{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:24:33Z","timestamp":1740122673221,"version":"3.37.3"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T00:00:00Z","timestamp":1681862400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T00:00:00Z","timestamp":1681862400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["GE 1163\/8-2"],"award-info":[{"award-number":["GE 1163\/8-2"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we derive error estimates for Runge\u2013Kutta schemes of optimal control problems subject to index one differential\u2013algebraic equations (DAEs). Usually, Runge\u2013Kutta methods applied to DAEs approximate the differential and algebraic state in an analogous manner. These schemes can be considered as discretizations of the index reduced system where the algebraic equation is solved for the algebraic variable to get an explicit ordinary differential equation. However, in optimal control this approach yields discrete necessary conditions that are not consistent with the continuous necessary conditions which are essential for deriving error estimates. Therefore, we suggest to treat the algebraic variable like a control, obtaining a new type of Runge\u2013Kutta scheme. For this method we derive consistent necessary conditions and compare the discrete and continuous systems to get error estimates up to order three for the states and control as well as the multipliers.<\/jats:p>","DOI":"10.1007\/s10589-023-00484-1","type":"journal-article","created":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T20:32:58Z","timestamp":1681936378000},"page":"1299-1325","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Error estimates for Runge\u2013Kutta schemes of optimal control problems with index 1 DAEs"],"prefix":"10.1007","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2439-1005","authenticated-orcid":false,"given":"Bj\u00f6rn","family":"Martens","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,19]]},"reference":[{"issue":"1","key":"484_CR1","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1080\/02331934.2011.568619","volume":"62","author":"W Alt","year":"2011","unstructured":"Alt, W., Baier, R., Lempio, F., Gerdts, M.: Approximations of linear control problems with bang\u2013bang solutions. Optimization 62(1), 9\u201332 (2011). https:\/\/doi.org\/10.1080\/02331934.2011.568619","journal-title":"Optimization"},{"issue":"3","key":"484_CR2","doi-asserted-by":"publisher","first-page":"547","DOI":"10.3934\/naco.2012.2.547","volume":"2","author":"W Alt","year":"2012","unstructured":"Alt, W., Baier, R., Gerdts, M., Lempio, F.: Error bounds for Euler approximation of linear-quadratic control problems with bang\u2013bang solutions. Numer. Algebra Contr. Optim. 2(3), 547\u2013570 (2012). https:\/\/doi.org\/10.3934\/naco.2012.2.547","journal-title":"Numer. Algebra Contr. Optim."},{"issue":"3","key":"484_CR3","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1080\/10556788.2013.821612","volume":"29","author":"W Alt","year":"2014","unstructured":"Alt, W., Seydenschwanz, M.: An implicit discretization scheme for linear-quadratic control problems with bang\u2013bang solutions. Optim. Methods Softw. 29(3), 535\u2013560 (2014). https:\/\/doi.org\/10.1080\/10556788.2013.821612","journal-title":"Optim. Methods Softw."},{"issue":"4","key":"484_CR4","doi-asserted-by":"publisher","first-page":"512","DOI":"10.1002\/oca.2126","volume":"36","author":"W Alt","year":"2015","unstructured":"Alt, W., Schneider, C.: Linear-quadratic control problems with $${L}^1$$-control cost. Optim. Contr. Appl. Methods 36(4), 512\u2013534 (2015). https:\/\/doi.org\/10.1002\/oca.2126","journal-title":"Optim. Contr. Appl. Methods"},{"issue":"5","key":"484_CR5","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1016\/j.amc.2016.04.028","volume":"287\u2013288","author":"W Alt","year":"2016","unstructured":"Alt, W., Schneider, C., Seydenschwanz, M.: Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang\u2013bang solutions. Appl. Math. Comput. 287\u2013288(5), 104\u2013124 (2016). https:\/\/doi.org\/10.1016\/j.amc.2016.04.028","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"484_CR6","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1007\/s10589-017-9969-7","volume":"69","author":"W Alt","year":"2018","unstructured":"Alt, W., Felgenhauer, U., Seydenschwanz, M.: Euler discretization for a class of nonlinear optimal control problems with control appearing linearly. Comput. Optim. Appl. 69(3), 825\u2013856 (2018). https:\/\/doi.org\/10.1007\/s10589-017-9969-7","journal-title":"Comput. Optim. Appl."},{"key":"484_CR7","doi-asserted-by":"publisher","unstructured":"Betts, J. T.: Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd edn. Advances in Design and Control. SIAM (2010). https:\/\/doi.org\/10.1137\/1.9780898718577","DOI":"10.1137\/1.9780898718577"},{"issue":"2","key":"484_CR8","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1137\/140999621","volume":"55","author":"JF Bonnans","year":"2017","unstructured":"Bonnans, J.F., Festa, A.: Error estimates for the Euler discretization of an optimal control problem with first-order state constraints. SIAM J. Numer. Anal. 55(2), 445\u2013471 (2017). https:\/\/doi.org\/10.1137\/140999621","journal-title":"SIAM J. Numer. Anal."},{"key":"484_CR9","doi-asserted-by":"publisher","unstructured":"Brennan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. volume 14 of Classics in Applied Mathematics. SIAM (1996). https:\/\/doi.org\/10.1137\/1.9781611971224","DOI":"10.1137\/1.9781611971224"},{"key":"484_CR10","doi-asserted-by":"publisher","unstructured":"Burger, M., Gerdts, M.: A survey on numerical methods for the simulation of initial value problems with sDAEs. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations IV. Differential-Algebraic Equations Forum. Springer, Berlin (2017). https:\/\/doi.org\/10.1007\/978-3-319-46618-7_5","DOI":"10.1007\/978-3-319-46618-7_5"},{"key":"484_CR11","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1007\/BF01215994","volume":"31","author":"AL Dontchev","year":"1995","unstructured":"Dontchev, A.L., Hager, W.W., Poore, A.B., Yang, B.: Optimality, stability, and convergence in nonlinear control. Appl. Math. Optim. 31, 297\u2013326 (1995). https:\/\/doi.org\/10.1007\/BF01215994","journal-title":"Appl. Math. Optim."},{"issue":"5\u20136","key":"484_CR12","doi-asserted-by":"publisher","first-page":"653","DOI":"10.1080\/01630560008816979","volume":"21","author":"AL Dontchev","year":"2000","unstructured":"Dontchev, A.L., Hager, W.W., Malanowski, K.: Error bounds for Euler approximation of a state and control constrained optimal control problem. Numer. Funct. Anal. Optim. 21(5\u20136), 653\u2013682 (2000). https:\/\/doi.org\/10.1080\/01630560008816979","journal-title":"Numer. Funct. Anal. Optim."},{"issue":"1","key":"484_CR13","doi-asserted-by":"publisher","first-page":"202","DOI":"10.1137\/S0036142999351765","volume":"38","author":"AL Dontchev","year":"2000","unstructured":"Dontchev, A.L., Hager, W.W., Veliov, V.M.: Second-order Runge\u2013Kutta approximations in control constrained optimal control. SIAM J. Numer. Anal. 38(1), 202\u2013226 (2000). https:\/\/doi.org\/10.1137\/S0036142999351765","journal-title":"SIAM J. Numer. Anal."},{"issue":"233","key":"484_CR14","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1090\/S0025-5718-00-01184-4","volume":"70","author":"AL Dontchev","year":"2001","unstructured":"Dontchev, A.L., Hager, W.W.: The Euler approximation in state constrained optimal control. Math. Comput. 70(233), 173\u2013203 (2001). https:\/\/doi.org\/10.1090\/S0025-5718-00-01184-4","journal-title":"Math. Comput."},{"issue":"6","key":"484_CR15","doi-asserted-by":"publisher","first-page":"1361","DOI":"10.1137\/0330072","volume":"30","author":"JC Dunn","year":"1992","unstructured":"Dunn, J.C., Tian, T.: Variants of the Kuhn\u2013Tucker sufficient conditions in cones of nonnegative functions. SIAM J. Contr. Optim. 30(6), 1361\u20131384 (1992). https:\/\/doi.org\/10.1137\/0330072","journal-title":"SIAM J. Contr. Optim."},{"key":"484_CR16","doi-asserted-by":"publisher","DOI":"10.1515\/9783110249996","volume-title":"Optimal Control of ODEs and DAEs","author":"M Gerdts","year":"2012","unstructured":"Gerdts, M.: Optimal Control of ODEs and DAEs. De Gruyter, Berlin (2012). https:\/\/doi.org\/10.1515\/9783110249996"},{"issue":"1","key":"484_CR17","doi-asserted-by":"publisher","first-page":"311","DOI":"10.3934\/JIMO.2014.10.311","volume":"10","author":"M Gerdts","year":"2014","unstructured":"Gerdts, M., Kunkel, M.: Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. J. Ind. Manag. Optim. 10(1), 311\u2013336 (2014). https:\/\/doi.org\/10.3934\/JIMO.2014.10.311","journal-title":"J. Ind. Manag. Optim."},{"issue":"2","key":"484_CR18","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1007\/s002110000178","volume":"87","author":"WW Hager","year":"2000","unstructured":"Hager, W.W.: Runge\u2013Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87(2), 247\u2013282 (2000). https:\/\/doi.org\/10.1007\/s002110000178","journal-title":"Numer. Math."},{"key":"484_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0093947","volume-title":"The Numerical Solution of Differential-Algebraic Systems by Runge\u2013Kutta Methods","author":"E Hairer","year":"1989","unstructured":"Hairer, E., Lubich, C., Roche, M.: The Numerical Solution of Differential-Algebraic Systems by Runge\u2013Kutta Methods, vol. 1409. Springer, Berlin (1989). https:\/\/doi.org\/10.1007\/BFb0093947"},{"key":"484_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-09947-6","volume-title":"Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics","author":"E Hairer","year":"1996","unstructured":"Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, vol. 14, 2nd edn. Springer, Berlin (1996). https:\/\/doi.org\/10.1007\/978-3-662-09947-6","edition":"2"},{"key":"484_CR21","doi-asserted-by":"publisher","unstructured":"Haunschmied, J.L., Pietrus, A., Veliov, V.M.: The Euler method for linear control systems revisited. In: Proceedings of the 9th International Conference on Large-Scale Scientific Computations, Sozopol, pp. 90\u201397 (2013). https:\/\/doi.org\/10.1007\/978-3-662-43880-0_9","DOI":"10.1007\/978-3-662-43880-0_9"},{"key":"484_CR22","volume-title":"Theory of Extremal Problems. Studies in Mathematics and Its Applications","author":"AD Ioffe","year":"1979","unstructured":"Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. Studies in Mathematics and Its Applications, vol. 6. North-Holland Publishing Company, Amsterdam (1979)"},{"key":"484_CR23","unstructured":"Kraft, D.: FORTRAN-Programme zur numerischen L\u00f6sung optimaler Steuerungsprobleme. DFVLR-Mitteilung, vol. 80. DFVLR (1980)"},{"key":"484_CR24","doi-asserted-by":"publisher","unstructured":"Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations: Analysis and Numerical Solution. EMS Textbooks in Mathematics. European Mathematical Society (2006). https:\/\/doi.org\/10.4171\/017","DOI":"10.4171\/017"},{"key":"484_CR25","doi-asserted-by":"publisher","DOI":"10.1201\/9781003072119-12","volume-title":"Mathematical programming with data perturbations","author":"K Malanowski","year":"1997","unstructured":"Malanowski, K., B\u00fcskens, C., Maurer, H.: Convergence of Approximations to Nonlinear Optimal Control Problems. Lecture Notes in Pure and Applied Mathematics. In: Fiacco, A.V. (ed.) Mathematical programming with data perturbations. CRC Press, Boca Raton (1997). https:\/\/doi.org\/10.1201\/9781003072119-12"},{"key":"484_CR26","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1007\/S11228-018-0471-X","volume":"27","author":"B Martens","year":"2019","unstructured":"Martens, B., Gerdts, M.: Convergence analysis of the implicit Euler-discretization and sufficient conditions for optimal control problems subject to index-one differential-algebraic equations. Set-Valued Var Anal 27, 405\u2013431 (2019). https:\/\/doi.org\/10.1007\/S11228-018-0471-X","journal-title":"Set-Valued Var Anal"},{"key":"484_CR27","doi-asserted-by":"publisher","unstructured":"Martens, B.: Necessary conditions, sufficient conditions, and convergence analysis for optimal control problems with differential-algebraic equations. PhD thesis, Universit\u00e4t der Bundeswehr M\u00fcnchen (2019) https:\/\/doi.org\/10.1007\/978-3-030-53905-4_10","DOI":"10.1007\/978-3-030-53905-4_10"},{"issue":"1","key":"484_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/18m1219382","volume":"58","author":"B Martens","year":"2020","unstructured":"Martens, B., Gerdts, M.: Convergence analysis for approximations of optimal control problems subject to higher index differential-algebraic equations and mixed control-state constraints. SIAM J. Contr. Optim. 58(1), 1\u201333 (2020). https:\/\/doi.org\/10.1137\/18m1219382","journal-title":"SIAM J. Contr. Optim."},{"key":"484_CR29","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-53905-4_10","volume-title":"Progress in differential-algebraic Equations II. Differential-algebraic equations forum","author":"B Martens","year":"2020","unstructured":"Martens, B., Gerdts, M.: Error analysis for the implicit Euler discretization of linear-quadratic control problems with higher index DAEs and bang-bang solutions. In: Reis, T., Grundel, S., Sch\u00f6ps, S. (eds.) Progress in differential-algebraic Equations II. Differential-algebraic equations forum. Springer, Berlin (2020). https:\/\/doi.org\/10.1007\/978-3-030-53905-4_10"},{"issue":"3","key":"484_CR30","doi-asserted-by":"publisher","first-page":"1903","DOI":"10.1137\/20M1353952","volume":"59","author":"B Martens","year":"2021","unstructured":"Martens, B., Gerdts, M.: Convergence analysis for approximations of optimal control problems subject to higher index differential-algebraic equations and pure state constraints. SIAM J. Contr. Optim. 59(3), 1903\u20131926 (2021). https:\/\/doi.org\/10.1137\/20M1353952","journal-title":"SIAM J. Contr. Optim."},{"issue":"6","key":"484_CR31","first-page":"1383","volume":"6","author":"B Martens","year":"2021","unstructured":"Martens, B., Schneider, C.: Error analysis for the implicit Euler discretization of affine optimal control problems with index two DAEs. Pure Appl. Funct. Anal. 6(6), 1383\u20131414 (2021)","journal-title":"Pure Appl. Funct. Anal."},{"key":"484_CR32","doi-asserted-by":"publisher","DOI":"10.1051\/COCV\/2019046","author":"NP Osmolovskii","year":"2020","unstructured":"Osmolovskii, N.P., Veliov, V.M.: Metric sub-regularity in optimal control of affine problems with free end state. ESAIM: COCV (2020). https:\/\/doi.org\/10.1051\/COCV\/2019046","journal-title":"ESAIM: COCV"},{"issue":"1","key":"484_CR33","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1137\/16M1079142","volume":"56","author":"A Pietrus","year":"2018","unstructured":"Pietrus, A., Scarinci, T., Veliov, V.M.: High order discrete approximations to Mayer\u2019s problems for linear systems. SIAM J. Contr. Optim. 56(1), 102\u2013119 (2018). https:\/\/doi.org\/10.1137\/16M1079142","journal-title":"SIAM J. Contr. Optim."},{"key":"484_CR34","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1007\/s10589-017-9948-z","volume":"69","author":"T Scarinci","year":"2018","unstructured":"Scarinci, T., Veliov, V.M.: Higher-order numerical scheme for linear-quadratic problems with bang-bang controls. Comput. Optim. Appl. 69, 403\u2013422 (2018). https:\/\/doi.org\/10.1007\/s10589-017-9948-z","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"484_CR35","doi-asserted-by":"publisher","first-page":"811","DOI":"10.1051\/COCV\/2017049","volume":"24","author":"C Schneider","year":"2018","unstructured":"Schneider, C., Wachsmuth, G.: Regularization and discretization error estimates for optimal control of ODEs with group sparsity. ESAIM: COCV 24(2), 811\u2013834 (2018). https:\/\/doi.org\/10.1051\/COCV\/2017049","journal-title":"ESAIM: COCV"},{"key":"484_CR36","doi-asserted-by":"publisher","first-page":"731","DOI":"10.1007\/s10589-015-9730-z","volume":"61","author":"M Seydenschwanz","year":"2015","unstructured":"Seydenschwanz, M.: Convergence results for the discrete regularization of linear-quadratic control problems with bang\u2013bang solutions. Comput. Optim. Appl. 61, 731\u2013760 (2015). https:\/\/doi.org\/10.1007\/s10589-015-9730-z","journal-title":"Comput. Optim. Appl."},{"key":"484_CR37","unstructured":"von Stryk, O.: Numerische L\u00f6sung optimaler Steuerungsprobleme: Diskretisierung, Parameteroptimierung und Berechnung der adjungierten Variablen. Ph.D. thesis, Technische Universit\u00e4t M\u00fcnchen (1994)"},{"issue":"3","key":"484_CR38","first-page":"967","volume":"34","author":"VM Veliov","year":"2005","unstructured":"Veliov, V.M.: Error analysis of discrete approximations to bang\u2013bang optimal control problems: the linear case. Contr. Cybern. 34(3), 967\u2013982 (2005)","journal-title":"Contr. Cybern."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-023-00484-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-023-00484-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-023-00484-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T12:13:59Z","timestamp":1699877639000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-023-00484-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,19]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["484"],"URL":"https:\/\/doi.org\/10.1007\/s10589-023-00484-1","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"type":"print","value":"0926-6003"},{"type":"electronic","value":"1573-2894"}],"subject":[],"published":{"date-parts":[[2023,4,19]]},"assertion":[{"value":"17 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 March 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author has no relevant financial or non-financial interests to disclose. All data generated or analyzed during this study are included in this published article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Statements and Declarations"}}]}}