{"id":"https://openalex.org/W3082163659","doi":"https://doi.org/10.1007/s10957-020-01727-5","title":"Convex Quadratic Equation","display_name":"Convex Quadratic Equation","publication_year":2020,"publication_date":"2020-08-27","ids":{"openalex":"https://openalex.org/W3082163659","doi":"https://doi.org/10.1007/s10957-020-01727-5","mag":"3082163659"},"language":"en","primary_location":{"id":"doi:10.1007/s10957-020-01727-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-020-01727-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-020-01727-5.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"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":"Journal of Optimization Theory 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/s10957-020-01727-5.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089628474","display_name":"Li-Gang Lin","orcid":"https://orcid.org/0000-0001-5732-9964"},"institutions":[{"id":"https://openalex.org/I22265921","display_name":"National Central University","ror":"https://ror.org/00944ve71","country_code":"TW","type":"education","lineage":["https://openalex.org/I22265921"]}],"countries":["TW"],"is_corresponding":true,"raw_author_name":"Li-Gang Lin","raw_affiliation_strings":["Department of Mechanical Engineering, National Central University, Taoyuan, Taiwan"],"raw_orcid":"https://orcid.org/0000-0001-5732-9964","affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, National Central University, Taoyuan, Taiwan","institution_ids":["https://openalex.org/I22265921"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5113498632","display_name":"Yew-Wen Liang","orcid":null},"institutions":[{"id":"https://openalex.org/I148366613","display_name":"National Yang Ming Chiao Tung University","ror":"https://ror.org/00se2k293","country_code":"TW","type":"education","lineage":["https://openalex.org/I148366613"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Yew-Wen Liang","raw_affiliation_strings":["Institute of Electrical Control Engineering, National Chiao Tung University, Hsinchu, Taiwan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Electrical Control Engineering, National Chiao Tung University, Hsinchu, Taiwan","institution_ids":["https://openalex.org/I148366613"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5108369231","display_name":"Wen-Yuan Hsieh","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wen-Yuan Hsieh","raw_affiliation_strings":["Sonix Technology Co., Ltd, Zhubei, Taiwan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Sonix Technology Co., Ltd, Zhubei, Taiwan","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5089628474"],"corresponding_institution_ids":["https://openalex.org/I22265921"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.2977,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.56018764,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"186","issue":"3","first_page":"1006","last_page":"1028"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12772","display_name":"Control and Stability of Dynamical Systems","score":0.996999979019165,"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"}},"topics":[{"id":"https://openalex.org/T12772","display_name":"Control and Stability of Dynamical Systems","score":0.996999979019165,"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/T10040","display_name":"Adaptive Control of Nonlinear Systems","score":0.9955000281333923,"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/T10791","display_name":"Advanced Control Systems Optimization","score":0.9954000115394592,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8560689687728882},{"id":"https://openalex.org/keywords/riccati-equation","display_name":"Riccati equation","score":0.6050577163696289},{"id":"https://openalex.org/keywords/hamilton\u2013jacobi-equation","display_name":"Hamilton\u2013Jacobi equation","score":0.5609186291694641},{"id":"https://openalex.org/keywords/bellman-equation","display_name":"Bellman equation","score":0.4870074391365051},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4793061316013336},{"id":"https://openalex.org/keywords/hessian-matrix","display_name":"Hessian matrix","score":0.4374557137489319},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.43604177236557007},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.4254763126373291},{"id":"https://openalex.org/keywords/algebraic-equation","display_name":"Algebraic equation","score":0.4190451502799988},{"id":"https://openalex.org/keywords/hamilton\u2013jacobi\u2013bellman-equation","display_name":"Hamilton\u2013Jacobi\u2013Bellman equation","score":0.41247329115867615},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.35868996381759644},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.33418020606040955},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.2953339219093323},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.21625763177871704},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.15078768134117126}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8560689687728882},{"id":"https://openalex.org/C45473103","wikidata":"https://www.wikidata.org/wiki/Q851503","display_name":"Riccati equation","level":3,"score":0.6050577163696289},{"id":"https://openalex.org/C2778860007","wikidata":"https://www.wikidata.org/wiki/Q1060137","display_name":"Hamilton\u2013Jacobi equation","level":2,"score":0.5609186291694641},{"id":"https://openalex.org/C14646407","wikidata":"https://www.wikidata.org/wiki/Q1430750","display_name":"Bellman equation","level":2,"score":0.4870074391365051},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4793061316013336},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.4374557137489319},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.43604177236557007},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.4254763126373291},{"id":"https://openalex.org/C23917780","wikidata":"https://www.wikidata.org/wiki/Q50698","display_name":"Algebraic equation","level":3,"score":0.4190451502799988},{"id":"https://openalex.org/C196978813","wikidata":"https://www.wikidata.org/wiki/Q3302775","display_name":"Hamilton\u2013Jacobi\u2013Bellman equation","level":3,"score":0.41247329115867615},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.35868996381759644},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.33418020606040955},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.2953339219093323},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.21625763177871704},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.15078768134117126},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s10957-020-01727-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-020-01727-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-020-01727-5.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"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":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:spr:joptap:v:186:y:2020:i:3:d:10.1007_s10957-020-01727-5","is_oa":true,"landing_page_url":"http://link.springer.com/10.1007/s10957-020-01727-5","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.1007/s10957-020-01727-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-020-01727-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-020-01727-5.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"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":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322434","display_name":"Max-Planck-Gesellschaft","ror":"https://ror.org/01hhn8329"},{"id":"https://openalex.org/F4320322795","display_name":"Ministry of Science and Technology, Taiwan","ror":"https://ror.org/02kv4zf79"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3082163659.pdf","grobid_xml":"https://content.openalex.org/works/W3082163659.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W1494474111","https://openalex.org/W1547930079","https://openalex.org/W1981745143","https://openalex.org/W1993170675","https://openalex.org/W2002663556","https://openalex.org/W2041106619","https://openalex.org/W2047090868","https://openalex.org/W2047397174","https://openalex.org/W2056583404","https://openalex.org/W2076312345","https://openalex.org/W2076896389","https://openalex.org/W2097002402","https://openalex.org/W2112384869","https://openalex.org/W2117175541","https://openalex.org/W2130238423","https://openalex.org/W2139849628","https://openalex.org/W2284006517","https://openalex.org/W2296319761","https://openalex.org/W2475591765","https://openalex.org/W2610857016","https://openalex.org/W2781741829","https://openalex.org/W2798909945","https://openalex.org/W2904441903","https://openalex.org/W2914764895","https://openalex.org/W2983212311","https://openalex.org/W2989986341","https://openalex.org/W3039025989","https://openalex.org/W3121855996","https://openalex.org/W4232591629","https://openalex.org/W4249667877","https://openalex.org/W4250589301","https://openalex.org/W4292230442","https://openalex.org/W6600753396","https://openalex.org/W6695674720"],"related_works":["https://openalex.org/W2032834442","https://openalex.org/W3153900688","https://openalex.org/W2115544975","https://openalex.org/W2902017027","https://openalex.org/W1654696814","https://openalex.org/W4291598237","https://openalex.org/W2766998270","https://openalex.org/W4255265352","https://openalex.org/W4226406511","https://openalex.org/W1996904445"],"abstract_inverted_index":{"Abstract":[0],"Two":[1],"main":[2,92],"results":[3],"(A)":[4,13],"and":[5,24,38,101,123],"(B)":[6],"are":[7,29,126],"presented":[8],"in":[9,35,39,150],"algebraic":[10],"closed":[11],"forms.":[12],"Regarding":[14],"the":[15,32,36,48,61,72,78,90,99,111,118,131,135,138,143,161,164,174],"convex":[16,113],"quadratic":[17,114],"equation,":[18,120],"an":[19],"analytical":[20],"equivalent":[21],"solvability":[22],"condition":[23],"parameterization":[25],"of":[26,66,109,137,142],"all":[27],"solutions":[28],"formulated,":[30],"for":[31,89,134,173],"first":[33],"time":[34],"literature":[37],"a":[40,54,82,156],"unified":[41],"framework.":[42],"The":[43],"philosophy":[44],"is":[45,75,85,171],"based":[46],"on":[47],"matrix":[49],"algebra,":[50],"while":[51],"facilitated":[52],"by":[53],"novel":[55],"equivalence/coordinate":[56],"transformation":[57],"(with":[58],"respect":[59],"to":[60,158],"much":[62],"more":[63],"challenging":[64],"case":[65],"rank-deficient":[67],"Hessian":[68],"matrix).":[69],"In":[70],"addition,":[71],"parameter-solution":[73],"bijection":[74],"verified.":[76],"From":[77],"perspective":[79],"via":[80],"(A),":[81,110],"major":[83],"application":[84],"re-examined":[86],"that":[87],"accounts":[88],"other":[91],"result":[93],"(B),":[94],"which":[95],"deals":[96],"with":[97,117],"both":[98],"infinite":[100],"finite-time":[102],"horizon":[103],"nonlinear":[104],"optimal":[105,139],"control.":[106],"By":[107],"virtue":[108],"underlying":[112],"equations":[115],"associated":[116,144],"Hamilton\u2013Jacobi":[119,121],"inequality,":[122],"Hamilton\u2013Jacobi\u2013Bellman":[124],"equation":[125,169],"explicitly":[127],"solved,":[128],"respectively.":[129],"Therefore,":[130],"long":[132],"quest":[133],"constituent":[136],"controller,":[140],"gradient":[141],"value":[145],"function,":[146],"can":[147],"be":[148],"captured":[149],"each":[151],"solution":[152],"set.":[153],"Moving":[154],"forward,":[155],"preliminary":[157],"exactly":[159],"locate":[160],"optimality":[162],"using":[163],"state-dependent":[165],"(resp.,":[166],"differential)":[167],"Riccati":[168],"scheme":[170],"prepared":[172],"remaining":[175],"symmetry":[176],"condition.":[177]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2022,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
