{
  "abstract": "The classical linear quadratic regulator (LQR) minimizes the expected cumulative return but fails to account for performance variability, rendering it inadequate for risk-aware applications. To address this, we introduce the variance of the cumulative return as a risk measure in LQR. We derive the first exact closed-form expression for the variance of the discounted in?finite horizon return within the discrete-time Distributional LQR framework, for i.i.d. disturbances with symmetric probability densities. For Gaussian disturbances, this expression elegantly simplifies to a form dependent only on the disturbance covariance. Leveraging these theoretical foundations, we formulate a mean-variance optimal control problem that explicitly manages the trade-off between expected return and performance variability. To address the resulting non-convex optimization problem, we propose a novel adjoint gradient descent algorithm for a penalized formulation of the original problem, and establish that all iterates remain stabilizing and converge to a stationary point of the penalized objective. The effectiveness of this framework and the inherent risk-performance trade-off? are demonstrated through numerical experiments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39981",
  "author_total": 3,
  "authors": [
    {
      "name": "Ruyi Teng"
    },
    {
      "name": "Zifan Wang"
    },
    {
      "name": "Yulong Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aazxzq4jtm7vy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39981",
  "fulltext_url": null,
  "id": "aazxzq4jtm7vy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39981",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.39981",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aazxzq4jtm7vy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39981",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39981"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "ExactVariance of Random Return in Distributional LQR and Its Application to Mean-Variance Optimal Control",
  "title_text": "ExactVariance of Random Return in Distributional LQR and Its Application to Mean-Variance Optimal Control",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/aazxzq4jtm7vy",
  "venue": "arXiv",
  "versions": []
}