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Within this architecture, each robot is assigned a cost function to ensure optimal performance while minimizing energy consumption. The FOE employs a distributed optimization formation control protocol to compute the optimal positions in finite-time, effectively mitigating the influence of disturbances and uncertainty. The FLC enables the actual position for each robot to track the estimated signal generated by the FOE, thereby achieving coordinated formation control. A Lyapunov-based stability analysis is conducted to derive sufficient conditions ensuring the finite-time convergence and stability of the proposed scheme. Simulation results validate the effectiveness and robustness of the proposed control strategy.<\/jats:p>","DOI":"10.1007\/s44163-025-00415-5","type":"journal-article","created":{"date-parts":[[2025,7,29]],"date-time":"2025-07-29T09:56:53Z","timestamp":1753783013000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Finite-time distributed optimization formation control of networked robots with time-varying reference signals under directed graphs"],"prefix":"10.1007","volume":"5","author":[{"given":"Weicheng","family":"Zhao","sequence":"first","affiliation":[]},{"given":"Qian","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Li","family":"Ding","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,29]]},"reference":[{"issue":"1","key":"415_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s44163-025-00300-1","volume":"5","author":"H Zhang","year":"2025","unstructured":"Zhang H, Shen DW. 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The studies and data included in this review were sourced from publicly available and previously published literature.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"All authors listed in this paper have reviewed and approved the final manuscript for submission to Discover Artificial Intelligence.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"184"}}