{"id":"https://openalex.org/W2001951765","doi":"https://doi.org/10.1155/2013/356746","title":"Stability of Nonlinear Stochastic Discrete-Time Systems","display_name":"Stability of Nonlinear Stochastic Discrete-Time Systems","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W2001951765","doi":"https://doi.org/10.1155/2013/356746","mag":"2001951765"},"language":"en","primary_location":{"id":"doi:10.1155/2013/356746","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/356746","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/356746.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"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 Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/jam/2013/356746.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5050422997","display_name":"Yan Li","orcid":"https://orcid.org/0000-0001-7491-759X"},"institutions":[{"id":"https://openalex.org/I80143920","display_name":"Shandong University of Science and Technology","ror":"https://ror.org/04gtjhw98","country_code":"CN","type":"education","lineage":["https://openalex.org/I80143920"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yan Li","raw_affiliation_strings":["College of Information Science & Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266590, China","College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao Shandong 266590, China"],"affiliations":[{"raw_affiliation_string":"College of Information Science & Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266590, China","institution_ids":["https://openalex.org/I80143920"]},{"raw_affiliation_string":"College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao Shandong 266590, China","institution_ids":["https://openalex.org/I80143920"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020929135","display_name":"Weihai Zhang","orcid":"https://orcid.org/0000-0001-7470-077X"},"institutions":[{"id":"https://openalex.org/I80143920","display_name":"Shandong University of Science and Technology","ror":"https://ror.org/04gtjhw98","country_code":"CN","type":"education","lineage":["https://openalex.org/I80143920"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Weihai Zhang","raw_affiliation_strings":["College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266590, China"],"affiliations":[{"raw_affiliation_string":"College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266590, China","institution_ids":["https://openalex.org/I80143920"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101488371","display_name":"Xikui Liu","orcid":"https://orcid.org/0000-0002-4509-9468"},"institutions":[{"id":"https://openalex.org/I80143920","display_name":"Shandong University of Science and Technology","ror":"https://ror.org/04gtjhw98","country_code":"CN","type":"education","lineage":["https://openalex.org/I80143920"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xikui Liu","raw_affiliation_strings":["College of Information Science & Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266590, China","College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao Shandong 266590, China"],"affiliations":[{"raw_affiliation_string":"College of Information Science & Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266590, China","institution_ids":["https://openalex.org/I80143920"]},{"raw_affiliation_string":"College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao Shandong 266590, China","institution_ids":["https://openalex.org/I80143920"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5050422997"],"corresponding_institution_ids":["https://openalex.org/I80143920"],"apc_list":{"value":1025,"currency":"USD","value_usd":1025},"apc_paid":{"value":1025,"currency":"USD","value_usd":1025},"fwci":1.5258,"has_fulltext":true,"cited_by_count":33,"citation_normalized_percentile":{"value":0.84547255,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"2013","issue":null,"first_page":"1","last_page":"8"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10046","display_name":"Stability and Control of Uncertain Systems","score":0.9994999766349792,"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/T10046","display_name":"Stability and Control of Uncertain Systems","score":0.9994999766349792,"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":0.9930999875068665,"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/T12772","display_name":"Control and Stability of Dynamical Systems","score":0.9886000156402588,"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/stability","display_name":"Stability (learning theory)","score":0.7491920590400696},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.678695559501648},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6614890694618225},{"id":"https://openalex.org/keywords/moment","display_name":"Moment (physics)","score":0.6576954126358032},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5176018476486206},{"id":"https://openalex.org/keywords/discrete-time-and-continuous-time","display_name":"Discrete time and continuous time","score":0.5137211084365845},{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.480642706155777},{"id":"https://openalex.org/keywords/exponential-stability","display_name":"Exponential stability","score":0.4287189245223999},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.4136106073856354},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.22312122583389282},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.07044380903244019},{"id":"https://openalex.org/keywords/control","display_name":"Control (management)","score":0.07038703560829163},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.03761774301528931}],"concepts":[{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.7491920590400696},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.678695559501648},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6614890694618225},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.6576954126358032},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5176018476486206},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.5137211084365845},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.480642706155777},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.4287189245223999},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.4136106073856354},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.22312122583389282},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.07044380903244019},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.07038703560829163},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.03761774301528931},{"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/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1155/2013/356746","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/356746","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/356746.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"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 Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jam/1394806128","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jam/1394806128","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:RePEc:hin:jnljam:356746","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/JAM/2013/356746.xml","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:doaj.org/article:329b5944e823435e8558a1466b9d790d","is_oa":true,"landing_page_url":"https://doaj.org/article/329b5944e823435e8558a1466b9d790d","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Applied Mathematics, Vol 2013 (2013)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2013/356746","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/356746","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/356746.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"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 Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1121271761","display_name":null,"funder_award_id":"Program","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G1380026940","display_name":null,"funder_award_id":"2011KYTD105","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G1467263865","display_name":null,"funder_award_id":"61174078","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2493111410","display_name":null,"funder_award_id":"61170054","funder_id":"https://openalex.org/F4320327677","funder_display_name":"State Key Laboratory Of Alternate Electrical Power System With Renewable Energy Sources"},{"id":"https://openalex.org/G2584917244","display_name":null,"funder_award_id":"20103718110006","funder_id":"https://openalex.org/F4320327677","funder_display_name":"State Key Laboratory Of Alternate Electrical Power System With Renewable Energy Sources"},{"id":"https://openalex.org/G2978943334","display_name":null,"funder_award_id":"117005","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G391238517","display_name":null,"funder_award_id":", and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4020255992","display_name":null,"funder_award_id":"Project","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4135714025","display_name":null,"funder_award_id":"LAPS13018","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4146523950","display_name":"Collaborative Proposal: Central States Mathematics Undergraduate Research Conferences","funder_award_id":"1811000","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4707332837","display_name":null,"funder_award_id":"61170054","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5644764193","display_name":null,"funder_award_id":"17005","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6017585462","display_name":null,"funder_award_id":"LAPS13018","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G633793928","display_name":null,"funder_award_id":"61174078","funder_id":"https://openalex.org/F4320327677","funder_display_name":"State Key Laboratory Of Alternate Electrical Power System With Renewable Energy Sources"},{"id":"https://openalex.org/G7033253288","display_name":null,"funder_award_id":"Grants","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7116322729","display_name":null,"funder_award_id":"20103718110006","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7270933721","display_name":null,"funder_award_id":"2011KYTD105","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7273148515","display_name":null,"funder_award_id":"61170054","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8297818335","display_name":null,"funder_award_id":"LAPS13018","funder_id":"https://openalex.org/F4320327677","funder_display_name":"State Key Laboratory Of Alternate Electrical Power System With Renewable Energy Sources"},{"id":"https://openalex.org/G8689763037","display_name":null,"funder_award_id":"2011KYTD105","funder_id":"https://openalex.org/F4320327677","funder_display_name":"State Key Laboratory Of Alternate Electrical Power System With Renewable Energy Sources"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322522","display_name":"Shandong University of Science and Technology","ror":"https://ror.org/04gtjhw98"},{"id":"https://openalex.org/F4320327677","display_name":"State Key Laboratory Of Alternate Electrical Power System With Renewable Energy Sources","ror":null},{"id":"https://openalex.org/F4320336024","display_name":"Specialized Research Fund for the Doctoral Program of Higher Education of China","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2001951765.pdf","grobid_xml":"https://content.openalex.org/works/W2001951765.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W56139714","https://openalex.org/W655163825","https://openalex.org/W1275718198","https://openalex.org/W1489872560","https://openalex.org/W1494628843","https://openalex.org/W1495711302","https://openalex.org/W1505057802","https://openalex.org/W1593496074","https://openalex.org/W1995502176","https://openalex.org/W2024033376","https://openalex.org/W2032693149","https://openalex.org/W2035734710","https://openalex.org/W2038413096","https://openalex.org/W2056822163","https://openalex.org/W2082256594","https://openalex.org/W2799664482"],"related_works":["https://openalex.org/W1593898832","https://openalex.org/W2142048021","https://openalex.org/W2756324413","https://openalex.org/W2164703465","https://openalex.org/W2168383058","https://openalex.org/W2143247663","https://openalex.org/W1965992231","https://openalex.org/W2050693590","https://openalex.org/W1990502710","https://openalex.org/W2159479095"],"abstract_inverted_index":{"This":[0],"paper":[1],"studies":[2],"the":[3,32,35,52,55],"stability":[4,16,43],"for":[5,41],"nonlinear":[6],"stochastic":[7,42],"discrete-time":[8],"systems.":[9],"First":[10],"of":[11,34,54],"all,":[12],"several":[13],"definitions":[14],"on":[15],"are":[17,44,48],"introduced,":[18],"such":[19],"as":[20],"stability,":[21,23],"asymptotical":[22],"and":[24],"p":[25],"th":[26],"moment":[27],"exponential":[28],"stability.":[29],"Moreover,":[30],"using":[31],"method":[33],"Lyapunov":[36],"functionals,":[37],"some":[38],"efficient":[39],"criteria":[40],"obtained.":[45],"Some":[46],"examples":[47],"presented":[49],"to":[50],"illustrate":[51],"effectiveness":[53],"proposed":[56],"theoretical":[57],"results.":[58]},"counts_by_year":[{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":7},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":7},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}