{"id":"https://openalex.org/W4206093983","doi":"https://doi.org/10.3233/jifs-210583","title":"A numerical method for approximating the solution of fuzzy fractional optimal control problems in caputo sense using legendre functions","display_name":"A numerical method for approximating the solution of fuzzy fractional optimal control problems in caputo sense using legendre functions","publication_year":2022,"publication_date":"2022-01-21","ids":{"openalex":"https://openalex.org/W4206093983","doi":"https://doi.org/10.3233/jifs-210583"},"language":"en","primary_location":{"id":"doi:10.3233/jifs-210583","is_oa":false,"landing_page_url":"https://doi.org/10.3233/jifs-210583","pdf_url":null,"source":{"id":"https://openalex.org/S179157397","display_name":"Journal of Intelligent & Fuzzy Systems","issn_l":"1064-1246","issn":["1064-1246","1875-8967"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310318577","host_organization_name":"IOS Press","host_organization_lineage":["https://openalex.org/P4310318577"],"host_organization_lineage_names":["IOS Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Intelligent &amp; Fuzzy Systems","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/A5078216661","display_name":"M. Mirvakili","orcid":null},"institutions":[{"id":"https://openalex.org/I55547365","display_name":"Payame Noor University","ror":"https://ror.org/031699d98","country_code":"IR","type":"education","lineage":["https://openalex.org/I55547365"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"M. Mirvakili","raw_affiliation_strings":["Department of Mathematics, Payame Noor University (PNU), Tehran, Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Payame Noor University (PNU), Tehran, Iran","institution_ids":["https://openalex.org/I55547365"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073095019","display_name":"Tofigh Allahviranloo","orcid":"https://orcid.org/0000-0002-6673-3560"},"institutions":[{"id":"https://openalex.org/I4210121333","display_name":"Istinye University","ror":"https://ror.org/03081nz23","country_code":"TR","type":"education","lineage":["https://openalex.org/I4210121333"]}],"countries":["TR"],"is_corresponding":true,"raw_author_name":"T. Allahviranloo","raw_affiliation_strings":["Faculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey","institution_ids":["https://openalex.org/I4210121333"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5062444216","display_name":"Fahimeh Soltanian","orcid":"https://orcid.org/0000-0003-3068-3378"},"institutions":[{"id":"https://openalex.org/I55547365","display_name":"Payame Noor University","ror":"https://ror.org/031699d98","country_code":"IR","type":"education","lineage":["https://openalex.org/I55547365"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"F. Soltanian","raw_affiliation_strings":["Department of Mathematics, Payame Noor University (PNU), Tehran, Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Payame Noor University (PNU), Tehran, Iran","institution_ids":["https://openalex.org/I55547365"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5073095019"],"corresponding_institution_ids":["https://openalex.org/I4210121333"],"apc_list":null,"apc_paid":null,"fwci":0.406,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.51370415,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":96},"biblio":{"volume":"43","issue":"3","first_page":"3827","last_page":"3858"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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":0.8289757370948792},{"id":"https://openalex.org/keywords/legendre-polynomials","display_name":"Legendre polynomials","score":0.5549238920211792},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5236789584159851},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.504104495048523},{"id":"https://openalex.org/keywords/legendre-wavelet","display_name":"Legendre wavelet","score":0.4875878691673279},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.4799914062023163},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.4434337913990021},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.43643414974212646},{"id":"https://openalex.org/keywords/lagrange-multiplier","display_name":"Lagrange multiplier","score":0.42706581950187683},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.4212125837802887},{"id":"https://openalex.org/keywords/state-variable","display_name":"State variable","score":0.41271206736564636},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.322613924741745},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3187909722328186},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10895407199859619}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8289757370948792},{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.5549238920211792},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5236789584159851},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.504104495048523},{"id":"https://openalex.org/C123769847","wikidata":"https://www.wikidata.org/wiki/Q6517888","display_name":"Legendre wavelet","level":5,"score":0.4875878691673279},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.4799914062023163},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.4434337913990021},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.43643414974212646},{"id":"https://openalex.org/C73684929","wikidata":"https://www.wikidata.org/wiki/Q598870","display_name":"Lagrange multiplier","level":2,"score":0.42706581950187683},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.4212125837802887},{"id":"https://openalex.org/C129537906","wikidata":"https://www.wikidata.org/wiki/Q7603913","display_name":"State variable","level":2,"score":0.41271206736564636},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.322613924741745},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3187909722328186},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10895407199859619},{"id":"https://openalex.org/C196216189","wikidata":"https://www.wikidata.org/wiki/Q2867","display_name":"Wavelet transform","level":3,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","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},{"id":"https://openalex.org/C46286280","wikidata":"https://www.wikidata.org/wiki/Q2414958","display_name":"Discrete wavelet transform","level":4,"score":0.0},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3233/jifs-210583","is_oa":false,"landing_page_url":"https://doi.org/10.3233/jifs-210583","pdf_url":null,"source":{"id":"https://openalex.org/S179157397","display_name":"Journal of Intelligent & Fuzzy Systems","issn_l":"1064-1246","issn":["1064-1246","1875-8967"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310318577","host_organization_name":"IOS Press","host_organization_lineage":["https://openalex.org/P4310318577"],"host_organization_lineage_names":["IOS Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Intelligent &amp; Fuzzy Systems","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W1606033442","https://openalex.org/W1850511188","https://openalex.org/W1969495063","https://openalex.org/W1995487946","https://openalex.org/W2008152104","https://openalex.org/W2009342598","https://openalex.org/W2011301580","https://openalex.org/W2056258420","https://openalex.org/W2062163287","https://openalex.org/W2065268523","https://openalex.org/W2082062104","https://openalex.org/W2087187650","https://openalex.org/W2122804283","https://openalex.org/W2156367245","https://openalex.org/W2587344048","https://openalex.org/W2753480125","https://openalex.org/W2756572656","https://openalex.org/W3018099102","https://openalex.org/W3024206168","https://openalex.org/W3048692941","https://openalex.org/W4247953052","https://openalex.org/W4256340197"],"related_works":["https://openalex.org/W4376278615","https://openalex.org/W3212740416","https://openalex.org/W4224216550","https://openalex.org/W1968819460","https://openalex.org/W3007167322","https://openalex.org/W3201136098","https://openalex.org/W1993087783","https://openalex.org/W2008599992","https://openalex.org/W2592565440","https://openalex.org/W4206093983"],"abstract_inverted_index":{"Fractional":[0,59,109],"order":[1],"differential":[2],"equations":[3,116,172],"accurately":[4],"model":[5],"dynamic":[6,100],"systems":[7],"and":[8,27,94,98,119,122,151,169,177,196,210,220],"processes.":[9],"In":[10,42],"some":[11],"of":[12,30,57,82,90,107,128,133,202,208],"the":[13,21,24,28,33,52,65,76,92,95,99,120,126,131,140,148,156,161,171,184,194,206,214,221,229],"fractional":[14,71],"optimal":[15],"control":[16,96,197,211,226],"problems":[17],"(FOCPs),":[18],"due":[19],"to":[20,32,38,50,155,167,182],"ambiguity":[22,31],"in":[23,75,144,173],"initial":[25],"conditions":[26,124],"transfer":[29],"solution,":[34],"it":[35],"is":[36,48,73,85,142,223],"necessary":[37,121],"use":[39],"fuzzy":[40,70],"mathematics.":[41],"this":[43],"paper,":[44],"a":[45,55,88,105,174,190],"numerical":[46],"method":[47,164],"presented":[49,185],"approximate":[51],"solution":[53],"for":[54,117,125,193,199,213,225,228],"class":[56],"Fuzzy":[58,108],"Optimal":[60],"Control":[61],"Problems":[62],"(FFOCPs)":[63],"using":[64,130],"Legendre":[66,203],"basis":[67],"functions.":[68],"The":[69,79],"derivative":[72],"described":[74],"Caputo":[77],"sense.":[78],"performance":[80],"index":[81],"an":[83,152],"FFOCP":[84],"considered":[86,143],"as":[87],"function":[89,150],"both":[91],"state":[93,195,209],"variables,":[97],"constraints":[101],"are":[102,165,180],"expressed":[103],"by":[104],"set":[106],"Differential":[110],"Equations":[111],"(FFDEs).":[112],"After":[113],"obtaining":[114],"Euler\u2013Lagrange":[115],"FFOCPs":[118],"sufficient":[123],"existence":[127],"solutions,":[129],"definition":[132],"generalized":[134],"Hukuhara":[135],"differentiability":[136],"(types":[137],"I,":[138],"II),":[139],"problem":[141],"two":[145],"cases.":[146],"Then":[147],"distance":[149],"approach":[153],"similar":[154,191],"variational":[157],"type":[158],"along":[159],"with":[160],"Lagrange":[162],"multiplier":[163],"used":[166],"formulate":[168],"solve":[170],"system.":[175,231],"Time-invariant":[176],"time-varying":[178,230],"examples":[179],"provided":[181],"assess":[183],"method.":[186],"Numerical":[187],"results":[188],"show":[189],"trend":[192],"variables":[198,212,227],"various":[200],"numbers":[201],"polynomials.":[204],"Also,":[205],"convergence":[207],"time-invariant":[215],"system":[216],"can":[217],"be":[218],"seen,":[219],"same":[222],"true":[224]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
