{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T06:25:19Z","timestamp":1772864719392,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,6,16]],"date-time":"2022-06-16T00:00:00Z","timestamp":1655337600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,6,16]],"date-time":"2022-06-16T00:00:00Z","timestamp":1655337600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Control Signals Syst."],"published-print":{"date-parts":[[2022,12]]},"DOI":"10.1007\/s00498-022-00325-2","type":"journal-article","created":{"date-parts":[[2022,6,16]],"date-time":"2022-06-16T09:02:57Z","timestamp":1655370177000},"page":"819-853","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The turnpike property and the longtime behavior of the Hamilton\u2013Jacobi\u2013Bellman equation for finite-dimensional LQ control problems"],"prefix":"10.1007","volume":"34","author":[{"given":"Carlos","family":"Esteve","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hicham","family":"Kouhkouh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5181-7350","authenticated-orcid":false,"given":"Dario","family":"Pighin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enrique","family":"Zuazua","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,6,16]]},"reference":[{"key":"325_CR1","volume-title":"Matrix Riccati equations in control and systems theory, systems and control: foundations and applications","author":"H Abou-Kandil","year":"2012","unstructured":"Abou-Kandil H, Freiling G, Ionescu V, Jank G (2012) Matrix Riccati equations in control and systems theory, systems and control: foundations and applications. Birkh\u00e4user, Basel"},{"key":"325_CR2","volume-title":"Optimal control: linear quadratic methods","author":"BD Anderson","year":"2007","unstructured":"Anderson BD, Moore JB (2007) Optimal control: linear quadratic methods. Courier Corporation, North Chelmsford"},{"key":"325_CR3","doi-asserted-by":"publisher","first-page":"1615","DOI":"10.1109\/TAC.2011.2179349","volume":"57","author":"D Angeli","year":"2011","unstructured":"Angeli D, Amrit R, Rawlings JB (2011) On average performance and stability of economic model predictive control. IEEE Trans Autom Control 57:1615\u20131626","journal-title":"IEEE Trans Autom Control"},{"key":"325_CR4","doi-asserted-by":"crossref","unstructured":"Arisawa M (1997) Ergodic problem for the Hamilton\u2013Jacobi\u2013Bellman equation. I. Existence of the ergodic attractor. In: Annales de l\u2019Institut Henri Poincare (C) non linear analysis, vol 14, Elsevier, pp 415\u2013438","DOI":"10.1016\/s0294-1449(97)80134-5"},{"key":"325_CR5","doi-asserted-by":"crossref","unstructured":"Arisawa M (1998) Ergodic problem for the Hamilton-Jacobi-Bellman equation. II, In: Annales de l\u2019Institut Henri Poincare (C) Non Linear Analysis, vol 15, Elsevier, pp. 1\u201324","DOI":"10.1016\/s0294-1449(99)80019-5"},{"key":"325_CR6","volume-title":"Optimal control and viscosity solutions of Hamilton\u2013Jacobi\u2013Bellman equations","author":"M Bardi","year":"2008","unstructured":"Bardi M, Capuzzo-Dolcetta I (2008) Optimal control and viscosity solutions of Hamilton\u2013Jacobi\u2013Bellman equations. Springer, Berlin"},{"key":"325_CR7","first-page":"1","volume":"112","author":"G Barles","year":"2019","unstructured":"Barles G, Ley O, Nguyen T-T, Phan TV (2019) Large time behavior of unbounded solutions of first-order Hamilton\u2013Jacobi equations in $${\\mathbb{R}}^N$$. Asymptot Anal 112:1\u201322","journal-title":"Asymptot Anal"},{"key":"325_CR8","doi-asserted-by":"publisher","first-page":"1209","DOI":"10.1080\/03605300500361461","volume":"31","author":"G Barles","year":"2006","unstructured":"Barles G, Roquejoffre J-M (2006) Ergodic type problems and large time behaviour of unbounded solutions of Hamilton\u2013Jacobi equations. Commun Partial Differ. Equ. 31:1209\u20131225","journal-title":"Commun Partial Differ. Equ."},{"key":"325_CR9","doi-asserted-by":"publisher","first-page":"925","DOI":"10.1137\/S0036141099350869","volume":"31","author":"G Barles","year":"2000","unstructured":"Barles G, Souganidis PE (2000) On the large time behavior of solutions of Hamilton\u2013Jacobi equations. SIAM J Math Anal 31:925\u2013939","journal-title":"SIAM J Math Anal"},{"key":"325_CR10","volume-title":"Representation and control of infinite dimensional systems, systems and control: foundations and applications","author":"A Bensoussan","year":"2011","unstructured":"Bensoussan A, Da Prato G, Delfour M, Mitter S (2011) Representation and control of infinite dimensional systems, systems and control: foundations and applications. Birkh\u00e4user, Boston"},{"key":"325_CR11","doi-asserted-by":"publisher","first-page":"1441","DOI":"10.1016\/j.matpur.2014.11.005","volume":"103","author":"A Bensoussan","year":"2015","unstructured":"Bensoussan A, Frehse J, Yam SCP (2015) The master equation in mean field theory. J Math\u00e9matiques Pures Appliqu\u00e9es 103:1441\u20131474","journal-title":"J Math\u00e9matiques Pures Appliqu\u00e9es"},{"key":"325_CR12","doi-asserted-by":"publisher","first-page":"2093","DOI":"10.1016\/j.spa.2016.10.004","volume":"127","author":"A Bensoussan","year":"2017","unstructured":"Bensoussan A, Frehse J, Yam SCP (2017) On the interpretation of the master equation. Stochas Process Appl 127:2093\u20132137","journal-title":"Stochas Process Appl"},{"key":"325_CR13","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1137\/0310026","volume":"10","author":"RF Brammer","year":"1972","unstructured":"Brammer RF (1972) Controllability in linear autonomous systems with positive controllers. SIAM J Control 10:339\u2013353","journal-title":"SIAM J Control"},{"key":"325_CR14","doi-asserted-by":"publisher","first-page":"1077","DOI":"10.1137\/18M1225811","volume":"58","author":"T Breiten","year":"2020","unstructured":"Breiten T, Pfeiffer L (2020) On the turnpike property and the receding-horizon method for linear-quadratic optimal control problems. SIAM J Control Optim 58:1077\u20131102","journal-title":"SIAM J Control Optim"},{"key":"325_CR15","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1006\/jmaa.1995.1169","volume":"192","author":"FM Callier","year":"1995","unstructured":"Callier FM, Winkin J (1995) Convergence of the time-invariant Riccati differential equation towards its strong solution for stabilizable systems. J Math Anal Appl 192:230\u2013257","journal-title":"J Math Anal Appl"},{"key":"325_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/b138356","volume-title":"Semiconcave functions, Hamilton\u2013Jacobi equations, and optimal control","author":"P Cannarsa","year":"2004","unstructured":"Cannarsa P, Sinestrari C (2004) Semiconcave functions, Hamilton\u2013Jacobi equations, and optimal control, vol 58. Springer, Berlin"},{"key":"325_CR17","doi-asserted-by":"publisher","first-page":"1397","DOI":"10.2140\/apde.2019.12.1397","volume":"12","author":"P Cardaliaguet","year":"2019","unstructured":"Cardaliaguet P, Porretta A (2019) Long time behavior of the master equation in mean field game theory. Anal PDE 12:1397\u20131453","journal-title":"Anal PDE"},{"key":"325_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0273-0979-1992-00266-5","volume":"27","author":"MG Crandall","year":"1992","unstructured":"Crandall MG, Ishii H, Lions P-L (1992) User\u2019s guide to viscosity solutions of second order partial differential equations. Bull Am Math Soc 27:1\u201367","journal-title":"Bull Am Math Soc"},{"key":"325_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0002-9947-1983-0690039-8","volume":"277","author":"MG Crandall","year":"1983","unstructured":"Crandall MG, Lions P-L (1983) Viscosity solutions of Hamilton\u2013Jacobi equations. Trans Am Math Soc 277:1\u201342","journal-title":"Trans Am Math Soc"},{"key":"325_CR20","unstructured":"Esteve C, Geshkovski B, Pighin D, Zuazua E (2020) Turnpike in lipschitz-nonlinear optimal control. arXiv:2011.11091"},{"key":"325_CR21","volume-title":"Partial differential equations","author":"LC Evans","year":"2010","unstructured":"Evans LC (2010) Partial differential equations, vol 19. American Mathematical Soc, Providence"},{"key":"325_CR22","doi-asserted-by":"publisher","first-page":"109907","DOI":"10.1016\/j.automatica.2021.109907","volume":"134","author":"T Faulwasser","year":"2021","unstructured":"Faulwasser T, Kellett CM (2021) On continuous-time infinite horizon optimal control-dissipativity, stability, and transversality. Automatica 134:109907","journal-title":"Automatica"},{"key":"325_CR23","doi-asserted-by":"publisher","first-page":"1671","DOI":"10.1512\/iumj.2006.55.2813","volume":"2006","author":"Y Fujita","year":"2006","unstructured":"Fujita Y, Ishii H, Loreti P (2006) Asymptotic solutions of Hamilton\u2013Jacobi equations in Euclidean n space. Indiana Univ Math J 2006:1671\u20131700","journal-title":"Indiana Univ Math J"},{"key":"325_CR24","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1365\/s13291-016-0134-5","volume":"118","author":"L Gr\u00fcne","year":"2016","unstructured":"Gr\u00fcne L (2016) Approximation properties of receding horizon optimal control. Jahresber Deutsch Math Verein 118:3\u201337","journal-title":"Jahresber Deutsch Math Verein"},{"key":"325_CR25","unstructured":"Gr\u00fcne L (2021) Dissipativity and optimal control. arXiv:2101.12606"},{"key":"325_CR26","doi-asserted-by":"publisher","first-page":"1282","DOI":"10.1137\/17M112350X","volume":"56","author":"L Gr\u00fcne","year":"2018","unstructured":"Gr\u00fcne L, Guglielmi R (2018) Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems. SIAM J Control Optim 56:1282\u20131302","journal-title":"SIAM J Control Optim"},{"key":"325_CR27","doi-asserted-by":"publisher","first-page":"169","DOI":"10.3934\/mcrf.2020032","volume":"11","author":"L Gr\u00fcne","year":"2021","unstructured":"Gr\u00fcne L, Guglielmi R (2021) On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Math Control Rel Fields 11:169","journal-title":"Math Control Rel Fields"},{"key":"325_CR28","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/j.sysconle.2016.01.003","volume":"90","author":"L Gr\u00fcne","year":"2016","unstructured":"Gr\u00fcne L, M\u00fcller MA (2016) On the relation between strict dissipativity and turnpike properties. Syst Control Lett 90:45\u201353","journal-title":"Syst Control Lett"},{"key":"325_CR29","doi-asserted-by":"crossref","unstructured":"Gr\u00fcne L, Pannek J (2017) Nonlinear model predictive control. In: Nonlinear Model Predictive Control, Springer, pp 45\u201369","DOI":"10.1007\/978-3-319-46024-6_3"},{"key":"325_CR30","first-page":"213","volume":"3","author":"H Ishii","year":"2006","unstructured":"Ishii H (2006) Asymptotic solutions for large time of Hamilton\u2013Jacobi equations. Int Congr Math 3:213\u2013227","journal-title":"Int Congr Math"},{"key":"325_CR31","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/j.anihpc.2006.09.002","volume":"25","author":"H Ishii","year":"2008","unstructured":"Ishii H (2008) Asymptotic solutions for large time of Hamilton\u2013Jacobi equations in Euclidean $$ n $$ space. Ann l\u2019IHP Analyse Non lin\u00e9aire 25:231\u2013266","journal-title":"Ann l\u2019IHP Analyse Non lin\u00e9aire"},{"key":"325_CR32","doi-asserted-by":"crossref","unstructured":"Ishii H (2013) A short introduction to viscosity solutions and the large time behavior of solutions of Hamilton\u2013Jacobi equations, In: Hamilton\u2013Jacobi Equations: Approximations, Numerical Analysis and Applications, Springer, pp 111\u2013249","DOI":"10.1007\/978-3-642-36433-4_3"},{"key":"325_CR33","unstructured":"Kouhkouh H (2018) Dynamic programming interpretation of turnpike and Hamilton\u2013Jacobi\u2013Bellman equation. Master thesis, Paris-Saclay University. http:\/\/bit.ly\/2R7soRx"},{"key":"325_CR34","volume-title":"Linear optimal control systems","author":"H Kwakernaak","year":"1972","unstructured":"Kwakernaak H, Sivan R (1972) Linear optimal control systems, vol 1. Wiley, New York"},{"key":"325_CR35","volume-title":"Foundations of optimal control theory","author":"EB Lee","year":"1967","unstructured":"Lee EB, Markus L (1967) Foundations of optimal control theory. Robert E. Krieger Publishing Company, Florida"},{"key":"325_CR36","volume-title":"Generalized solutions of Hamilton\u2013Jacobi equations","author":"P-L Lions","year":"1982","unstructured":"Lions P-L (1982) Generalized solutions of Hamilton\u2013Jacobi equations, vol 69. Pitman, London"},{"key":"325_CR37","unstructured":"Pighin D (2020) Nonuniqueness of minimizers for semilinear optimal control problems. arXiv:2002.04485"},{"key":"325_CR38","doi-asserted-by":"crossref","unstructured":"Pighin D (2020) The turnpike property in semilinear control. arXiv:2004.03269","DOI":"10.1051\/cocv\/2021036"},{"key":"325_CR39","unstructured":"Pighin D, Sakamoto N (2020) The turnpike with lack of observability. arXiv:2007.14081"},{"key":"325_CR40","doi-asserted-by":"publisher","first-page":"4242","DOI":"10.1137\/130907239","volume":"51","author":"A Porretta","year":"2013","unstructured":"Porretta A, Zuazua E (2013) Long time versus steady state optimal control. SIAM J Control Optim 51:4242\u20134273","journal-title":"SIAM J Control Optim"},{"key":"325_CR41","doi-asserted-by":"crossref","unstructured":"Porretta A, Zuazua E (2016) Remarks on long time versus steady state optimal control. In: Mathematical paradigms of climate science, Springer, pp 67\u201389","DOI":"10.1007\/978-3-319-39092-5_5"},{"key":"325_CR42","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/S0021-7824(00)01183-1","volume":"80","author":"J-M Roquejoffre","year":"2001","unstructured":"Roquejoffre J-M (2001) Convergence to steady states or periodic solutions in a class of Hamilton\u2013Jacobi equations. J Math\u00e9matiques pures et appliqu\u00e9es 80:85\u2013104","journal-title":"J Math\u00e9matiques pures et appliqu\u00e9es"},{"key":"325_CR43","doi-asserted-by":"publisher","first-page":"2335","DOI":"10.1109\/TAC.2008.2006113","volume":"53","author":"N Sakamoto","year":"2008","unstructured":"Sakamoto N, van der Schaft AJ (2008) Analytical approximation methods for the stabilizing solution of the Hamilton\u2013Jacobi equation. IEEE Trans Autom Control 53:2335\u20132350","journal-title":"IEEE Trans Autom Control"},{"key":"325_CR44","volume-title":"Contr\u00f4le optimal: th\u00e9orie & applications","author":"E Tr\u00e9lat","year":"2005","unstructured":"Tr\u00e9lat E (2005) Contr\u00f4le optimal: th\u00e9orie & applications, vol 36. Vuibert, Paris"},{"key":"325_CR45","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s00498-018-0209-1","volume":"30","author":"E Tr\u00e9lat","year":"2018","unstructured":"Tr\u00e9lat E, Zhang C (2018) Integral and measure-turnpike properties for infinite-dimensional optimal control systems. Math Control Signals Syst 30:3","journal-title":"Math Control Signals Syst"},{"key":"325_CR46","doi-asserted-by":"publisher","first-page":"1222","DOI":"10.1137\/16M1097638","volume":"56","author":"E Tr\u00e9lat","year":"2018","unstructured":"Tr\u00e9lat E, Zhang C, Zuazua E (2018) Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces. SIAM J Control Optim 56:1222\u20131252","journal-title":"SIAM J Control Optim"},{"key":"325_CR47","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/j.jde.2014.09.005","volume":"258","author":"E Tr\u00e9lat","year":"2015","unstructured":"Tr\u00e9lat E, Zuazua E (2015) The turnpike property in finite-dimensional nonlinear optimal control. J Differ Equ 258:81\u2013114","journal-title":"J Differ Equ"},{"key":"325_CR48","doi-asserted-by":"publisher","first-page":"621","DOI":"10.1109\/TAC.1971.1099831","volume":"16","author":"J Willems","year":"1971","unstructured":"Willems J (1971) Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans Autom Control 16:621\u2013634","journal-title":"IEEE Trans Autom Control"},{"key":"325_CR49","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jprocont.2017.12.005","volume":"63","author":"M Zanon","year":"2018","unstructured":"Zanon M, Faulwasser T (2018) Economic MPC without terminal constraints: gradient-correcting end penalties enforce asymptotic stability. J Process Control 63:1\u201314","journal-title":"J Process Control"},{"key":"325_CR50","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1016\/j.jprocont.2016.06.006","volume":"45","author":"M Zanon","year":"2016","unstructured":"Zanon M, Gros S, Diehl M (2016) A tracking MPC formulation that is locally equivalent to economic MPC. J Process Control 45:30\u201342","journal-title":"J Process Control"}],"container-title":["Mathematics of Control, Signals, and Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00498-022-00325-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00498-022-00325-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00498-022-00325-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,23]],"date-time":"2022-11-23T00:15:30Z","timestamp":1669162530000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00498-022-00325-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,16]]},"references-count":50,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,12]]}},"alternative-id":["325"],"URL":"https:\/\/doi.org\/10.1007\/s00498-022-00325-2","relation":{},"ISSN":["0932-4194","1435-568X"],"issn-type":[{"value":"0932-4194","type":"print"},{"value":"1435-568X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,16]]},"assertion":[{"value":"30 June 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 June 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}