{
  "abstract": "In this paper, we study standard and distributionally robust $\\mathcal{H}_2$ synthesis problem of a stabilizing state-feedback controller for discrete-time linear time-invariant systems. Without requiring a nonsingular disturbance controllability Gramian or a positive-definite control penalty, we establish that stationarity of a stabilizing gain in standard $\\mathcal{H}_2$ synthesis is equivalent to global optimality and derive an exact Pythagorean identity for the performance difference. We then generalize the $\\mathcal{H}_2$ synthesis problem by considering i.i.d. disturbances with zero mean and uniformly bounded second moments, and extend the stationarity-optimality equivalence to the corresponding distributionally robust $\\mathcal{H}_2$ synthesis. Moreover, we show that every standard $\\mathcal{H}_2$-optimal gain is simultaneously optimal for every such distributionally robust problem. Finally, we specialize the framework to Frobenius, Kullback--Leibler, and Wasserstein-2 ambiguity sets and obtain explicit characterizations of their worst-case covariances, which, in turn, provide performance certificates for a common optimal controller.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24421",
  "author_total": 4,
  "authors": [
    {
      "name": "Arman Sharifi Kolarijani"
    },
    {
      "name": "Peyman Mohajerin Esfahani"
    },
    {
      "name": "Tamás Keviczky"
    },
    {
      "name": "Mohamad Amin Sharifi Kolarijani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e5vbeg6q7e9qp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24421",
  "fulltext_url": null,
  "id": "e5vbeg6q7e9qp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24421",
  "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",
      "cs.SY"
    ],
    [
      "arxiv",
      "eess.SY"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2207|Stability and Control of Uncertain Systems"
    ]
  ],
  "openalex": "W7214013688",
  "pdf_url": "https://arxiv.org/pdf/2609.24421",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e5vbeg6q7e9qp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24421",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24421"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Benign Geometry and Distributional Robustness of $\\mathcal{H}_2$ Synthesis",
  "title_text": "Benign Geometry and Distributional Robustness of ℋ_2 Synthesis",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/e5vbeg6q7e9qp",
  "venue": "arXiv",
  "versions": []
}