{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T15:55:30Z","timestamp":1770911730252,"version":"3.50.1"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,5,11]],"date-time":"2022-05-11T00:00:00Z","timestamp":1652227200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,5,11]],"date-time":"2022-05-11T00:00:00Z","timestamp":1652227200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001652","name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001652","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2022,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval. We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systems.<\/jats:p>","DOI":"10.1007\/s00211-022-01290-3","type":"journal-article","created":{"date-parts":[[2022,5,11]],"date-time":"2022-05-11T21:06:20Z","timestamp":1652303180000},"page":"495-549","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A framework for randomized time-splitting in linear-quadratic optimal control"],"prefix":"10.1007","volume":"151","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0767-2710","authenticated-orcid":false,"given":"D. W. M.","family":"Veldman","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1377-0958","authenticated-orcid":false,"given":"E.","family":"Zuazua","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,5,11]]},"reference":[{"key":"1290_CR1","doi-asserted-by":"publisher","unstructured":"Apel, T., Flaig, T.G.: Crank\u2013Nicolson schemes for optimal control problems with evolution equations. SIAM J. Numer. Anal. 50(3), 1484\u20131512 (2012). ISSN 0036-1429. https:\/\/doi.org\/10.1137\/100819333","DOI":"10.1137\/100819333"},{"key":"1290_CR2","doi-asserted-by":"publisher","unstructured":"Azmi, B., Kunisch, K.: On the stabilizability of the Burgers equation by receding horizon control. SIAM J. Control Optim. 54(3), 1378\u20131405 (2016). ISSN 0363-0129. https:\/\/doi.org\/10.1137\/15M1030352","DOI":"10.1137\/15M1030352"},{"key":"1290_CR3","doi-asserted-by":"publisher","unstructured":"Azmi, B., Kunisch, K.: Receding horizon control for the stabilization of the wave equation. Discrete Contin. Dyn. Syst. 38(2), 449\u2013484 (2018). ISSN 1078-0947. https:\/\/doi.org\/10.3934\/dcds.2018021","DOI":"10.3934\/dcds.2018021"},{"key":"1290_CR4","doi-asserted-by":"publisher","unstructured":"Benning, M., Celledoni, E., Ehrhardt, M.J., Owren, B., Sch\u00f6nlieb, C.-B.: Deep learning as optimal control problems: models and numerical methods. J. Comput. Dyn. 6(2), 171\u2013198 (2019). ISSN 2158-2491. https:\/\/doi.org\/10.3934\/jcd.2019009","DOI":"10.3934\/jcd.2019009"},{"key":"1290_CR5","doi-asserted-by":"publisher","unstructured":"Biccari, U., Hern\u00e1ndez-Santamar\u00eda, V.: Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects. IMA J. Math. Control Inf. 36(4), 1199\u20131235 (2018). ISSN 0265-0754. https:\/\/doi.org\/10.1093\/imamci\/dny025","DOI":"10.1093\/imamci\/dny025"},{"key":"1290_CR6","doi-asserted-by":"publisher","unstructured":"Bottou, L., Curtis, F.E., Nocedal, J.: Optimization methods for large-scale machine learning. SIAM Rev. 60(2), 223\u2013311 (2018). ISSN 0036-1445. https:\/\/doi.org\/10.1137\/16M1080173","DOI":"10.1137\/16M1080173"},{"key":"1290_CR7","doi-asserted-by":"publisher","unstructured":"Brauer, F.: Perturbations of nonlinear systems of differential equations. J. Math. Anal. Appl. 14, 198\u2013206 (1966). ISSN 0022-247X. https:\/\/doi.org\/10.1016\/0022-247X(66)90021-7","DOI":"10.1016\/0022-247X(66)90021-7"},{"key":"1290_CR8","doi-asserted-by":"publisher","unstructured":"Dolean, V., Jolivet, P., Nataf, F.: An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2015). ISBN 978-1-611974-05-8. https:\/\/doi.org\/10.1137\/1.9781611974065.ch1","DOI":"10.1137\/1.9781611974065.ch1"},{"key":"1290_CR9","unstructured":"Esteve, C., Geshkovski, B.: Sparse approximation in learning via neural ODEs (2021)"},{"key":"1290_CR10","unstructured":"Esteve, C., Geshkovski, B., Pighin, D., Zuazua, E.: Large-time asymptotics in deep learning (2021)"},{"key":"1290_CR11","doi-asserted-by":"publisher","unstructured":"Gr\u00fcne, L., Pannek, J.: Nonlinear Model Predictive Control. Communications and Control Engineering Series. Springer, Cham (2017). ISBN 978-3-319-46023-9; 978-3-319-46024-6. https:\/\/doi.org\/10.1007\/978-3-319-46024-6. Theory and algorithms, Second edition [of MR3155076]","DOI":"10.1007\/978-3-319-46024-6"},{"key":"1290_CR12","doi-asserted-by":"publisher","unstructured":"Hansen, E., Ostermann, A.: Dimension splitting for evolution equations. Numer. Math. 108(4), 557\u2013570 (2008). ISSN 0029-599X. https:\/\/doi.org\/10.1007\/s00211-007-0129-3","DOI":"10.1007\/s00211-007-0129-3"},{"key":"1290_CR13","doi-asserted-by":"publisher","unstructured":"Ignat, L.I.: A splitting method for the nonlinear Schr\u00f6dinger equation. J. Differ. Equ. 250(7), 3022\u20133046 (2011). ISSN 0022-0396. https:\/\/doi.org\/10.1016\/j.jde.2011.01.028","DOI":"10.1016\/j.jde.2011.01.028"},{"key":"1290_CR14","doi-asserted-by":"publisher","unstructured":"Jin, S., Li, L., Liu, J.-G.: Random batch methods (RBM) for interacting particle systems. J. Comput. Phys. 400, 108877 (2020). ISSN 0021-9991. https:\/\/doi.org\/10.1016\/j.jcp.2019.108877","DOI":"10.1016\/j.jcp.2019.108877"},{"key":"1290_CR15","doi-asserted-by":"crossref","unstructured":"Jin, S., Li, L., Liu, J.-G.: Convergence of random batch method for interacting particles with disparate species and weights (2020)","DOI":"10.1137\/20M1327641"},{"key":"1290_CR16","unstructured":"Kato, T.: Trotter\u2019s product formula for an arbitrary pair of self-adjoint contraction semigroups. In: Topics in Functional Analysis (Essays Dedicated to M. G. Kre\u012dn on the Occasion of his 70th birthday), Adv. in Math. Suppl. Stud., vol. 3, pp. 185\u2013195. Academic Press, New York (1978)"},{"key":"1290_CR17","unstructured":"Kirk, D.E.: Optimal Control Theory: An Introduction. Dover (2004)"},{"key":"1290_CR18","doi-asserted-by":"publisher","unstructured":"Ko, D., Zuazua, E.: Model predictive control with random batch methods for a guiding problem. Math. Models Methods Appl. Sci. 31(8), 1569\u20131592 (2021). ISSN 0218-2025. https:\/\/doi.org\/10.1142\/S0218202521500329","DOI":"10.1142\/S0218202521500329"},{"key":"1290_CR19","doi-asserted-by":"publisher","unstructured":"Lapidus, M.L.: Generalization of the Trotter-Lie formula. Integral Equ. Oper. Theory 4(3), 366\u2013415 (1981). ISSN 0378-620X. https:\/\/doi.org\/10.1007\/BF01697972","DOI":"10.1007\/BF01697972"},{"key":"1290_CR20","volume-title":"Foundations of Optimal Control Theory","author":"EB Lee","year":"1967","unstructured":"Lee, E.B., Markus, L.: Foundations of Optimal Control Theory. John Wiley & Sons Inc, New York (1967)"},{"key":"1290_CR21","doi-asserted-by":"publisher","unstructured":"Li, L., Xu, Z., Zhao, Y.: A random-batch Monte Carlo method for many-body systems with singular kernels. SIAM J. Sci. Comput. 42(3), A1486\u2013A1509 (2020). ISSN 1064-8275. https:\/\/doi.org\/10.1137\/19M1302077","DOI":"10.1137\/19M1302077"},{"key":"1290_CR22","unstructured":"Minoux, M., Vajda, S.: Mathematical Programming: Theory and Algorithms. A Wiley-Interscience publication, Wiley (1986). ISBN 9780471901709. https:\/\/books.google.de\/books?id=5kDvAAAAMAAJ"},{"key":"1290_CR23","doi-asserted-by":"publisher","unstructured":"Neidhardt, H., Zagrebnov, V.A.: On error estimates for the Trotter-Kato product formula. Lett. Math. Phys. 44(3), 169\u2013186 (1998). ISSN 0377-9017. https:\/\/doi.org\/10.1023\/A:1007494816401","DOI":"10.1023\/A:1007494816401"},{"key":"1290_CR24","doi-asserted-by":"publisher","unstructured":"Ostermann, A., Schratz, K.: Stability of exponential operator splitting methods for noncontractive semigroups. SIAM J. Numer. Anal. 51(1), 191\u2013203 (2013). ISSN 0036-1429. https:\/\/doi.org\/10.1137\/110846580","DOI":"10.1137\/110846580"},{"key":"1290_CR25","doi-asserted-by":"publisher","unstructured":"Reble, M., Allg\u00f6wer, F.: Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems. Automatica J. IFAC 48(8), 1812\u20131817 (2012). ISSN 0005-1098. https:\/\/doi.org\/10.1016\/j.automatica.2012.05.067","DOI":"10.1016\/j.automatica.2012.05.067"},{"key":"1290_CR26","doi-asserted-by":"publisher","unstructured":"Rohatgi, V.K., Ehsanes Saleh, A.K.M.: An Introduction to Probability and Statistics, 3rd edn. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken (2015). ISBN 978-1-118-79964-2. https:\/\/doi.org\/10.1002\/9781118799635","DOI":"10.1002\/9781118799635"},{"key":"1290_CR27","unstructured":"Ruiz-Balet, D., Zuazua, E.: Neural ODE control for classification, approximation and transport (2021)"},{"key":"1290_CR28","doi-asserted-by":"publisher","unstructured":"Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545\u2013551 (1959). ISSN 0002-9939. https:\/\/doi.org\/10.2307\/2033649","DOI":"10.2307\/2033649"},{"key":"1290_CR29","doi-asserted-by":"publisher","unstructured":"E, W.: A proposal on machine learning via dynamical systems. Commun. Math. Stat. 5(1), 1\u201311 (2017). ISSN 2194-6701. https:\/\/doi.org\/10.1007\/s40304-017-0103-z","DOI":"10.1007\/s40304-017-0103-z"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01290-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-022-01290-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01290-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,1]],"date-time":"2022-06-01T23:04:33Z","timestamp":1654124673000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-022-01290-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,11]]},"references-count":29,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,6]]}},"alternative-id":["1290"],"URL":"https:\/\/doi.org\/10.1007\/s00211-022-01290-3","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,11]]},"assertion":[{"value":"27 September 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 March 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 April 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 May 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}