{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:15:20Z","timestamp":1760228120779,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,7]],"date-time":"2022-05-07T00:00:00Z","timestamp":1651881600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Hunan Province","award":["2019JJ50490","2020JJ3029","18C0455","190XQD056"],"award-info":[{"award-number":["2019JJ50490","2020JJ3029","18C0455","190XQD056"]}]},{"DOI":"10.13039\/100014472","name":"Scientific Research Foundation of Education Bureau of Hunan Province","doi-asserted-by":"publisher","award":["2019JJ50490","2020JJ3029","18C0455","190XQD056"],"award-info":[{"award-number":["2019JJ50490","2020JJ3029","18C0455","190XQD056"]}],"id":[{"id":"10.13039\/100014472","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Doctoral Scientific Research Foundation of University of South China","award":["2019JJ50490","2020JJ3029","18C0455","190XQD056"],"award-info":[{"award-number":["2019JJ50490","2020JJ3029","18C0455","190XQD056"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The synchronous genetic regulatory networks model includes the drive system and response system, and the drive-response system is symmetric. From a biological point of view, this model illustrates the dynamic behaviors in gene-to-protein processes, in terms of transcription and translation. This paper introduces a model of genetic regulatory networks with time delay. The fixed-time synchronization control problem of the proposed model is studied based on fixed-time stability theory and the Lyapunov method. Concretely, the authors first propose a way to switch from the given model to matrix form. Then, two types of novel controllers are designed and the corresponding sufficient conditions are investigated respectively to ensure the fixed-time synchronization goal. Moreover, the settling times of fixed-time synchronization are estimated for the addressed discontinuous controllers, which are not dependent on the initial or delayed states of the model. Finally, numerical simulations are presented and compared to illustrate the benefits of the theoretical results.<\/jats:p>","DOI":"10.3390\/sym14050951","type":"journal-article","created":{"date-parts":[[2022,5,8]],"date-time":"2022-05-08T23:27:25Z","timestamp":1652052445000},"page":"951","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Fixed-Time Synchronization Analysis of Genetic Regulatory Network Model with Time-Delay"],"prefix":"10.3390","volume":"14","author":[{"given":"Yajun","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, University of South China, Hengyang 421001, China"},{"name":"School of Primary Education, Hunan Vocational College for Nationalities, Yueyang 414000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8402-3477","authenticated-orcid":false,"given":"You","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, University of South China, Hengyang 421001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1002\/cplx.6130010612","article-title":"Modeling the complexity of genetic networks: Understanding multigenic and pleiotropic regulation","volume":"1","author":"Somogyi","year":"1996","journal-title":"Complexity"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1109\/MSP.2007.323279","article-title":"A systems biology perspective on signal processing in ge-netic network motifs","volume":"24","author":"Li","year":"2007","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1002\/hep.21682","article-title":"Genome-level analysis of genetic regulation of liver gene expres-sion networks","volume":"46","author":"Chesler","year":"2007","journal-title":"Hepatology"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/S0092-8674(02)00625-6","article-title":"Apoptosis: A link between cancer genetics and chemotherapy","volume":"108","author":"Johnstone","year":"2002","journal-title":"Cell"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1038\/35002131","article-title":"Construction of a genetic toggle switch in escherichia coli","volume":"403","author":"Gardner","year":"2000","journal-title":"Nature"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2448","DOI":"10.1109\/TAC.2008.2007862","article-title":"Filtering for nonlinear genetic reg-ulatory networks with stochastic disturbances","volume":"53","author":"Wang","year":"2008","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1080\/00207720903141434","article-title":"Robust state estimation for stochastic genetic regulatory networks","volume":"41","author":"Liang","year":"2010","journal-title":"Int. J. Syst. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1425","DOI":"10.1109\/TSMCB.2009.2039478","article-title":"Robust state estimation for neural networks with discontinuous activations","volume":"40","author":"Liu","year":"2010","journal-title":"IEEE Trans. Syst. Man Cybern. Part B (Cybern.)"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1109\/TCSI.2002.1001949","article-title":"Stability of genetic regulatory networks with time delay","volume":"49","author":"Chen","year":"2002","journal-title":"IEEE Trans. Circuits Syst. I Fundam. Theory Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4060","DOI":"10.1016\/j.cnsns.2011.02.009","article-title":"Robust distributed state estimation for genetic regulatory networks with Markovian jumping parameters","volume":"16","author":"Lv","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1962","DOI":"10.1016\/j.automatica.2011.05.013","article-title":"Finite-time distributed consensus via binary control protocols","volume":"47","author":"Chen","year":"2011","journal-title":"Automatica"},{"key":"ref_12","first-page":"737","article-title":"Modeling method of gene regulation network","volume":"7","author":"Yu","year":"2006","journal-title":"J. Second Mil. Med. Univ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"840","DOI":"10.1126\/science.1074560","article-title":"Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop","volume":"298","author":"Hirata","year":"2002","journal-title":"Science"},{"key":"ref_14","first-page":"315","article-title":"Stability and chaos in an inertial two neuron sys-tem","volume":"411","author":"Wheeler","year":"1997","journal-title":"Am. Inst. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"9540548","DOI":"10.1155\/2021\/9540548","article-title":"Finite-Time Stability Analysis of Switched Genetic Regulatory Networks with Time-Varying Delays via Wirtinger\u2019s Integral Inequality","volume":"2021","author":"Saravanan","year":"2021","journal-title":"Complexity"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4328","DOI":"10.1016\/j.physleta.2009.09.055","article-title":"State estimation for Markov-type genetic regulatory networks with delays and uncertain mode transition rates","volume":"373","author":"Liang","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1219","DOI":"10.1007\/s11063-017-9695-2","article-title":"Decentralized Event-Triggered Exponential Stability for Un-certain Delayed Genetic Regulatory Networks with Markov Jump Parameters and Distributed Delays","volume":"47","author":"Ali","year":"2018","journal-title":"Neural Process Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2479","DOI":"10.1016\/j.cnsns.2013.11.002","article-title":"State estimation for markovian jumping genetic regulatory networks with random delays","volume":"19","author":"Liu","year":"2014","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1007\/s40995-017-0336-4","article-title":"Analytical solution for optimal control by the second kind Chebyshev polynomials expansion","volume":"41","author":"Razmjooy","year":"2017","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"824","DOI":"10.1002\/asjc.1777","article-title":"Uncertain method for optimal control prob-lems with uncertainties using Chebyshev inclusion functions","volume":"21","author":"Razmjooy","year":"2018","journal-title":"Asian J. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2106","DOI":"10.1109\/TAC.2011.2179869","article-title":"Nonlinear feedback design for fixed-time stabilization of linear control systems","volume":"57","author":"Polyakov","year":"2012","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1109\/TNNLS.2014.2387355","article-title":"Lag synchronization of switched neural networks via neural activation function and applications in image encryption","volume":"26","author":"Wen","year":"2015","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2571","DOI":"10.1142\/S0218127496001648","article-title":"Synchronization of chaotic neural networks and applications to communications","volume":"6","author":"Zaghloul","year":"1996","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Wang, Z., Song, C., Yan, A., and Wang, G. (2022). Complete Synchronization and Partial An-ti-Synchronization of Complex L\u00fc Chaotic Systems by the UDE-Based Control Meth-od. Symmetry, 14.","DOI":"10.3390\/sym14030517"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1016\/j.neucom.2015.04.064","article-title":"Finite-time stochastic synchronization of genetic regulatory networks","volume":"167","author":"Jiang","year":"2015","journal-title":"Neurocomputing"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.apm.2017.03.024","article-title":"Finite-time synchronization of master\u2013slave neural net-works with time-delays and discontinuous activations","volume":"47","author":"Cai","year":"2017","journal-title":"J. Appl. Math. Model."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"858","DOI":"10.1109\/TAC.2006.875006","article-title":"Adaptive finite-time control of nonlinear systems with parametric uncertainty","volume":"51","author":"Hong","year":"2006","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1993","DOI":"10.1016\/j.automatica.2006.06.015","article-title":"Finite-time convergent gradient flows with applications to network consensus","volume":"42","year":"2006","journal-title":"Automatica"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.neucom.2016.09.017","article-title":"Finite-time stability of ge-netic regulatory networks with impulsive effects","volume":"219","author":"Qiu","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_30","first-page":"126035","article-title":"New study on fixed-time synchroniza-tion control of delayed inertial memristive neural networks","volume":"399","author":"Dong","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Filippov, A. (1988). Differential Equations with Discontinuous Right-Hand Side, Kluwer Academic.","DOI":"10.1007\/978-94-015-7793-9"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.physd.2005.12.006","article-title":"Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-lipschitz activations","volume":"214","author":"Forti","year":"2006","journal-title":"Physics D"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.neunet.2020.04.020","article-title":"Fixed-time synchronization of delayed Co-hen-Grossberg neural networks based on a novel sliding mode","volume":"128","author":"Xiao","year":"2020","journal-title":"Neural Netw."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.neunet.2015.10.009","article-title":"Robust fixed-time synchronization of delayed Cohen-Grossberg neural networks","volume":"73","author":"Wan","year":"2016","journal-title":"Neural Netw."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.neunet.2017.02.001","article-title":"Fixed-time stability of dynamical sys-tems and fixed-time synchronization of coupled discontinuous neural networks","volume":"89","author":"Hu","year":"2017","journal-title":"Neural Netw."},{"key":"ref_36","unstructured":"Hardy, G., Littlewood, J., and Polya, G. (1988). Inequalities, Cambridge University Press."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1109\/TNB.2014.2327582","article-title":"Stochastic Stability of Switched Genetic Regulatory Net-works With Time-Varying Delays","volume":"13","author":"Zhang","year":"2014","journal-title":"IEEE Trans. Nanobiosci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1007\/s12591-017-0349-7","article-title":"Dissipative Control of Markovian Jumping Genetic Regula-tory Networks with Time-Varying Delays and Reaction-Diffusion Driven by Fractional Brownian Motion","volume":"28","author":"Ma","year":"2020","journal-title":"Differ. Equ. Dyn. 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