{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T18:46:03Z","timestamp":1785177963897,"version":"3.55.0"},"reference-count":22,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,10]],"date-time":"2022-10-10T00:00:00Z","timestamp":1665360000000},"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":["61872204"],"award-info":[{"award-number":["61872204"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["145109143"],"award-info":[{"award-number":["145109143"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Scientific Research Project of Heilongjiang Provincial Universities","award":["61872204"],"award-info":[{"award-number":["61872204"]}]},{"name":"Scientific Research Project of Heilongjiang Provincial Universities","award":["145109143"],"award-info":[{"award-number":["145109143"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, the distributed optimal control method is used to study the cooperative formation of heterogeneous multi-agents in the air\u2013ground environment. The considered system consists of an unmanned aerial vehicle (UAV) and an unmanned ground vehicle (UGV). The optimal control theory is introduced into the formation control protocol, the distributed optimal formation control protocol is designed, and the stability is verified by graph theory. Furthermore, the cooperative optimal formation control protocol is designed, and the stability is analyzed using a block Kronecker product and matrix transformation theory. Through the comparison of simulation results, the introduction of optimal control theory shortens the formation time of the system and accelerates the convergence speed of the system.<\/jats:p>","DOI":"10.3390\/e24101440","type":"journal-article","created":{"date-parts":[[2022,10,10]],"date-time":"2022-10-10T21:07:11Z","timestamp":1665436031000},"page":"1440","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Collaborative Optimal Formation Control for Heterogeneous Multi-Agent Systems"],"prefix":"10.3390","volume":"24","author":[{"given":"Yandong","family":"Li","sequence":"first","affiliation":[{"name":"College of Computer and Control Engineering, Qiqihar University, Qiqihar 161000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Meichen","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Computer and Control Engineering, Qiqihar University, Qiqihar 161000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiya","family":"Lian","sequence":"additional","affiliation":[{"name":"College of Computer and Control Engineering, Qiqihar University, Qiqihar 161000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuan","family":"Guo","sequence":"additional","affiliation":[{"name":"College of Computer and Control Engineering, Qiqihar University, Qiqihar 161000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,10]]},"reference":[{"key":"ref_1","first-page":"381","article-title":"UAV-UGV Collaboration with a PackBot UGV and Raven SUAV for Pursuit and Tracking of a Dynamic Target","volume":"Volume 6962","author":"Cheung","year":"2008","journal-title":"Unmanned Systems Technology X"},{"key":"ref_2","first-page":"30","article-title":"Integrated long-range UAV\/UGV collaborative target tracking","volume":"Volume 7332","author":"Moseley","year":"2009","journal-title":"Unmanned Systems Technology XI"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1117\/12.553047","article-title":"Project stork UAV\/UGV collaborative initiative","volume":"Volume 5422","author":"Schulteis","year":"2004","journal-title":"Unmanned Ground Vehicle Technology VI"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1109\/TII.2012.2219061","article-title":"An overview of recent progress in the study of distributed multi-agent coordination","volume":"9","author":"Cao","year":"2013","journal-title":"IEEE Trans. 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