{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T22:23:41Z","timestamp":1775082221893,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T00:00:00Z","timestamp":1761609600000},"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":"crossref","award":["62201102"],"award-info":[{"award-number":["62201102"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["62003065"],"award-info":[{"award-number":["62003065"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Basic and Frontier Research Project of Chongqing","award":["CSTB2024NSCQ-MSX0008"],"award-info":[{"award-number":["CSTB2024NSCQ-MSX0008"]}]},{"name":"Project of Science and Technology Research Program of Chongqing Education Commission of China","award":["KJZD-K202400801"],"award-info":[{"award-number":["KJZD-K202400801"]}]},{"name":"Project of Science and Technology Research Program of Chongqing Education Commission of China","award":["KJQN202200822"],"award-info":[{"award-number":["KJQN202200822"]}]},{"name":"Scientific Research Project of Chongqing Technology and Business University in China","award":["1956013"],"award-info":[{"award-number":["1956013"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study addresses the problem of fast fixed-time synchronization (FDTS) in multi-layer networks (MLNs), focusing on both intra-layer and inter-layer dynamic synchronization. First, we propose a more representative MLN framework in which each layer has a distinct topological structure, and node-to-node connections are established between corresponding nodes across layers. Second, we design a fast fixed-time controller to achieve synchronization of MLNs and estimate the synchronization time. Third, by applying the Lyapunov method and comparison theorem, we derive several novel synchronization conditions that are independent of initial states. Finally, numerical simulations are conducted to validate the theoretical findings.<\/jats:p>","DOI":"10.3390\/axioms14110795","type":"journal-article","created":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T03:44:39Z","timestamp":1761795879000},"page":"795","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Fast Fixed-Time Control for Intra- and Inter-Layer Synchronization of Multi-Layer Networks"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7002-8936","authenticated-orcid":false,"given":"Shiju","family":"Yang","sequence":"first","affiliation":[{"name":"School of Artificial Intelligence, Chongqing Technology and Business University, Chongqing 400067, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-7492-0117","authenticated-orcid":false,"given":"Hongsen","family":"He","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Chongqing Technology and Business University, Chongqing 400067, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7753-0136","authenticated-orcid":false,"given":"Man-Fai","family":"Leung","sequence":"additional","affiliation":[{"name":"School of Computing and Information Science, Faculty of Science and Engineering, Anglia Ruskin University, Cambridge CB1 1PT, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiting","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Chongqing Technology and Business University, Chongqing 400067, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Chongqing Technology and Business University, Chongqing 400067, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2809","DOI":"10.1109\/TFUZZ.2022.3229048","article-title":"Global exponential synchronization of delayed fuzzy neural networks with reaction diffusions","volume":"31","author":"Sheng","year":"2023","journal-title":"IEEE Trans. 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