{"id":"https://openalex.org/W2118291459","doi":"https://doi.org/10.1109/tac.2010.2047742","title":"Smooth Regularization of Bang-Bang Optimal Control Problems","display_name":"Smooth Regularization of Bang-Bang Optimal Control Problems","publication_year":2010,"publication_date":"2010-04-09","ids":{"openalex":"https://openalex.org/W2118291459","doi":"https://doi.org/10.1109/tac.2010.2047742","mag":"2118291459"},"language":"en","primary_location":{"id":"doi:10.1109/tac.2010.2047742","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tac.2010.2047742","pdf_url":null,"source":{"id":"https://openalex.org/S184954342","display_name":"IEEE Transactions on Automatic Control","issn_l":"0018-9286","issn":["0018-9286","1558-2523","2334-3303"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Automatic Control","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/A5038856616","display_name":"Cristiana J. Silva","orcid":"https://orcid.org/0000-0002-7238-546X"},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I4210131315","display_name":"Laboratoire de Math\u00e9matiques Analyse, Probabilit\u00e9s, Mod\u00e9lisation Orl\u00e9ans","ror":"https://ror.org/03d6rdf69","country_code":"FR","type":"facility","lineage":["https://openalex.org/I12449238","https://openalex.org/I1294671590","https://openalex.org/I4210131315","https://openalex.org/I4210141950"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Cristiana Silva","raw_affiliation_strings":["CNRS UMR 6628, Laboratoire MAPMO, France","CNRS UMR 6628, Laboratoire MAPMO, Orl\u00e9ans, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"CNRS UMR 6628, Laboratoire MAPMO, France","institution_ids":["https://openalex.org/I4210131315","https://openalex.org/I1294671590"]},{"raw_affiliation_string":"CNRS UMR 6628, Laboratoire MAPMO, Orl\u00e9ans, France","institution_ids":["https://openalex.org/I4210131315","https://openalex.org/I1294671590"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5089339704","display_name":"Emmanuel Tr\u00e9lat","orcid":"https://orcid.org/0000-0002-0656-7003"},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I4210131315","display_name":"Laboratoire de Math\u00e9matiques Analyse, Probabilit\u00e9s, Mod\u00e9lisation Orl\u00e9ans","ror":"https://ror.org/03d6rdf69","country_code":"FR","type":"facility","lineage":["https://openalex.org/I12449238","https://openalex.org/I1294671590","https://openalex.org/I4210131315","https://openalex.org/I4210141950"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Emmanuel Trelat","raw_affiliation_strings":["CNRS UMR 6628, Laboratoire MAPMO, France","CNRS UMR 6628, Laboratoire MAPMO, Orl\u00e9ans, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"CNRS UMR 6628, Laboratoire MAPMO, France","institution_ids":["https://openalex.org/I4210131315","https://openalex.org/I1294671590"]},{"raw_affiliation_string":"CNRS UMR 6628, Laboratoire MAPMO, Orl\u00e9ans, France","institution_ids":["https://openalex.org/I4210131315","https://openalex.org/I1294671590"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.779,"has_fulltext":false,"cited_by_count":86,"citation_normalized_percentile":{"value":0.93340728,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"55","issue":"11","first_page":"2488","last_page":"2499"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13855","display_name":"Aerospace Engineering and Control Systems","score":0.993399977684021,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace 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/T12809","display_name":"Contact Mechanics and Variational Inequalities","score":0.9828000068664551,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4399162828922272},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.40963631868362427}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4399162828922272},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.40963631868362427}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tac.2010.2047742","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tac.2010.2047742","pdf_url":null,"source":{"id":"https://openalex.org/S184954342","display_name":"IEEE Transactions on Automatic Control","issn_l":"0018-9286","issn":["0018-9286","1558-2523","2334-3303"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Automatic Control","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.406.4398","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.406.4398","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://hal.archives-ouvertes.fr/docs/00/45/82/77/PDF/reg.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":34,"referenced_works":["https://openalex.org/W28854931","https://openalex.org/W230866137","https://openalex.org/W613218979","https://openalex.org/W1494380031","https://openalex.org/W1513745936","https://openalex.org/W1519191737","https://openalex.org/W1527244075","https://openalex.org/W1558492509","https://openalex.org/W1571935615","https://openalex.org/W1587799944","https://openalex.org/W1595765050","https://openalex.org/W1864105480","https://openalex.org/W1895367271","https://openalex.org/W1965895153","https://openalex.org/W1971947347","https://openalex.org/W1991977833","https://openalex.org/W2009353114","https://openalex.org/W2010853770","https://openalex.org/W2046576092","https://openalex.org/W2068717022","https://openalex.org/W2080415776","https://openalex.org/W2082637521","https://openalex.org/W2088809818","https://openalex.org/W2090718664","https://openalex.org/W2092078242","https://openalex.org/W2121948997","https://openalex.org/W2125696921","https://openalex.org/W2334591577","https://openalex.org/W2571558868","https://openalex.org/W2949123337","https://openalex.org/W3206777520","https://openalex.org/W4213059792","https://openalex.org/W4302620424","https://openalex.org/W6634225431"],"related_works":["https://openalex.org/W2748952813","https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W2390279801","https://openalex.org/W2358668433","https://openalex.org/W2376932109","https://openalex.org/W2001405890","https://openalex.org/W2382290278","https://openalex.org/W2350741829","https://openalex.org/W2530322880"],"abstract_inverted_index":{"Consider":[0],"the":[1,31,40,68,71,78,87,98,103,143,164,203,206,210,213],"minimal":[2,99],"time":[3,100],"control":[4,33,59,79,104],"problem":[5,101,208],"for":[6,53,102],"a":[7,50,198],"single-input":[8],"control-affine":[9],"system":[10,105],"\u1e8b":[11,106],"=":[12,107,155],"X(x)":[13,108],"+":[14,109,122],"u":[15,34,110,130,146,173],"<sub":[16,20,35,43,111,118,124,131,138,147,167,174,181],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[17,21,27,36,44,112,115,119,125,128,132,135,139,148,151,168,171,175,178,182,185,189],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">1</sub>":[18,22,37,45,113,120],"Y":[19,117,137],"(x)":[23,121],"in":[24],"R":[25],"<sup":[26,114,127,134,150,170,177,184,188],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n</sup>":[28],",":[29,157],"where":[30,142],"scalar":[32,144],"(\u00b7)":[38],"satisfies":[39],"constraint":[41,165],"lu":[42],"(\u00b7)l":[46],"\u2264.":[47],"When":[48],"applying":[49],"shooting":[51,72],"method":[52],"solving":[54],"this":[55,83],"kind":[56],"of":[57,202,205,212],"optimal":[58],"problem,":[60],"one":[61],"may":[62],"encounter":[63],"numerical":[64],"problems":[65],"due":[66],"to":[67,209],"fact":[69],"that":[70],"function":[73],"is":[74,80],"not":[75],"smooth":[76],"whenever":[77],"bang-bang.":[81],"In":[82],"article":[84],"we":[85,96],"propose":[86],"following":[88],"smoothing":[89],"procedure.":[90],"For":[91],"\u03b5":[92],">":[93],"0":[94],"small,":[95],"consider":[97],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u03b5</sup>":[116,136,152,179,186],"\u03b5\u03a3":[123],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">i=2</sub>":[126],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">m</sup>":[129,172],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">i</sub>":[133,140,149,176,183],"(x),":[141],"controls":[145],"(\u00b7),":[153],"i":[154],"1,...":[156],"m,":[158],"with":[159],"m":[160],"\u2265":[161],"2,":[162],"satisfy":[163],"\u03a3":[166],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">i=1</sub>":[169],"(u":[180],"(t))":[187],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sup>":[190],"\u2264":[191],"1.":[192],"We":[193],"prove,":[194],"under":[195],"appropriate":[196],"assumptions,":[197],"strong":[199],"convergence":[200],"result":[201],"solution":[204,211],"regularized":[207],"initial":[214],"problem.":[215]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":8},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":9},{"year":2019,"cited_by_count":8},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":11},{"year":2016,"cited_by_count":7},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}