{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T18:04:40Z","timestamp":1771524280194,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,17]],"date-time":"2025-08-17T00:00:00Z","timestamp":1755388800000},"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":["62466019"],"award-info":[{"award-number":["62466019"]}],"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":["2023JJ30485"],"award-info":[{"award-number":["2023JJ30485"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Hunan Province of China","award":["62466019"],"award-info":[{"award-number":["62466019"]}]},{"name":"Natural Science Foundation of Hunan Province of China","award":["2023JJ30485"],"award-info":[{"award-number":["2023JJ30485"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Time-variant nonlinear problems have always been a kind of complex research object in the field of control. The accuracy and efficiency of settling time-variant nonlinear inequality-equation (NIE) systems are often affected by the nonlinearity degree of the systems, and there are currently no complete algorithms to settle the time-variant NIE systems effectively. To settle this class of complex systems effectively, time-variant NIE systems are first equivalently transformed into a time-variant equation by introducing a nonnegative variable. Then, through the idea of zeroing neural network (ZNN) and the role of time-variant parameter-gain functions, a parameter-gain accelerated ZNN (PGAZNN) model is proposed to solve time-variant NIE systems. Theoretically, the stability of the proposed PGAZNN model is proved by strict mathematical analysis. In addition, the PGAZNN model can achieve fixed-time convergence, and the upper-bound of convergence time is estimated. Finally, numerical simulation example and symmetry trajectory tracking are given to verify the validity and correctness of the proposed PGAZNN model.<\/jats:p>","DOI":"10.3390\/sym17081342","type":"journal-article","created":{"date-parts":[[2025,8,18]],"date-time":"2025-08-18T07:53:12Z","timestamp":1755503592000},"page":"1342","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Parameter-Gain Accelerated ZNN Model for Solving Time-Variant Nonlinear Inequality-Equation Systems and Application on Tracking Symmetrical Trajectory"],"prefix":"10.3390","volume":"17","author":[{"given":"Yihui","family":"Lei","sequence":"first","affiliation":[{"name":"College of Mathematics and Statistics, Jishou University, Jishou 416000, China"}]},{"given":"Longyi","family":"Xu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Jishou University, Jishou 416000, China"}]},{"given":"Jialiang","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Computer Science and Engineering, Jishou University, Jishou 416000, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1142\/S0129065704001905","article-title":"Nonlinear system modelling via optimal design of neural trees","volume":"14","author":"Chen","year":"2004","journal-title":"Int. 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