{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T00:44:14Z","timestamp":1777423454261,"version":"3.51.4"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T00:00:00Z","timestamp":1718668800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T00:00:00Z","timestamp":1718668800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62103176"],"award-info":[{"award-number":["62103176"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"\u201cGuangyue Young Scholar Innovation Team\" of Liaocheng University","award":["LCUGYTD2022-01"],"award-info":[{"award-number":["LCUGYTD2022-01"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2024,10]]},"DOI":"10.1007\/s12190-024-02158-5","type":"journal-article","created":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T18:12:44Z","timestamp":1718734364000},"page":"4827-4850","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A matrix approach to the analysis and computation of robust cycles for Boolean control networks with disturbances"],"prefix":"10.1007","volume":"70","author":[{"given":"Lei","family":"Deng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shihua","family":"Fu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinling","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianjun","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,6,18]]},"reference":[{"issue":"3","key":"2158_CR1","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1016\/0022-5193(69)90015-0","volume":"22","author":"S Kauffman","year":"1969","unstructured":"Kauffman, S.: Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22(3), 437\u2013467 (1969)","journal-title":"J. Theor. Biol."},{"key":"2158_CR2","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1016\/j.pmcj.2014.03.005","volume":"13","author":"A Nazi","year":"2014","unstructured":"Nazi, A., Raj, M., Francesco, M., Ghoshet, P., Das, S.: Deployment of robust wireless sensor networks using gene regulatory networks: an isomorphism-based approach. Pervasive Mobile Comput. 13, 246\u2013257 (2014)","journal-title":"Pervasive Mobile Comput."},{"issue":"23","key":"2158_CR3","doi-asserted-by":"publisher","first-page":"9467","DOI":"10.1073\/pnas.87.23.9467","volume":"87","author":"L Wang","year":"1991","unstructured":"Wang, L., Pichler, E., Ross, J.: Oscillations and chaos in neural networks: an exactly solvable model. Proc. Natl. Acad. Sci. 87(23), 9467\u20139471 (1991)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"2158_CR4","doi-asserted-by":"publisher","first-page":"3384","DOI":"10.1016\/j.automatica.2013.08.008","volume":"49","author":"P Guo","year":"2013","unstructured":"Guo, P., Wang, Y., Li, H.: Algebraic formulation and strategy optimization for a class of evolutionary networked games via semi-tensor product method. Automatica 49, 3384\u20133389 (2013)","journal-title":"Automatica"},{"key":"2158_CR5","doi-asserted-by":"crossref","unstructured":"Bridoux, F., Gaze-Maillot, C., Perrot, K., Sen, S.: Complexity of limit-cycle problems in Boolean networks. In: Proceedings of the 47th International Conference on Current Trends in Theory and Practice of Computer Science, 135-146 (2021)","DOI":"10.1007\/978-3-030-67731-2_10"},{"key":"2158_CR6","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/j.cam.2018.10.003","volume":"350","author":"H Li","year":"2019","unstructured":"Li, H., Zheng, Y., Alsaadi, F.: Algebraic formulation and topological structure of Boolean networks with state-dependent delay. J. Comput. Appl. Math. 350, 87\u201397 (2019)","journal-title":"J. Comput. Appl. Math."},{"issue":"8","key":"2158_CR7","doi-asserted-by":"publisher","first-page":"1766","DOI":"10.1109\/TAC.2010.2092290","volume":"56","author":"Y Zhao","year":"2011","unstructured":"Zhao, Y., Li, Z., Cheng, D.: Optimal control of logical control networks. IEEE Trans. Autom. Control 56(8), 1766\u20131776 (2011)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"2","key":"2158_CR8","doi-asserted-by":"publisher","first-page":"348","DOI":"10.1109\/TNN.2004.824262","volume":"15","author":"C Farrow","year":"2004","unstructured":"Farrow, C., Heidel, J., Maloney, J., Rogers, J.: Scalar equations for synchronous Boolean networks with biological applications. IEEE Trans. Neural Netw. 15(2), 348\u2013354 (2004)","journal-title":"IEEE Trans. Neural Netw."},{"key":"2158_CR9","doi-asserted-by":"publisher","first-page":"1025","DOI":"10.1016\/S0092-8240(03)00061-2","volume":"65","author":"Y Devloo","year":"2003","unstructured":"Devloo, Y., Hansen, P., Labb\u00e9, M.: Identification of all steady states in large networks by logical analysis. Bull. Math. Biol. 65, 1025\u20131051 (2003)","journal-title":"Bull. Math. Biol."},{"key":"2158_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-85729-097-7","volume-title":"Analysis and Control of Boolean Networks: A Semi-tensor Product Approach","author":"D Cheng","year":"2011","unstructured":"Cheng, D., Qi, H., Li, Z.: Analysis and Control of Boolean Networks: A Semi-tensor Product Approach. Springer, London (2011)"},{"issue":"2","key":"2158_CR11","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1002\/rnc.1581","volume":"21","author":"D Cheng","year":"2011","unstructured":"Cheng, D., Qi, H., Li, Z., Liu, J.: Stability and stabilization of Boolean networks. Int. J. Robust Nonlinear Control 21(2), 134\u2013156 (2011)","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"2158_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2023.111397","volume":"159","author":"H Li","year":"2024","unstructured":"Li, H., Pang, X.: Stability analysis of large-scale Boolean networks via compositional method. Automatica 159, 111397 (2024)","journal-title":"Automatica"},{"issue":"9","key":"2158_CR13","doi-asserted-by":"publisher","first-page":"5461","DOI":"10.1109\/TAC.2022.3226701","volume":"68","author":"S Zhu","year":"2023","unstructured":"Zhu, S., Lu, J., Azuma, S., Zheng, W.: Strong structural controllability of Boolean networks: polynomial-time criteria, minimal node control, and distributed pinning strategies. IEEE Trans. Autom. Control 68(9), 5461\u20135476 (2023)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"6","key":"2158_CR14","doi-asserted-by":"publisher","first-page":"3632","DOI":"10.1137\/19M1281332","volume":"58","author":"H Li","year":"2020","unstructured":"Li, H., Wang, S., Li, X., Zhao, G.: Perturbation analysis for controllability of logical control networks. SIAM J. Control. Optim. 58(6), 3632\u20133657 (2020)","journal-title":"SIAM J. Control. Optim."},{"issue":"5","key":"2158_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/s11432-021-3365-2","volume":"65","author":"Y Liu","year":"2022","unstructured":"Liu, Y., Zhong, J., Ho, D.W.C., Gui, W.: Minimal observability of Boolean networks. Sci. China Inf. Sci. 65(5), 152203 (2022)","journal-title":"Sci. China Inf. Sci."},{"issue":"8","key":"2158_CR16","doi-asserted-by":"publisher","first-page":"4162","DOI":"10.1109\/TAC.2021.3110165","volume":"67","author":"S Zhu","year":"2022","unstructured":"Zhu, S., Lu, J., Zhong, J., Liu, Y., Cao, J.: On the sensors construction of large Boolean networks via pinning observability. IEEE Trans. Autom. Control 67(8), 4162\u20134169 (2022)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"6","key":"2158_CR17","doi-asserted-by":"publisher","first-page":"2910","DOI":"10.1109\/TNNLS.2020.3008960","volume":"32","author":"Y Wu","year":"2021","unstructured":"Wu, Y., Guo, Y., Toyoda, M.: Policy iteration approach to the infinite horizon average optimal control of probabilistic Boolean networks. IEEE Trans. Neural Netw. Learn. Syst. 32(6), 2910\u20132924 (2021)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"2158_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2023.110970","volume":"152","author":"H Li","year":"2023","unstructured":"Li, H., Yang, X.: Robust optimal control of logical control networks with function perturbation. Automatica 152, 110970 (2023)","journal-title":"Automatica"},{"issue":"4","key":"2158_CR19","doi-asserted-by":"publisher","first-page":"2278","DOI":"10.1109\/TCYB.2019.2939761","volume":"51","author":"J Lu","year":"2021","unstructured":"Lu, J., Yang, J., Lou, J., Qiu, J.: Event-triggered sampled feedback synchronization in an array of output-coupled Boolean control networks. IEEE Trans. Cybern. 51(4), 2278\u20132283 (2021)","journal-title":"IEEE Trans. Cybern."},{"issue":"3","key":"2158_CR20","doi-asserted-by":"publisher","first-page":"612","DOI":"10.1109\/TNNLS.2015.2497708","volume":"28","author":"H Chen","year":"2017","unstructured":"Chen, H., Liang, J., Huang, T., Cao, J.: Synchronization of arbitrarily switched Boolean networks. IEEE Trans. Neural Netw. Learn. Syst. 28(3), 612\u2013619 (2017)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"17","key":"2158_CR21","doi-asserted-by":"publisher","first-page":"8576","DOI":"10.1016\/j.jfranklin.2018.09.004","volume":"355","author":"S Fu","year":"2018","unstructured":"Fu, S., Zhao, J., Wang, J.: Input-output decoupling control design for switched Boolean control networks. J. Franklin Inst. 355(17), 8576\u20138596 (2018)","journal-title":"J. Franklin Inst."},{"issue":"1","key":"2158_CR22","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1109\/TAC.2010.2050161","volume":"56","author":"D Cheng","year":"2011","unstructured":"Cheng, D.: Disturbance decoupling of Boolean control networks. IEEE Trans. Autom. Control 56(1), 2\u201310 (2011)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"13","key":"2158_CR23","doi-asserted-by":"publisher","first-page":"2040","DOI":"10.1049\/iet-cta.2016.1659","volume":"11","author":"J Lu","year":"2017","unstructured":"Lu, J., Li, H., Liu, Y., Li, F.: Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems. Iet Control Theory and Appl. 11(13), 2040\u20132047 (2017)","journal-title":"Iet Control Theory and Appl."},{"issue":"4","key":"2158_CR24","doi-asserted-by":"publisher","first-page":"818","DOI":"10.1109\/JAS.2016.7510259","volume":"54","author":"G Zhao","year":"2018","unstructured":"Zhao, G., Wang, Y., Li, H.: A matrix approach to the modeling and analysis of networked evolutionary games with time delays. IEEE\/CAA J. Autom. Sin. 54(4), 818\u2013826 (2018)","journal-title":"IEEE\/CAA J. Autom. Sin."},{"issue":"1","key":"2158_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/s11432-017-9269-6","volume":"61","author":"J Lu","year":"2018","unstructured":"Lu, J., Li, M., Liu, Y., Ho, D.W.C., Kurths, J.: Nonsingularity of Grain-like cascade FSRs via semi-tensor product. Sci. China Inf. Sci. 61(1), 010204 (2018)","journal-title":"Sci. China Inf. Sci."},{"issue":"1","key":"2158_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/s11432-022-3538-4","volume":"66","author":"Y Yan","year":"2023","unstructured":"Yan, Y., Cheng, D., Feng, J., Li, H., Yue, J.: Survey on applications of algebraic state space theory of logical systems to finite state machines. Sci. China Inf. Sci. 66(1), 111201 (2023)","journal-title":"Sci. China Inf. Sci."},{"issue":"10","key":"2158_CR27","doi-asserted-by":"publisher","first-page":"2251","DOI":"10.1109\/TAC.2010.2043294","volume":"55","author":"D Cheng","year":"2010","unstructured":"Cheng, D., Qi, H.: A linear representation of dynamics of Boolean networks. IEEE Trans. Autom. Control 55(10), 2251\u20132258 (2010)","journal-title":"IEEE Trans. Autom. Control"},{"key":"2158_CR28","doi-asserted-by":"publisher","first-page":"1506","DOI":"10.1016\/j.automatica.2013.02.027","volume":"49","author":"E Fornasini","year":"2013","unstructured":"Fornasini, E., Valcher, M.: On the periodic trajectories of Boolean control networks. Automatica 49, 1506\u20131509 (2013)","journal-title":"Automatica"},{"key":"2158_CR29","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/j.automatica.2014.10.071","volume":"51","author":"Y Zou","year":"2015","unstructured":"Zou, Y., Zhu, J.: Cycles of periodically time-variant Boolean networks. Automatica 51, 175\u2013179 (2015)","journal-title":"Automatica"},{"key":"2158_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2021.104878","volume":"149","author":"Y Liu","year":"2021","unstructured":"Liu, Y., Li, H.: Logical matrix factorization towards topological structure and stability of probabilistic Boolean networks. Syst. Control Lett. 149, 104878 (2021)","journal-title":"Syst. Control Lett."},{"issue":"15","key":"2158_CR31","doi-asserted-by":"publisher","first-page":"7077","DOI":"10.1016\/j.jfranklin.2017.08.017","volume":"354","author":"L Tong","year":"2017","unstructured":"Tong, L., Liu, Y., Alsaadi, F., Hayat, T.: Robust sampled-data control invariance for Boolean control networks. J. Franklin Inst. 354(15), 7077\u20137087 (2017)","journal-title":"J. Franklin Inst."},{"key":"2158_CR32","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1016\/j.automatica.2017.07.013","volume":"84","author":"J Zhong","year":"2017","unstructured":"Zhong, J., Ho, D., Lu, J., Xu, W.: Global robust stability and stabilization of Boolean network with disturbances. Automatica 84, 142\u2013148 (2017)","journal-title":"Automatica"},{"issue":"3","key":"2158_CR33","doi-asserted-by":"publisher","first-page":"1231","DOI":"10.1109\/TAC.2020.2997282","volume":"66","author":"H Li","year":"2021","unstructured":"Li, H., Yang, X., Wang, S.: Robustness for stability and stabilization of Boolean networks with stochastic function perturbations. IEEE Trans. Autom. Control 66(3), 1231\u20131237 (2021)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"4","key":"2158_CR34","doi-asserted-by":"publisher","first-page":"553","DOI":"10.15388\/NA.2018.4.6","volume":"23","author":"X Xu","year":"2018","unstructured":"Xu, X., Liu, Y., Li, H., Alsaadi, F.: Robust set stabilization of Boolean control networks with impulsive effects. Nonlinear Anal.: Model. Control 23(4), 553\u2013567 (2018)","journal-title":"Nonlinear Anal.: Model. Control"},{"issue":"13","key":"2158_CR35","doi-asserted-by":"publisher","first-page":"2089","DOI":"10.1049\/iet-cta.2017.0260","volume":"11","author":"J Zhong","year":"2017","unstructured":"Zhong, J., Ho, D., Lu, J., Xu, W.: Switching-signal-triggered pinning control for output tracking of switched Boolean networks. IET Control Theory Appl. 11(13), 2089\u20132096 (2017)","journal-title":"IET Control Theory Appl."},{"key":"2158_CR36","doi-asserted-by":"publisher","first-page":"670","DOI":"10.1016\/j.jtbi.2006.09.023","volume":"244","author":"T Akutsu","year":"2007","unstructured":"Akutsu, T., Hayashida, M., Ching, W.K., Ng, M.K.: Control of Boolean networks: hardness results and algorithms for tree structured networks. J. Theor. Biol. 244, 670\u2013679 (2007)","journal-title":"J. Theor. Biol."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-024-02158-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-024-02158-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-024-02158-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,26]],"date-time":"2024-09-26T12:30:39Z","timestamp":1727353839000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-024-02158-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,18]]},"references-count":36,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["2158"],"URL":"https:\/\/doi.org\/10.1007\/s12190-024-02158-5","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,18]]},"assertion":[{"value":"21 November 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 June 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 June 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 June 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}