On the variational discretization of optimal control problems for Lagrangian dynamics

On the variational discretization of optimal control problems for Lagrangian dynamics | 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 On the variational discretization of optimal control problems for Lagrangian dynamics Michael Konopik, Sigrid Leyendecker, Sofya Maslovskaya, Sina Ober-Blöbaum, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6566751/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 09 Jan, 2026 Read the published version in Multibody System Dynamics → Version 1 posted 2 You are reading this latest preprint version Abstract We derive the necessary optimality conditions of an optimal control problem with dynamical constraints described by forced Euler-Lagrange equations, using a recently proposed new Lagrangian approach [1] and use variational integrators to solve them. We show that for a family of low-order discretizations the resulting numerical schemes are ‘doubly symplectic’, meaning they provide forced symplectic integrators for the underlying controlled mechanical system and symplectic integrators in the state-adjoint space. This paves the way for variational error analysis to be used in the optimal control setting to derive the order of convergence of the resulting numerical schemes and further the possibility to apply discrete Noether’s theorem for the calculation of conserved quantities there. The schemes derived behave well numerically, benefiting from this ’double-symplecticity’. A multi-body example is solved and the benefits of this ‘double-symplecticity’ illustrated. The example demonstrates the convergence and also the ability to preserve first integrals associated to symmetries in the considered optimal control problem. Mathematics Subject Classification: 37M15, 49K15, 49M05, 49M25, 65K10, 65P10, 70H15, 70G45, 70G65, 70Q05. discretization discrete mechanics mechanical systems necessary conditions numerical integration optimal control symmetries symplectic methods variational methods Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 09 Jan, 2026 Read the published version in Multibody System Dynamics → Version 1 posted Submission checks completed at journal 01 May, 2025 First submitted to journal 30 Apr, 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. 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-6566751","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":450616053,"identity":"3588ca0e-a153-40cf-a8c5-6bca6ada9416","order_by":0,"name":"Michael Konopik","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8UlEQVRIiWNgGAWjYDACZgY2IGnBIAHmGSTIsTcAaR7CWiTgWox5DhDSwoCihSEhsYeQFt125mcPflRIMEi29x58XFCQlt7DfsaA4U0Fbi1mh9nMDXvOSDBI85xLNp5hkJPbw5NjwDjnDD4tPGwSvG0SDHISOWbSPAYVufsZcgyYedvwa5H8+w+hJZ2H/w1Qyz/8WqR5G4AOg2jJSeCRANnSgNcvZtIyxyR4JHvOGAP9kmbYI/Gs4OCcY3i0nD/8TPJNjY2cxPEew8cFf5LlefiTNz54U4NbCwyAI4IZxjtAWAMUMBNWMgpGwSgYBSMRAAAcvEMsfLbQtAAAAABJRU5ErkJggg==","orcid":"","institution":"University of Erlangen-Nuremberg","correspondingAuthor":true,"prefix":"","firstName":"Michael","middleName":"","lastName":"Konopik","suffix":""},{"id":450616054,"identity":"676e1ae8-ddc7-4ffa-b946-f770960e758c","order_by":1,"name":"Sigrid Leyendecker","email":"","orcid":"","institution":"University of Erlangen-Nuremberg","correspondingAuthor":false,"prefix":"","firstName":"Sigrid","middleName":"","lastName":"Leyendecker","suffix":""},{"id":450616055,"identity":"d333d336-a50e-48c8-b600-dfd042651cd7","order_by":2,"name":"Sofya Maslovskaya","email":"","orcid":"","institution":"University of Paderborn","correspondingAuthor":false,"prefix":"","firstName":"Sofya","middleName":"","lastName":"Maslovskaya","suffix":""},{"id":450616056,"identity":"d551257a-78bb-44ec-9db0-2fa221da4ace","order_by":3,"name":"Sina Ober-Blöbaum","email":"","orcid":"","institution":"University of Paderborn","correspondingAuthor":false,"prefix":"","firstName":"Sina","middleName":"","lastName":"Ober-Blöbaum","suffix":""},{"id":450616057,"identity":"c4ab4411-7d6f-4dab-8d6d-df1bc6aa5e73","order_by":4,"name":"Rodrigo T. Sato Martin de Almagro","email":"","orcid":"","institution":"University of Erlangen-Nuremberg","correspondingAuthor":false,"prefix":"","firstName":"Rodrigo","middleName":"T. Sato Martin","lastName":"de Almagro","suffix":""}],"badges":[],"createdAt":"2025-04-30 16:38:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6566751/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6566751/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11044-025-10138-1","type":"published","date":"2026-01-09T15:58:51+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":100070013,"identity":"96489f34-12c8-49f5-afba-e1b83d31c377","added_by":"auto","created_at":"2026-01-12 16:15:55","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":471607,"visible":true,"origin":"","legend":"","description":"","filename":"forcedLagrangianvariationalintegrator2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6566751/v1_covered_5094db2d-57ac-47b6-a65c-b7d092f38b13.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"On the variational discretization of optimal control problems for Lagrangian dynamics","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"multibody-system-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"mubo","sideBox":"Learn more about [Multibody System Dynamics](http://link.springer.com/journal/11044)","snPcode":"11044","submissionUrl":"https://submission.nature.com/new-submission/11044/3","title":"Multibody System Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"discretization, discrete mechanics, mechanical systems, necessary conditions, numerical integration, optimal control, symmetries, symplectic methods, variational methods","lastPublishedDoi":"10.21203/rs.3.rs-6566751/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6566751/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe derive the necessary optimality conditions of an optimal control problem with dynamical constraints described by forced Euler-Lagrange equations, using a recently proposed new Lagrangian approach [1] and use variational integrators to solve them. We show that for a family of low-order discretizations the resulting numerical schemes are ‘doubly symplectic’, meaning they provide forced symplectic integrators for the underlying controlled mechanical system and symplectic integrators in the state-adjoint space. This paves the way for variational error analysis to be used in the optimal control setting to derive the order of convergence of the resulting numerical schemes and further the possibility to apply discrete Noether’s theorem for the calculation of conserved quantities there. The schemes derived behave well numerically, benefiting from this ’double-symplecticity’. A multi-body example is solved and the benefits of this ‘double-symplecticity’ illustrated. The example demonstrates the convergence and also the ability to preserve first integrals associated to symmetries in the considered optimal control problem.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMathematics Subject Classification:\u003c/strong\u003e 37M15, 49K15, 49M05, 49M25, 65K10, 65P10, 70H15, 70G45, 70G65, 70Q05.\u003c/p\u003e","manuscriptTitle":"On the variational discretization of optimal control problems for Lagrangian dynamics","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-05-02 02:36:34","doi":"10.21203/rs.3.rs-6566751/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"checksComplete","content":"","date":"2025-05-01T10:57:18+00:00","index":"","fulltext":""},{"type":"submitted","content":"Multibody System Dynamics","date":"2025-04-30T16:25:56+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"multibody-system-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"mubo","sideBox":"Learn more about [Multibody System Dynamics](http://link.springer.com/journal/11044)","snPcode":"11044","submissionUrl":"https://submission.nature.com/new-submission/11044/3","title":"Multibody System Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"41d1fd76-ed2f-429e-9d33-c826bfa7e4dd","owner":[],"postedDate":"May 2nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2026-01-12T16:09:36+00:00","versionOfRecord":{"articleIdentity":"rs-6566751","link":"https://doi.org/10.1007/s11044-025-10138-1","journal":{"identity":"multibody-system-dynamics","isVorOnly":false,"title":"Multibody System Dynamics"},"publishedOn":"2026-01-09 15:58:51","publishedOnDateReadable":"January 9th, 2026"},"versionCreatedAt":"2025-05-02 02:36:34","video":"","vorDoi":"10.1007/s11044-025-10138-1","vorDoiUrl":"https://doi.org/10.1007/s11044-025-10138-1","workflowStages":[]},"version":"v1","identity":"rs-6566751","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6566751","identity":"rs-6566751","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}