{"id":"https://openalex.org/W4385235890","doi":"https://doi.org/10.1007/s10589-023-00507-x","title":"A space\u2013time variational method for optimal control problems: well-posedness, stability and numerical solution","display_name":"A space\u2013time variational method for optimal control problems: well-posedness, stability and numerical solution","publication_year":2023,"publication_date":"2023-07-25","ids":{"openalex":"https://openalex.org/W4385235890","doi":"https://doi.org/10.1007/s10589-023-00507-x"},"language":"en","primary_location":{"id":"doi:10.1007/s10589-023-00507-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-023-00507-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-023-00507-x.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10589-023-00507-x.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5049224850","display_name":"Nina Beranek","orcid":null},"institutions":[{"id":"https://openalex.org/I4210129763","display_name":"Helmholtz-Institute Ulm","ror":"https://ror.org/034rhsb33","country_code":"DE","type":"government","lineage":["https://openalex.org/I102335020","https://openalex.org/I1305996414","https://openalex.org/I1305996414","https://openalex.org/I4210129763"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Nina Beranek","raw_affiliation_strings":["Institute for Numerical Mathematics, Ulm University, Helmholtzstr. 20, 89081, Ulm, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Numerical Mathematics, Ulm University, Helmholtzstr. 20, 89081, Ulm, Germany","institution_ids":["https://openalex.org/I4210129763"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065003133","display_name":"M. Alexander Reinhold","orcid":"https://orcid.org/0000-0002-7842-2516"},"institutions":[{"id":"https://openalex.org/I4210129763","display_name":"Helmholtz-Institute Ulm","ror":"https://ror.org/034rhsb33","country_code":"DE","type":"government","lineage":["https://openalex.org/I102335020","https://openalex.org/I1305996414","https://openalex.org/I1305996414","https://openalex.org/I4210129763"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Martin Alexander Reinhold","raw_affiliation_strings":["Institute for Numerical Mathematics, Ulm University, Helmholtzstr. 20, 89081, Ulm, Germany"],"raw_orcid":"https://orcid.org/0000-0002-7842-2516","affiliations":[{"raw_affiliation_string":"Institute for Numerical Mathematics, Ulm University, Helmholtzstr. 20, 89081, Ulm, Germany","institution_ids":["https://openalex.org/I4210129763"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5008601082","display_name":"Karsten Urban","orcid":"https://orcid.org/0000-0002-6262-7955"},"institutions":[{"id":"https://openalex.org/I4210129763","display_name":"Helmholtz-Institute Ulm","ror":"https://ror.org/034rhsb33","country_code":"DE","type":"government","lineage":["https://openalex.org/I102335020","https://openalex.org/I1305996414","https://openalex.org/I1305996414","https://openalex.org/I4210129763"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Karsten Urban","raw_affiliation_strings":["Institute for Numerical Mathematics, Ulm University, Helmholtzstr. 20, 89081, Ulm, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Numerical Mathematics, Ulm University, Helmholtzstr. 20, 89081, Ulm, Germany","institution_ids":["https://openalex.org/I4210129763"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5049224850"],"corresponding_institution_ids":["https://openalex.org/I4210129763"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":1.4494,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.78288278,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"86","issue":"2","first_page":"767","last_page":"794"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.993399977684021,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9926999807357788,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8798768520355225},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.5736718773841858},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5662904977798462},{"id":"https://openalex.org/keywords/function-space","display_name":"Function space","score":0.5291980504989624},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5041071176528931},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.47714802622795105},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4394339323043823},{"id":"https://openalex.org/keywords/invertible-matrix","display_name":"Invertible matrix","score":0.410508394241333},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.33640217781066895},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.16318103671073914},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08481726050376892}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8798768520355225},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.5736718773841858},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5662904977798462},{"id":"https://openalex.org/C142730499","wikidata":"https://www.wikidata.org/wiki/Q934367","display_name":"Function space","level":2,"score":0.5291980504989624},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5041071176528931},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.47714802622795105},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4394339323043823},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.410508394241333},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.33640217781066895},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.16318103671073914},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08481726050376892},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s10589-023-00507-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-023-00507-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-023-00507-x.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:spr:coopap:v:86:y:2023:i:2:d:10.1007_s10589-023-00507-x","is_oa":false,"landing_page_url":"http://link.springer.com/10.1007/s10589-023-00507-x","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:econstor.eu:10419/311910","is_oa":true,"landing_page_url":"https://hdl.handle.net/10419/311910","pdf_url":null,"source":{"id":"https://openalex.org/S4306401696","display_name":"Econstor (Econstor)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"doc-type:article"}],"best_oa_location":{"id":"doi:10.1007/s10589-023-00507-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-023-00507-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-023-00507-x.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16","score":0.7400000095367432}],"awards":[],"funders":[{"id":"https://openalex.org/F4320325530","display_name":"Universit\u00e4t Ulm","ror":"https://ror.org/032000t02"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4385235890.pdf"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W596189563","https://openalex.org/W1580899028","https://openalex.org/W1976385614","https://openalex.org/W1981159109","https://openalex.org/W1993072133","https://openalex.org/W2006697861","https://openalex.org/W2008621392","https://openalex.org/W2012560715","https://openalex.org/W2021411519","https://openalex.org/W2034797962","https://openalex.org/W2038695002","https://openalex.org/W2047133076","https://openalex.org/W2060219309","https://openalex.org/W2083310254","https://openalex.org/W2093049475","https://openalex.org/W2133041109","https://openalex.org/W2265232287","https://openalex.org/W2402468387","https://openalex.org/W2725298667","https://openalex.org/W2753365189","https://openalex.org/W2775987586","https://openalex.org/W2964299872","https://openalex.org/W3118722523","https://openalex.org/W3135337288","https://openalex.org/W4224248463","https://openalex.org/W4238719544","https://openalex.org/W4298228534","https://openalex.org/W4301208692","https://openalex.org/W4302819252","https://openalex.org/W4321430205"],"related_works":["https://openalex.org/W4251428025","https://openalex.org/W2006251942","https://openalex.org/W2364741597","https://openalex.org/W4289867043","https://openalex.org/W1492103595","https://openalex.org/W2784467519","https://openalex.org/W1864774435","https://openalex.org/W946352265","https://openalex.org/W3020787026","https://openalex.org/W2732220824"],"abstract_inverted_index":{"Abstract":[0],"We":[1,18,146],"consider":[2],"an":[3],"optimal":[4,38],"control":[5,39],"problem":[6,40],"constrained":[7],"by":[8],"a":[9,20,28,49,94,133],"parabolic":[10,138],"partial":[11],"differential":[12],"equation":[13,63],"with":[14,141,150],"Robin":[15],"boundary":[16],"conditions.":[17],"use":[19],"space\u2013time":[21,55,99,154],"variational":[22,56],"formulation":[23,35],"in":[24,48,54,64,106,116,166],"Lebesgue\u2013Bochner":[25,65,108],"spaces":[26],"yielding":[27],"boundedly":[29,72,89],"invertible":[30],"solution":[31],"operator.":[32],"The":[33,153],"abstract":[34],"of":[36,58,102,113,164],"the":[37,42,59,82,103,170],"yields":[41],"Lagrange":[43],"function":[44],"and":[45,61,75,81,118,122,160],"Karush\u2013Kuhn\u2013Tucker":[46],"conditions":[47,78],"natural":[50],"manner.":[51],"This":[52],"results":[53],"formulations":[57],"adjoint":[60],"gradient":[62],"spaces,":[66,124],"which":[67],"are":[68,79,144],"proven":[69],"to":[70,87,129,132,168],"be":[71,88,130],"invertible.":[73,90],"Necessary":[74],"sufficient":[76],"optimality":[77,83,104],"formulated":[80],"system":[84,105],"is":[85,127],"shown":[86],"Next,":[91],"we":[92],"introduce":[93],"conforming":[95],"uniformly":[96],"stable":[97],"simultaneous":[98],"(tensorproduct)":[100],"discretization":[101],"these":[107],"spaces.":[109],"Using":[110],"finite":[111],"elements":[112],"appropriate":[114],"orders":[115],"space":[117],"time":[119,167],"for":[120,137],"trial":[121],"test":[123],"this":[125],"setting":[126],"known":[128],"equivalent":[131],"Crank\u2013Nicolson":[134],"time-stepping":[135,151],"scheme":[136],"problems.":[139],"Comparisons":[140],"existing":[142],"methods":[143],"detailed.":[145],"show":[147],"numerical":[148],"comparisons":[149],"methods.":[152],"method":[155],"shows":[156],"good":[157],"stability":[158],"properties":[159],"requires":[161],"fewer":[162],"degrees":[163],"freedom":[165],"reach":[169],"same":[171],"accuracy.":[172]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":2}],"updated_date":"2026-01-21T23:30:37.877113","created_date":"2025-10-10T00:00:00"}