{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T21:25:59Z","timestamp":1778793959110,"version":"3.51.4"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T00:00:00Z","timestamp":1732492800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T00:00:00Z","timestamp":1732492800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100021856","name":"Ministero dell'Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["CUP H53D2300892000"],"award-info":[{"award-number":["CUP H53D2300892000"]}],"id":[{"id":"10.13039\/501100021856","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100021856","name":"Ministero dell'Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["2022238YY5"],"award-info":[{"award-number":["2022238YY5"]}],"id":[{"id":"10.13039\/501100021856","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Dyn Games Appl"],"published-print":{"date-parts":[[2025,5]]},"DOI":"10.1007\/s13235-024-00605-3","type":"journal-article","created":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T13:36:18Z","timestamp":1732541778000},"page":"481-502","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A POD Approach to Identify and Control PDEs Online Through State Dependent Riccati Equations"],"prefix":"10.1007","volume":"15","author":[{"given":"Alessandro","family":"Alla","sequence":"first","affiliation":[]},{"given":"Agnese","family":"Pacifico","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,11,25]]},"reference":[{"key":"605_CR1","doi-asserted-by":"crossref","unstructured":"Pacifico A, Pesare A, Falcone M (2021) A new algorithm for the LQR problem with partially unknown dynamics. In: 2021 Large-Scale Scientific Computing (LSSC) . Springer International Publishing","DOI":"10.1007\/978-3-030-97549-4_37"},{"key":"605_CR2","doi-asserted-by":"publisher","unstructured":"Alla A, Pacifico A, Palladino M, Pesare Andrea A (2024) Online identification and control of PDEs via reinforcement learning methods. Adv Comput Math. 50(4). https:\/\/doi.org\/10.1007\/s10444-024-10167-y","DOI":"10.1007\/s10444-024-10167-y"},{"issue":"1","key":"605_CR3","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1002\/gamm.201310003","volume":"36","author":"Peter Benner","year":"2013","unstructured":"Benner Peter, Saak Jens (2013) Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey. GAMM-Mitteilungen 36(1):32\u201352. https:\/\/doi.org\/10.1002\/gamm.201310003","journal-title":"GAMM-Mitteilungen"},{"issue":"9","key":"605_CR4","doi-asserted-by":"publisher","first-page":"755","DOI":"10.1002\/nla.622","volume":"15","author":"P Benner","year":"2008","unstructured":"Benner P, Li J-R, Penzl T (2008) Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems. Numer. Linear Algebra Appl. 15(9):755\u2013777","journal-title":"Numer. Linear Algebra Appl."},{"issue":"3","key":"605_CR5","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1137\/130912839","volume":"58","author":"V Simoncini","year":"2016","unstructured":"Simoncini V (2016) Computational methods for linear matrix equations. SIAM Rev 58(3):377\u2013441","journal-title":"SIAM Rev"},{"issue":"3","key":"605_CR6","doi-asserted-by":"publisher","first-page":"904","DOI":"10.1093\/imanum\/drt015","volume":"34","author":"V Simoncini","year":"2014","unstructured":"Simoncini V, Szyld DB, Monsalve M (2014) On two numerical methods for the solution of large-scale algebraic Riccati equations. IMA J Numer Anal 34(3):904\u2013920","journal-title":"IMA J Numer Anal"},{"issue":"4","key":"605_CR7","doi-asserted-by":"publisher","first-page":"1655","DOI":"10.1137\/16M1059382","volume":"37","author":"V Simoncini","year":"2016","unstructured":"Simoncini V (2016) Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations. SIAM J Matrix Anal Appl 37(4):1655\u20131674. https:\/\/doi.org\/10.1137\/16M1059382","journal-title":"SIAM J Matrix Anal Appl"},{"key":"605_CR8","volume-title":"Numerical Solution of Algebraic Riccati Equations","author":"DA Bini","year":"2012","unstructured":"Bini DA, Iannazzo B, Meini B (2012) Numerical Solution of Algebraic Riccati Equations. SIAM, Philadelphia"},{"issue":"1","key":"605_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1023\/A:1004607114958","volume":"107","author":"SC Beeler","year":"2000","unstructured":"Beeler SC, Tran HT, Banks HT (2000) Feedback control methodologies for nonlinear systems. J Optim Theory Appl 107(1):1\u201333","journal-title":"J Optim Theory Appl"},{"issue":"2","key":"605_CR10","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s10589-007-9015-2","volume":"37","author":"HT Banks","year":"2007","unstructured":"Banks HT, Lewis BM, Tran HT (2007) Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach. Comput Optim Appl 37(2):177\u2013218","journal-title":"Comput Optim Appl"},{"key":"605_CR11","volume-title":"Optimal Control: Linear Quadratic Methods","author":"BDO Anderson","year":"1999","unstructured":"Anderson BDO, Moore JB (1999) Optimal Control: Linear Quadratic Methods. Prentice-Hall, Hoboken"},{"issue":"4","key":"605_CR12","doi-asserted-by":"publisher","first-page":"1218","DOI":"10.1002\/rnc.3949","volume":"28","author":"P Benner","year":"2018","unstructured":"Benner P, Heiland J (2018) Exponential stability and stabilization of extended linearizations via continuous updates of Riccati-based feedback. Int J Robust Nonlinear Control 28(4):1218\u20131232. https:\/\/doi.org\/10.1002\/rnc.3949","journal-title":"Int J Robust Nonlinear Control"},{"key":"605_CR13","doi-asserted-by":"publisher","unstructured":"Alla A, Kalise D, Simoncini V (2023) State-dependent Riccati equation feedback stabilization for nonlinear PDEs. Adv Comput Math. 49(1). https:\/\/doi.org\/10.1007\/s10444-022-09998-4","DOI":"10.1007\/s10444-022-09998-4"},{"key":"605_CR14","doi-asserted-by":"publisher","unstructured":"Atwell JA, King BB (2001) Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Mathematical and Computer Modelling 33(1), 1\u201319 https:\/\/doi.org\/10.1016\/S0895-7177(00)00225-9 . Computation and control VI proceedings of the sixth Bozeman conference","DOI":"10.1016\/S0895-7177(00)00225-9"},{"issue":"1","key":"605_CR15","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1146\/annurev.fl.25.010193.002543","volume":"25","author":"G Berkooz","year":"1993","unstructured":"Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech 25(1):539\u2013575. https:\/\/doi.org\/10.1146\/annurev.fl.25.010193.002543","journal-title":"Annu Rev Fluid Mech"},{"key":"605_CR16","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1090\/qam\/910462","volume":"45","author":"L Sirovich","year":"1987","unstructured":"Sirovich L (1987) Turbulence and the dynamics of coherent structures. i. coherent structures. J Quart Appl. Math 45:561\u2013571. https:\/\/doi.org\/10.1090\/qam\/910462","journal-title":"J Quart Appl. Math"},{"issue":"4","key":"605_CR17","doi-asserted-by":"publisher","first-page":"413","DOI":"10.3934\/naco.2016018","volume":"6","author":"B Kramer","year":"2016","unstructured":"Kramer B, Singler JR (2016) A pod projection method for large-scale algebraic Riccati equations. Num Algebra Control Optimiz 6(4):413\u2013435. https:\/\/doi.org\/10.3934\/naco.2016018","journal-title":"Num Algebra Control Optimiz"},{"key":"605_CR18","doi-asserted-by":"publisher","unstructured":"(2009) Optimization with PDE Constraints. Springer, Dordrecht, Netherlands. https:\/\/doi.org\/10.1007\/978-1-4020-8839-1","DOI":"10.1007\/978-1-4020-8839-1"},{"key":"605_CR19","doi-asserted-by":"publisher","unstructured":"Alla A, Simoncini V (2018). In: Falcone M, Ferretti R, Gr\u00fcne L, McEneaney WM (eds) Order Reduction Approaches for the Algebraic Riccati Equation and the LQR Problem. Springer, Cham. https:\/\/doi.org\/10.1007\/978-3-030-01959-4_5","DOI":"10.1007\/978-3-030-01959-4_5"},{"issue":"1","key":"605_CR20","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1051\/cocv\/2017011","volume":"24","author":"Andreas Schmidt","year":"2018","unstructured":"Schmidt Andreas, Haasdonk Bernard (2018) Reduced basis approximation of large scale parametric algebraic Riccati equations. ESAIM: COCV 24(1):129\u2013151. https:\/\/doi.org\/10.1051\/cocv\/2017011","journal-title":"ESAIM: COCV"},{"key":"605_CR21","doi-asserted-by":"publisher","DOI":"10.1109\/LCSYS.2023.3291231","author":"J Heiland","year":"2023","unstructured":"Heiland J, Werner SWR (2023) Low-complexity linear parameter-varying approximations of incompressible Navier-Stokes equations for truncated state-dependent Riccati feedback. IEEE Control Syst Lett. https:\/\/doi.org\/10.1109\/LCSYS.2023.3291231","journal-title":"IEEE Control Syst Lett"},{"issue":"5","key":"605_CR22","doi-asserted-by":"publisher","first-page":"2737","DOI":"10.1137\/090766498","volume":"32","author":"Saifon Chaturantabut","year":"2010","unstructured":"Chaturantabut Saifon, Sorensen Danny C. (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32(5):2737\u20132764. https:\/\/doi.org\/10.1137\/090766498","journal-title":"SIAM J Sci Comput"},{"key":"605_CR23","doi-asserted-by":"publisher","unstructured":"Studinger A, Volkwein S (2013) In: Bredies, K., Clason, C., Kunisch, K., Winckel, G. (eds.) Numerical Analysis of POD A-posteriori Error Estimation for Optimal Control, pp. 137\u2013158. Springer, Basel . https:\/\/doi.org\/10.1007\/978-3-0348-0631-2_8","DOI":"10.1007\/978-3-0348-0631-2_8"},{"key":"605_CR24","volume-title":"Bayesian Statistics and Marketing","author":"PE Rossi","year":"2012","unstructured":"Rossi PE, Allenby GM, McCulloch R (2012) Bayesian Statistics and Marketing. John Wiley & Sons, Chichester"},{"key":"605_CR25","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.1602614","author":"SH Rudy","year":"2017","unstructured":"Rudy SH, Brunton SL, Proctor JL, Kutz JN (2017) Data-driven discovery of partial differential equations. Sci Adv. https:\/\/doi.org\/10.1126\/sciadv.1602614","journal-title":"Sci Adv"},{"issue":"2","key":"605_CR26","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1137\/18M1191944","volume":"18","author":"S Rudy","year":"2019","unstructured":"Rudy S, Alla A, Brunton SL, Kutz JN (2019) Data-driven identification of parametric partial differential equations. SIAM J Appl Dyn Syst 18(2):643\u2013660. https:\/\/doi.org\/10.1137\/18M1191944","journal-title":"SIAM J Appl Dyn Syst"},{"key":"605_CR27","doi-asserted-by":"publisher","first-page":"3932","DOI":"10.1073\/pnas.1517384113","volume":"115","author":"SL Brunton","year":"2016","unstructured":"Brunton SL, Proctor JL, Kutz JN (2016) Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc Natl Acad Sci USA 115:3932\u20133937","journal-title":"Proc Natl Acad Sci USA"},{"issue":"2219","key":"605_CR28","doi-asserted-by":"publisher","first-page":"20180335","DOI":"10.1098\/rspa.2018.0335","volume":"474","author":"E Kaiser","year":"2018","unstructured":"Kaiser E, Kutz JN, Brunton SL (2018) Sparse identification of nonlinear dynamics for model predictive control in the low-data limit. Proc Royal Soc A Math Phys Eng Sci 474(2219):20180335. https:\/\/doi.org\/10.1098\/rspa.2018.0335","journal-title":"Proc Royal Soc A Math Phys Eng Sci"},{"key":"605_CR29","doi-asserted-by":"publisher","unstructured":"Krstic M, Smyshlyaev A (2007) Adaptive control of pdes. IFAC Proceedings Volumes 40(13), 20\u201331 https:\/\/doi.org\/10.3182\/20070829-3-RU-4911.00004 . 9th IFAC Workshop on Adaptation and Learning in Control and Signal Processing","DOI":"10.3182\/20070829-3-RU-4911.00004"},{"issue":"2","key":"605_CR30","doi-asserted-by":"publisher","first-page":"8130","DOI":"10.1016\/j.ifacol.2020.12.2294","volume":"53","author":"AB Martinsen","year":"2020","unstructured":"Martinsen AB, Lekkas AM, Gros S (2020) Combining system identification with reinforcement learning-based MPC. IFAC-PapersOnLine 53(2):8130\u20138135. https:\/\/doi.org\/10.1016\/j.ifacol.2020.12.2294","journal-title":"IFAC-PapersOnLine"},{"key":"605_CR31","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/978-3-031-11818-0_21","volume-title":"Progress in Industrial Mathematics at ECMI 2021","author":"S Casper","year":"2022","unstructured":"Casper S, Fuertinger DH, Kotanko P, Mechelli L, Rohleff J, Volkwein S (2022) Data-driven modeling and control of complex dynamical systems arising in renal anemia therapy. In: Ehrhardt M, G\u00fcnther M (eds) Progress in Industrial Mathematics at ECMI 2021. Springer, Cham, pp 155\u2013161"},{"key":"605_CR32","doi-asserted-by":"crossref","unstructured":"Cloutier JR (1997) State-dependent Riccati equation techniques: an overview. In: Proceedings of the 1997 American Control Conference (Cat. No.97CH36041), vol. 2, pp. 932\u20139362","DOI":"10.1109\/ACC.1997.609663"},{"key":"605_CR33","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1137\/15M1019271","volume":"38","author":"Z Drmac","year":"2016","unstructured":"Drmac Z, Gugercin S (2016) A new selection operator for the discrete empirical interpolation method - improved a priori error bound and extensions. SIAM J. Sci. Comput. 38:631\u2013648. https:\/\/doi.org\/10.1137\/15M1019271","journal-title":"SIAM J. Sci. Comput."},{"key":"605_CR34","doi-asserted-by":"publisher","unstructured":"Alla A, Schmidt A, Haasdonk B (2017) In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K (eds.) Model Order Reduction Approaches for Infinite Horizon Optimal Control Problems via the HJB Equation, pp. 333\u2013347. Springer, Cham . https:\/\/doi.org\/10.1007\/978-3-319-58786-8_21","DOI":"10.1007\/978-3-319-58786-8_21"},{"key":"605_CR35","unstructured":"The MathWorks, Inc.: MATLAB 7, R2017b edn. (2017). The MathWorks, Inc"},{"key":"605_CR36","doi-asserted-by":"crossref","unstructured":"Rasmussen C, Williams C (2006) Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning, p. 248. MIT Press, Cambridge, MA, USA . Max-Planck-Gesellschaft","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"605_CR37","doi-asserted-by":"crossref","unstructured":"Alla A, Bertaglia G, Calzola E (2024) A PINN approach for the online identification and control of unknown PDEs.https:\/\/arxiv.org\/abs\/2408.03456","DOI":"10.1007\/s10957-025-02686-5"}],"container-title":["Dynamic Games and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-024-00605-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s13235-024-00605-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-024-00605-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,19]],"date-time":"2025-05-19T07:11:03Z","timestamp":1747638663000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s13235-024-00605-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,25]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["605"],"URL":"https:\/\/doi.org\/10.1007\/s13235-024-00605-3","relation":{},"ISSN":["2153-0785","2153-0793"],"issn-type":[{"value":"2153-0785","type":"print"},{"value":"2153-0793","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,25]]},"assertion":[{"value":"26 October 2024","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 November 2024","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"No conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}}]}}