Symmetry-Structured Constrained Allocation via Operator Decomposition in Stochastic Dynamical Systems

Symmetry-Structured Constrained Allocation via Operator Decomposition in Stochastic Dynamical Systems | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Symmetry-Structured Constrained Allocation via Operator Decomposition in Stochastic Dynamical Systems Nam Anh Quach This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8994655/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We introduce a symmetry-structured allocation framework for constrained distributed control systems subject to stochastic excitation. The central observation is that when the plant admits a discrete reflection symmetry, the control action can be decomposed into invariant (symmetric) and anti-invariant (antisymmetric) subspaces induced by the symmetry operator. We show that performing allocation separately within these orthogonal subspaces yields systematic suppression of residual state excursions under actuator magnitude and rate constraints. Formally, the method constructs parity projectors associated with a reflection operator acting on the state space and induces a corresponding decomposition of the control effectiveness operator. The resulting structured allocation is implemented via damped pseudo-inverse operators under hard actuator constraints. Performance is evaluated under colored stochastic excitation modeled as an AR( 1 ) process, with operational risk quantified by exceedance probabilities and tail distribution functionals of the residual norm. Numerical experiments on a reduced-order aeroelastic model demonstrate consistent reductions in peak excursions, RMS residuals, and high-quantile exceedance probabilities relative to unstructured allocation, including robustness under effectiveness uncertainty. The results indicate that symmetry-induced operator decomposition provides a general structural mechanism for reducing extreme responses in constrained networked dynamical systems admitting discrete symmetry actions. Applied Mathematics Aeronautics and Astronautics Symmetry-structured control Operator decomposition Constrained allocation Reflection invariance Pseudo-inverse methods Stochastic excitation Tail-risk suppression Distributed dynamical systems Robust optimization Parity decomposition Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8994655","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":598499651,"identity":"c3dce6a4-7f2e-4c2b-bb84-33d035383b0b","order_by":0,"name":"Nam Anh Quach","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9UlEQVRIiWNgGAWjYLACxgYQycP4+E8FkGZmbiBaC7MBzxmQFkbitbBJ8LYhuDiBOfvpxM+FO6wT+9vPHpCQnFcbzd8O1PKjYhtOLZY9uZulZ55JT5xxJi/BwHDb8dwZhxkbGHvO3MapxeBA7gZp3rbDiQ03eAwSErcdy20AamFmbMOj5fzbzb9BWuYDtRw4OOdY7nyCWm7kbgPbsuEGj2FjY0NN7gbCWt5us+ZtSzfeeCbHmJnh2IHcjUAtB/H65Xzu5tu8bday846fMf/NUFOXO+/84YMPflTg1gIFzDDGYTB5gJB6ZC11RCgeBaNgFIyCkQYAJ8Fh8iX1/zgAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0000-7962-6695","institution":"Hong Bang International University","correspondingAuthor":true,"prefix":"","firstName":"Nam","middleName":"Anh","lastName":"Quach","suffix":""}],"badges":[],"createdAt":"2026-02-28 10:55:20","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8994655/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8994655/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103854522,"identity":"1c7b794b-c739-470e-bf49-d861cb68a2fb","added_by":"auto","created_at":"2026-03-03 17:40:06","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1567533,"visible":true,"origin":"","legend":"","description":"","filename":"SymmetryStructuredConstrainedAllocation.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8994655/v1_covered_6ee22ccd-362b-471e-8306-ea6a73cb7d78.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eSymmetry-Structured Constrained Allocation via Operator Decomposition in Stochastic Dynamical Systems\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Symmetry-structured control, Operator decomposition, Constrained allocation, Reflection invariance, Pseudo-inverse methods, Stochastic excitation, Tail-risk suppression, Distributed dynamical systems, Robust optimization, Parity decomposition","lastPublishedDoi":"10.21203/rs.3.rs-8994655/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8994655/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe introduce a symmetry-structured allocation framework for constrained distributed control systems subject to stochastic excitation. The central observation is that when the plant admits a discrete reflection symmetry, the control action can be decomposed into invariant (symmetric) and anti-invariant (antisymmetric) subspaces induced by the symmetry operator. We show that performing allocation separately within these orthogonal subspaces yields systematic suppression of residual state excursions under actuator magnitude and rate constraints.\u003c/p\u003e \u003cp\u003eFormally, the method constructs parity projectors associated with a reflection operator acting on the state space and induces a corresponding decomposition of the control effectiveness operator. The resulting structured allocation is implemented via damped pseudo-inverse operators under hard actuator constraints. 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