{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T08:19:34Z","timestamp":1772525974677,"version":"3.50.1"},"reference-count":21,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SPP 1253"],"award-info":[{"award-number":["SPP 1253"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper we discuss the use of implicit Runge\u2013Kutta schemes for the time discretization\nof optimal control problems with evolution equations.\nThe specialty of the considered discretizations is that\nthe discretizations schemes for the state and adjoint state are chosen\nsuch that discretization and optimization commute.\nIt is well known that for Runge\u2013Kutta schemes with this property additional order conditions are necessary.\nWe give sufficient conditions for which class of schemes these additional order condition are automatically fulfilled.\nThe focus is especially on implicit Runge\u2013Kutta schemes of\nGauss, Radau IA, Radau IIA, Lobatto IIIA, Lobatto IIIB and Lobatto IIIC collocation type up to order six.\nFurthermore, we also use a SDIRK (singly diagonally implicit Runge\u2013Kutta) method to demonstrate, that for general implicit Runge\u2013Kutta methods the additional order conditions are not automatically fulfilled.\nNumerical examples illustrate the predicted convergence rates.<\/jats:p>","DOI":"10.1515\/cmam-2022-0097","type":"journal-article","created":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T08:36:50Z","timestamp":1680251810000},"page":"917-952","source":"Crossref","is-referenced-by-count":2,"title":["Implicit Runge\u2013Kutta Schemes for Optimal Control Problems with Evolution Equations"],"prefix":"10.1515","volume":"23","author":[{"given":"Thomas G.","family":"Flaig","sequence":"first","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Bauingenieurwesen und Umweltwissenschaften , Universit\u00e4t der Bundeswehr M\u00fcnchen , Neubiberg , Germany"}]}],"member":"374","published-online":{"date-parts":[[2023,4,1]]},"reference":[{"key":"2023100411150426208_j_cmam-2022-0097_ref_001","doi-asserted-by":"crossref","unstructured":"T.  Apel and T. G.  Flaig,\nCrank\u2013Nicolson schemes for optimal control problems with evolution equations,\nSIAM J. Numer. Anal. 50 (2012), no. 3, 1484\u20131512.","DOI":"10.1137\/100819333"},{"key":"2023100411150426208_j_cmam-2022-0097_ref_002","doi-asserted-by":"crossref","unstructured":"R.  Becker, D.  Meidner and B.  Vexler,\nEfficient numerical solution of parabolic optimization problems by finite element methods,\nOptim. Methods Softw. 22 (2007), no. 5, 813\u2013833.","DOI":"10.1080\/10556780701228532"},{"key":"2023100411150426208_j_cmam-2022-0097_ref_003","unstructured":"J. F.  Bonnans and J.  Laurent-Varin,\nComputation of order conditions for symplectic partitioned Runge\u2013Kutta schemes with application to optimal control,\nRapport de recherche RR\u20135398 2004, http:\/\/hal.inria.fr\/docs\/00\/07\/06\/05\/PDF\/RR-5398.pdf."},{"key":"2023100411150426208_j_cmam-2022-0097_ref_004","doi-asserted-by":"crossref","unstructured":"J. F.  Bonnans and J.  Laurent-Varin,\nComputation of order conditions for symplectic partitioned Runge\u2013Kutta schemes with application to optimal control,\nNumer. Math. 103 (2006), no. 1, 1\u201310.","DOI":"10.1007\/s00211-005-0661-y"},{"key":"2023100411150426208_j_cmam-2022-0097_ref_005","doi-asserted-by":"crossref","unstructured":"T.  Dupont and R.  Scott,\nPolynomial approximation of functions in Sobolev spaces,\nMath. Comp. 34 (1980), no. 150, 441\u2013463.","DOI":"10.1090\/S0025-5718-1980-0559195-7"},{"key":"2023100411150426208_j_cmam-2022-0097_ref_006","unstructured":"T. G.  Flaig,\nDiscretization strategies for optimal control problems with parabolic partial differential equations,\nPhD thesis, Universit\u00e4t der Bundeswehr M\u00fcnchen, 2013."},{"key":"2023100411150426208_j_cmam-2022-0097_ref_007","doi-asserted-by":"crossref","unstructured":"W. W.  Hager,\nRates of convergence for discrete approximations to unconstrained control problems,\nSIAM J. Numer. 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