Dynamic Defense Strategies for Cyber-Physical Systems Using Stackelberg Games and Deep Reinforcement Learning in Discrete and Continuous Time

Dynamic Defense Strategies for Cyber-Physical Systems Using Stackelberg Games and Deep Reinforcement Learning in Discrete and Continuous Time | 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 Dynamic Defense Strategies for Cyber-Physical Systems Using Stackelberg Games and Deep Reinforcement Learning in Discrete and Continuous Time Anastasios Raptis, Stefanos Gritzalis, Athanasios Yannacopoulos This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7065795/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 11 You are reading this latest preprint version Abstract As cyber threats to power grid infrastructures escalate, the urgency of understanding how to protect cyber-physical systems (CPS) has never been greater. These systems, which integrate physical processes with digital control, are increasingly susceptible to sophisticated cyberattacks that can lead to widespread disruption. While most existing defense models function within either discrete or continuous-time frameworks, this research tackles a significant limitation: the absence of a unified strategy that encompasses both temporal domains.This study presents a hybrid defense framework that combines Stackelberg game theory with Deep Reinforcement Learning (DRL), aiming to provide flexible and adaptive protection. The objective of this framework is to facilitate proactive defense decisions that can anticipate and respond to attacks with strategic precision.We conducted extensive simulations using Python to assess the proposed model in both discrete and continuous time scenarios. Our approach was rigorously tested under realistic adversarial conditions to confirm its resilience and cost-effectiveness.Key findings indicate that defender-first strategies in discrete time effectively minimize system damage and alleviate computational burdens, while continuous-time responses, although immediate, demand significantly higher resource investment.This dual-domain solution offers a robust, adaptable toolset for real-time CPS defense. It supports infrastructure operators in navigating nonlinear, dynamic environments by combining theoretical rigor with practical impact—an urgently needed step in an increasingly complex threat landscape. Mathematics Subject Classification (2010) 68M10 · 91A23 · 93C10 Adaptive Mechanisms Defense Strategies Dynamic Modeling Nonlinear Control Power Grid Security Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 04 Feb, 2026 Reviews received at journal 03 Feb, 2026 Reviews received at journal 24 Jan, 2026 Reviewers agreed at journal 08 Jan, 2026 Reviewers agreed at journal 06 Jan, 2026 Reviews received at journal 25 Sep, 2025 Reviewers agreed at journal 28 Jul, 2025 Reviewers invited by journal 28 Jul, 2025 Editor assigned by journal 11 Jul, 2025 Submission checks completed at journal 11 Jul, 2025 First submitted to journal 07 Jul, 2025 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. 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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-7065795","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":491765906,"identity":"e038d3ee-2cd4-4286-b278-2a03a1434f67","order_by":0,"name":"Anastasios Raptis","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVRIiWNgGAWjYFADZgbGBwwMEgxsRKk+AMQ8zAzMBiRqYWBgkyBKNf/s5sefP+6xy7NnZ39WzbvDIo+Pgf3aB3xaJO4cM5M48Cy5mIeZx+w27xmJYjYGnuIZeK25kWDGcOAAc2IPMw/bbd42icQ2Bp5kvDrkb6R//nDgQD1QC/uzYqK0GNzIMZA4cOAwUAuDGTNEC/thvFoMb+SUSZw5cLyY5zCPseTcNqBfmIHhjQ/I3Ujf/KHiQHUee//xhx/ettXlybe3P8arBQYSEAxmHgNStTCwPyBKyygYBaNgFIwYAACFFESMreAAHAAAAABJRU5ErkJggg==","orcid":"","institution":"University of Piraeus","correspondingAuthor":true,"prefix":"","firstName":"Anastasios","middleName":"","lastName":"Raptis","suffix":""},{"id":491765907,"identity":"6d6169ae-c746-4da2-8ec2-06df30aceab1","order_by":1,"name":"Stefanos Gritzalis","email":"","orcid":"","institution":"University of Piraeus","correspondingAuthor":false,"prefix":"","firstName":"Stefanos","middleName":"","lastName":"Gritzalis","suffix":""},{"id":491765908,"identity":"4730ff9c-4eed-4048-9476-5a98dbfa81fe","order_by":2,"name":"Athanasios Yannacopoulos","email":"","orcid":"","institution":"Athens University of Economics and Business","correspondingAuthor":false,"prefix":"","firstName":"Athanasios","middleName":"","lastName":"Yannacopoulos","suffix":""}],"badges":[],"createdAt":"2025-07-07 13:08:31","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7065795/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7065795/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":87958893,"identity":"dbbd4b1e-4873-4825-82dd-c871e58a6085","added_by":"auto","created_at":"2025-07-30 20:19:30","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1667128,"visible":true,"origin":"","legend":"","description":"","filename":"RaptisManuscriptUpdatedIJIS2025.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7065795/v1_covered_b0e62376-7798-4bd2-aca6-0675c041c847.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Dynamic Defense Strategies for Cyber-Physical Systems Using Stackelberg Games and Deep Reinforcement Learning in Discrete and Continuous Time","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"international-journal-of-information-security","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ijis","sideBox":"Learn more about [International Journal of Information Security](http://link.springer.com/journal/10207)","snPcode":"10207","submissionUrl":"https://submission.nature.com/new-submission/10207/3","title":"International Journal of Information Security","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Adaptive Mechanisms, Defense Strategies, Dynamic Modeling, Nonlinear Control, Power Grid Security","lastPublishedDoi":"10.21203/rs.3.rs-7065795/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7065795/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAs cyber threats to power grid infrastructures escalate, the urgency of understanding how to protect cyber-physical systems (CPS) has never been greater. 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