "Dual Representations and Proxy Characterization of H_2 Norm for Partial Integral Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/bkbmg7frtqk1g",
  "venue": "arXiv",
  "versions": []
}