{"id":"https://openalex.org/W4281870568","doi":"https://doi.org/10.1017/s0962492922000046","title":"Turnpike in optimal control of PDEs, ResNets, and beyond","display_name":"Turnpike in optimal control of PDEs, ResNets, and beyond","publication_year":2022,"publication_date":"2022-05-01","ids":{"openalex":"https://openalex.org/W4281870568","doi":"https://doi.org/10.1017/s0962492922000046"},"language":"en","primary_location":{"id":"doi:10.1017/s0962492922000046","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0962492922000046","pdf_url":null,"source":{"id":"https://openalex.org/S124973578","display_name":"Acta Numerica","issn_l":"0962-4929","issn":["0962-4929","1474-0508"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Acta Numerica","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5049531040","display_name":"Borjan Geshkovski","orcid":"https://orcid.org/0000-0002-7890-3352"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Borjan Geshkovski","raw_affiliation_strings":["Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA E-mail:"],"raw_orcid":"https://orcid.org/0000-0002-7890-3352","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA E-mail:","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5088702266","display_name":"Enrique Zuazua","orcid":"https://orcid.org/0000-0002-1377-0958"},"institutions":[{"id":"https://openalex.org/I136040515","display_name":"Universidad de Deusto","ror":"https://ror.org/00ne6sr39","country_code":"ES","type":"education","lineage":["https://openalex.org/I136040515"]},{"id":"https://openalex.org/I181369854","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57","country_code":"DE","type":"education","lineage":["https://openalex.org/I181369854"]},{"id":"https://openalex.org/I63634437","display_name":"Universidad Aut\u00f3noma de Madrid","ror":"https://ror.org/01cby8j38","country_code":"ES","type":"education","lineage":["https://openalex.org/I63634437"]}],"countries":["DE","ES"],"is_corresponding":false,"raw_author_name":"Enrique Zuazua","raw_affiliation_strings":["Chair in Dynamics, Control, and Numerics, Alexander von Humboldt-Professorship, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg, Cauerstrasse 11, 91052Erlangen, Germany, Chair of Computational Mathematics, Fundaci\u00f3n Deusto, Av. de las Universidades 24, 48007 Bilbao, Basque Country, Spain, and Departamento de Matem\u00e1ticas, Universidad Aut\u00f3noma de Madrid, 28049Madrid, Spain E-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Chair in Dynamics, Control, and Numerics, Alexander von Humboldt-Professorship, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg, Cauerstrasse 11, 91052Erlangen, Germany, Chair of Computational Mathematics, Fundaci\u00f3n Deusto, Av. de las Universidades 24, 48007 Bilbao, Basque Country, Spain, and Departamento de Matem\u00e1ticas, Universidad Aut\u00f3noma de Madrid, 28049Madrid, Spain E-mail:","institution_ids":["https://openalex.org/I136040515","https://openalex.org/I63634437","https://openalex.org/I181369854"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5049531040"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":6.5661,"has_fulltext":false,"cited_by_count":35,"citation_normalized_percentile":{"value":0.97273603,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"31","issue":null,"first_page":"135","last_page":"263"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9965000152587891,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9957000017166138,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.7530155181884766},{"id":"https://openalex.org/keywords/controllability","display_name":"Controllability","score":0.5294047594070435},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4962840676307678},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.49271324276924133},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.47173893451690674},{"id":"https://openalex.org/keywords/time-horizon","display_name":"Time horizon","score":0.45768702030181885},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.4459710717201233},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.44288402795791626},{"id":"https://openalex.org/keywords/path","display_name":"Path (computing)","score":0.4201827049255371},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3206542730331421},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.28837257623672485},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09534400701522827}],"concepts":[{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.7530155181884766},{"id":"https://openalex.org/C48209547","wikidata":"https://www.wikidata.org/wiki/Q1331104","display_name":"Controllability","level":2,"score":0.5294047594070435},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4962840676307678},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.49271324276924133},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.47173893451690674},{"id":"https://openalex.org/C28761237","wikidata":"https://www.wikidata.org/wiki/Q7805321","display_name":"Time horizon","level":2,"score":0.45768702030181885},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.4459710717201233},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.44288402795791626},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.4201827049255371},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3206542730331421},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.28837257623672485},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09534400701522827},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1017/s0962492922000046","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0962492922000046","pdf_url":null,"source":{"id":"https://openalex.org/S124973578","display_name":"Acta Numerica","issn_l":"0962-4929","issn":["0962-4929","1474-0508"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Acta Numerica","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/8","display_name":"Decent work and economic growth","score":0.4000000059604645}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":177,"referenced_works":["https://openalex.org/W102428431","https://openalex.org/W203646022","https://openalex.org/W205642032","https://openalex.org/W569156507","https://openalex.org/W569478347","https://openalex.org/W588902295","https://openalex.org/W630764638","https://openalex.org/W831507090","https://openalex.org/W879905685","https://openalex.org/W1209334740","https://openalex.org/W1410907832","https://openalex.org/W1496662771","https://openalex.org/W1511008224","https://openalex.org/W1554854591","https://openalex.org/W1579781026","https://openalex.org/W1580724525","https://openalex.org/W1583837637","https://openalex.org/W1598902920","https://openalex.org/W1600371943","https://openalex.org/W1604226557","https://openalex.org/W1833425995","https://openalex.org/W1880408395","https://openalex.org/W1963883253","https://openalex.org/W1974961323","https://openalex.org/W1981740375","https://openalex.org/W1982143211","https://openalex.org/W1982372749","https://openalex.org/W1988313155","https://openalex.org/W1996270146","https://openalex.org/W1998297320","https://openalex.org/W1998986126","https://openalex.org/W2000961292","https://openalex.org/W2001981441","https://openalex.org/W2002306296","https://openalex.org/W2002744405","https://openalex.org/W2003588897","https://openalex.org/W2008103464","https://openalex.org/W2011121645","https://openalex.org/W2015953351","https://openalex.org/W2024891059","https://openalex.org/W2025779039","https://openalex.org/W2026382324","https://openalex.org/W2029123910","https://openalex.org/W2031521898","https://openalex.org/W2033102642","https://openalex.org/W2033517841","https://openalex.org/W2037842665","https://openalex.org/W2038686546","https://openalex.org/W2043233479","https://openalex.org/W2043357744","https://openalex.org/W2048164789","https://openalex.org/W2052810255","https://openalex.org/W2053068479","https://openalex.org/W2054012096","https://openalex.org/W2056261501","https://openalex.org/W2060088282","https://openalex.org/W2060921867","https://openalex.org/W2063686245","https://openalex.org/W2072234220","https://openalex.org/W2073787051","https://openalex.org/W2077425610","https://openalex.org/W2077743708","https://openalex.org/W2079261393","https://openalex.org/W2079331586","https://openalex.org/W2094143568","https://openalex.org/W2095477585","https://openalex.org/W2097237536","https://openalex.org/W2102040002","https://openalex.org/W2103496339","https://openalex.org/W2116995996","https://openalex.org/W2119073761","https://openalex.org/W2121743042","https://openalex.org/W2135870560","https://openalex.org/W2141354966","https://openalex.org/W2141568824","https://openalex.org/W2149633325","https://openalex.org/W2162798807","https://openalex.org/W2171568556","https://openalex.org/W2194775991","https://openalex.org/W2221321873","https://openalex.org/W2234752102","https://openalex.org/W2302591207","https://openalex.org/W2487893106","https://openalex.org/W2498073803","https://openalex.org/W2503933042","https://openalex.org/W2505127555","https://openalex.org/W2527774453","https://openalex.org/W2529725405","https://openalex.org/W2541668901","https://openalex.org/W2556018921","https://openalex.org/W2557272457","https://openalex.org/W2566079294","https://openalex.org/W2587214152","https://openalex.org/W2600297185","https://openalex.org/W2604272474","https://openalex.org/W2604640182","https://openalex.org/W2606449094","https://openalex.org/W2611298687","https://openalex.org/W2612252966","https://openalex.org/W2613213659","https://openalex.org/W2613633283","https://openalex.org/W2613694435","https://openalex.org/W2755331706","https://openalex.org/W2765410998","https://openalex.org/W2795744704","https://openalex.org/W2796221713","https://openalex.org/W2796933306","https://openalex.org/W2796938752","https://openalex.org/W2804375568","https://openalex.org/W2822752092","https://openalex.org/W2898464076","https://openalex.org/W2898906188","https://openalex.org/W2899059679","https://openalex.org/W2901851481","https://openalex.org/W2904139432","https://openalex.org/W2904698172","https://openalex.org/W2922504883","https://openalex.org/W2924132978","https://openalex.org/W2947156405","https://openalex.org/W2963337560","https://openalex.org/W2963559690","https://openalex.org/W2964198179","https://openalex.org/W2964338264","https://openalex.org/W2966348706","https://openalex.org/W2990121169","https://openalex.org/W2991649084","https://openalex.org/W2995206221","https://openalex.org/W3009342461","https://openalex.org/W3035613866","https://openalex.org/W3036842247","https://openalex.org/W3037904587","https://openalex.org/W3080719302","https://openalex.org/W3088967319","https://openalex.org/W3089885363","https://openalex.org/W3094699692","https://openalex.org/W3098004101","https://openalex.org/W3098011980","https://openalex.org/W3101977391","https://openalex.org/W3102271101","https://openalex.org/W3102997397","https://openalex.org/W3104707106","https://openalex.org/W3105248177","https://openalex.org/W3105398875","https://openalex.org/W3105872988","https://openalex.org/W3106233363","https://openalex.org/W3109748076","https://openalex.org/W3125484093","https://openalex.org/W3134515582","https://openalex.org/W3137907199","https://openalex.org/W3150803659","https://openalex.org/W3150807214","https://openalex.org/W3160459473","https://openalex.org/W3161859084","https://openalex.org/W3171937494","https://openalex.org/W3172037580","https://openalex.org/W3173952933","https://openalex.org/W3180237281","https://openalex.org/W3183153896","https://openalex.org/W3193494219","https://openalex.org/W3206761256","https://openalex.org/W3207108121","https://openalex.org/W4210624614","https://openalex.org/W4212774754","https://openalex.org/W4213123361","https://openalex.org/W4213299226","https://openalex.org/W4242731857","https://openalex.org/W4248816015","https://openalex.org/W4252517507","https://openalex.org/W4292078458","https://openalex.org/W4298228534","https://openalex.org/W6603108611","https://openalex.org/W6641315968","https://openalex.org/W6648698375","https://openalex.org/W6677870239","https://openalex.org/W6731291048","https://openalex.org/W6744834936","https://openalex.org/W6770847983"],"related_works":["https://openalex.org/W2036697162","https://openalex.org/W2140812588","https://openalex.org/W2411113819","https://openalex.org/W2072887367","https://openalex.org/W2332386680","https://openalex.org/W2561315646","https://openalex.org/W2248621902","https://openalex.org/W2003779889","https://openalex.org/W2164760767","https://openalex.org/W4205698120"],"abstract_inverted_index":{"The":[0],"turnpike":[1,98,120],"property":[2,99],"in":[3,83,114,132,269],"contemporary":[4],"macroeconomics":[5],"asserts":[6],"that":[7,60,190],"if":[8],"an":[9,15,127,234],"economic":[10],"planner":[11,33],"seeks":[12],"to":[13,22,38,42,46,57,75,78,104,156,186,196,242],"move":[14,40],"economy":[16],"from":[17,124],"one":[18],"level":[19,44],"of":[20,109,146,205,218,229,263],"capital":[21,56],"another,":[23],"then":[24,53],"the":[25,32,47,63,72,79,94,97,119,147,151,157,161,165,173,187,203,206,219,223,227,230,261],"most":[26,145],"efficient":[27],"path,":[28,52],"as":[29,31,88],"long":[30,199],"has":[34,100],"enough":[35],"time,":[36,148],"is":[37,66,233],"rapidly":[39],"stock":[41,73],"a":[43,89,133,183,216],"close":[45,142],"optimal":[48,128,136,158,168,192,250],"stationary":[49,167,208],"or":[50],"constant":[51,177],"allow":[54],"for":[55,106,126,164,210],"develop":[58],"along":[59],"path":[61],"until":[62],"desired":[64],"term":[65],"nearly":[67,249],"reached,":[68],"at":[69],"which":[70,247],"point":[71],"ought":[74],"be":[76,240],"moved":[77],"final":[80,154],"target.":[81],"Motivated":[82],"part":[84],"by":[85],"its":[86],"nature":[87],"resource":[90],"allocation":[91],"strategy,":[92],"over":[93,178,198,222],"past":[95,224],"decade,":[96],"also":[101],"been":[102],"shown":[103],"hold":[105],"several":[107,254],"classes":[108],"partial":[110],"differential":[111],"equations":[112],"arising":[113],"mechanics.":[115],"When":[116],"formalized":[117],"mathematically,":[118],"theory":[121,220],"corroborates":[122],"insights":[123],"economics:":[125],"control":[129,159,169,193],"problem":[130,209],"set":[131],"finite-time":[134],"horizon,":[135],"controls":[137],"and":[138,153,160,212,237,252,266],"corresponding":[139,162,207],"states":[140],"are":[141,175,248],"(often":[143],"exponentially)":[144],"except":[149],"near":[150],"initial":[152],"times,":[155],"state":[163],"associated":[166],"problem.":[170],"In":[171],"particular,":[172],"former":[174],"mostly":[176],"time.":[179],"This":[180],"fact":[181],"provides":[182],"rigorous":[184],"meaning":[185],"asymptotic":[188],"simplification":[189],"some":[191],"problems":[194],"appear":[195],"enjoy":[197],"time":[200],"intervals,":[201],"allowing":[202],"consideration":[204],"computing":[211],"applications.":[213],"We":[214],"review":[215],"slice":[217],"developed":[221],"decade":[225],"\u2013":[226,251],"controllability":[228],"underlying":[231],"system":[232],"important":[235],"ingredient,":[236],"can":[238],"even":[239],"used":[241],"devise":[243],"simple":[244],"turnpike-like":[245],"strategies":[246],"present":[253],"novel":[255],"applications,":[256],"including,":[257],"among":[258],"many":[259],"others,":[260],"characterization":[262],"Hamilton\u2013Jacobi\u2013Bellman":[264],"asymptotics,":[265],"stability":[267],"estimates":[268],"deep":[270],"learning":[271],"via":[272],"residual":[273],"neural":[274],"networks.":[275]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":9},{"year":2023,"cited_by_count":11},{"year":2022,"cited_by_count":2}],"updated_date":"2026-05-23T08:51:43.019350","created_date":"2025-10-10T00:00:00"}