{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:03:07Z","timestamp":1760234587967,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,21]],"date-time":"2021-05-21T00:00:00Z","timestamp":1621555200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61625304; 62073209; 61991415"],"award-info":[{"award-number":["61625304; 62073209; 61991415"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2020KA005"],"award-info":[{"award-number":["ZR2020KA005"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the leader-following regional multiple-bipartite consensus problems of networked Lagrangian systems (NLSs) in coopetition networks. Our framework expands the application scopes of traditional regional consensus in cooperative networks. With the aid of a novel auxiliary variable embedded in the control protocols, the final states of NLSs are guaranteed to realise multi-regional symmetry in the constructed multiple symmetric regions. By utilising the characteristic of acyclic topology in the structurally balanced graph, the stability of the closed system is performed by perturbation analysis theory, nonlinear control theory, functional analysis theory, and so on. Finally, the effectiveness of our approach is verified by numerical simulations.<\/jats:p>","DOI":"10.3390\/sym13060920","type":"journal-article","created":{"date-parts":[[2021,5,24]],"date-time":"2021-05-24T02:32:43Z","timestamp":1621823563000},"page":"920","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Leader-Following Regional Multiple-Bipartite Consensus for Networked Lagrangian Systems with Coopetition Interactions"],"prefix":"10.3390","volume":"13","author":[{"given":"Tiehui","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China"},{"name":"College of Elementary Education, Jining University, Jining 273155, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2243-5908","authenticated-orcid":false,"given":"Hengyu","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhaoyan","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Jining University, Jining 273155, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shaorong","family":"Xie","sequence":"additional","affiliation":[{"name":"School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1109\/TRO.2011.2168690","article-title":"Controlled synchronization of heterogeneous robotic manipulators in the task space","volume":"23","author":"Liu","year":"2012","journal-title":"IEEE Trans. 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