{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:39:22Z","timestamp":1760236762419,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,19]],"date-time":"2021-12-19T00:00:00Z","timestamp":1639872000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Hebei Province of China","award":["F2016203507, A2019209005"],"award-info":[{"award-number":["F2016203507, A2019209005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Motivated by the two strategies of intermittent control and discrete feedback control, this paper aims to introduce a periodically intermittent discrete feedback control in the drift part to stabilize an unstable Markov jumping stochastic differential system. It is illustrated that, by the approach of comparison principle, this can be achieved in the sense of almost sure exponential stability. Further, the stabilization theory is applied to Markov jumping stochastic recurrent neural networks.<\/jats:p>","DOI":"10.3390\/sym13122447","type":"journal-article","created":{"date-parts":[[2021,12,20]],"date-time":"2021-12-20T02:40:32Z","timestamp":1639968032000},"page":"2447","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Stabilization of Periodical Discrete Feedback Control for Markov Jumping Stochastic Systems"],"prefix":"10.3390","volume":"13","author":[{"given":"Zhiyou","family":"Liu","sequence":"first","affiliation":[{"name":"Key Lab of Power Electronics for Energy Conservation and Motor Drive of Hebei Province, Department of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China"},{"name":"College of Science, North China University of Science and Technology, Tangshan 063000, China"}]},{"given":"Lichao","family":"Feng","sequence":"additional","affiliation":[{"name":"College of Science, North China University of Science and Technology, Tangshan 063000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1116-963X","authenticated-orcid":false,"given":"Xinbin","family":"Li","sequence":"additional","affiliation":[{"name":"Key Lab of Power Electronics for Energy Conservation and Motor Drive of Hebei Province, Department of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China"}]},{"given":"Zhigang","family":"Lu","sequence":"additional","affiliation":[{"name":"Key Lab of Power Electronics for Energy Conservation and Motor Drive of Hebei Province, Department of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China"}]},{"given":"Xianhui","family":"Meng","sequence":"additional","affiliation":[{"name":"Key Lab of Power Electronics for Energy Conservation and Motor Drive of Hebei Province, Department of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China"},{"name":"Department of Electromechanical Engineering, Tangshan University, Tangshan 063000, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1137\/S0363012999356325","article-title":"Stock trading: An optimal selling rule","volume":"40","author":"Zhang","year":"2001","journal-title":"Siam J. Control Optim."},{"key":"ref_2","unstructured":"Mariton, M. (1990). Jump Linear Systems in Automatic Control, Marcel Dekker."},{"key":"ref_3","unstructured":"Skorokhod, A.V. (1989). Asymptotic Methods in the Theory of Stochastic Differential Equations, American Mathematical Society."},{"key":"ref_4","first-page":"180","article-title":"Stability of stochastic hereditary systems with Markov switching","volume":"2","author":"Shaikhet","year":"1996","journal-title":"Theory Stoch. Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/S0304-4149(98)00070-2","article-title":"Stability of stochastic differential equations with Markovian switching","volume":"79","author":"Mao","year":"1999","journal-title":"Stoch. Process. Their Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Mao, X., and Yuan, C. (2006). Stochastic Differential Equations with Markovian Switching, Imperial College Press.","DOI":"10.1142\/p473"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2435","DOI":"10.1016\/j.automatica.2014.08.006","article-title":"Stability analysis for stochastic hybrid systems: A survey","volume":"50","author":"Teel","year":"2014","journal-title":"Automatica"},{"key":"ref_8","first-page":"45","article-title":"A note on stability of hybrid stochastic differential equations","volume":"299","author":"Li","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.nahs.2019.03.008","article-title":"Asymptotic stability of nonlinear hybrid stochastic systems driven by linear discrete time noises","volume":"33","author":"Feng","year":"2019","journal-title":"Nonlinear Anal. Hybrid Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"106468","DOI":"10.1016\/j.aml.2020.106468","article-title":"Exponential stability for nonlinear hybrid stochastic systems with time varying delays of neutral type","volume":"107","author":"Feng","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"104556","DOI":"10.1016\/j.sysconle.2019.104556","article-title":"Almost sure stability of hybrid stochastic systems under asynchronous Markovian switching","volume":"133","author":"Luo","year":"2019","journal-title":"Syst. Control Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/j.automatica.2006.09.006","article-title":"Stabilization and destabilization of hybrid systems of stochastic differential equations","volume":"43","author":"Mao","year":"2007","journal-title":"Automatica"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/j.automatica.2012.01.022","article-title":"Noise suppresses explosive solutions of differential systems with coefficients satisfying the polynomial growth condition","volume":"48","author":"Liu","year":"2012","journal-title":"Automatica"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2321","DOI":"10.1016\/j.automatica.2012.06.044","article-title":"Stochastic stabilization of hybrid differential equations","volume":"48","author":"Deng","year":"2012","journal-title":"Automatica"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1619","DOI":"10.1109\/TAC.2015.2471696","article-title":"Almost sure exponential stabilization by discrete-time stochastic feedback control","volume":"61","author":"Mao","year":"2016","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1007\/s11432-017-9340-4","article-title":"Suppression of explosion by polynomial noise for nonlinear differential systems","volume":"61","author":"Feng","year":"2018","journal-title":"Sci. China Inf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3677","DOI":"10.1016\/j.automatica.2013.09.005","article-title":"Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control","volume":"49","author":"Mao","year":"2013","journal-title":"Automatica"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.sysconle.2014.08.011","article-title":"Stabilization of hybrid stochastic differential equations by feedback control based on discrete time state observations","volume":"73","author":"Mao","year":"2014","journal-title":"Syst. Control Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1137\/140985779","article-title":"Stabilization of hybrid systems by feedback control based on discrete-time state observations","volume":"53","author":"You","year":"2015","journal-title":"SIAM J. Control Optim."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2899","DOI":"10.1109\/TAC.2019.2933604","article-title":"Stabilization of highly nonlinear hybrid systems by feedback control based on discrete-time state observations","volume":"65","author":"Fei","year":"2020","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_21","first-page":"126771","article-title":"Stabilization in general decay rate of discrete feedback control for non-autonomous Markov jump stochastic systems","volume":"417","author":"Feng","year":"2022","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/S0167-2789(00)00112-3","article-title":"Intermittent dynamical control","volume":"145","author":"Zochowski","year":"2000","journal-title":"Phys. D"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1109\/TCSII.2007.903205","article-title":"Stabilization of nonlinear systems via periodically intermittent control","volume":"54","author":"Li","year":"2007","journal-title":"IEEE Trans. Circuits Syst. Express Briefs"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"013120","DOI":"10.1063\/1.3071933","article-title":"Pinning synchronization of delayed dynamical networks via periodically intermittent control","volume":"19","author":"Xia","year":"2009","journal-title":"Chaos"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3259","DOI":"10.1016\/j.physleta.2009.07.013","article-title":"Stochastic synchronization of coupled neural networks with intermittent control","volume":"373","author":"Yang","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.neunet.2015.12.003","article-title":"Exponential stabilization and synchronization for fuzzy model of memristive neural networks by periodically intermittent control","volume":"75","author":"Yang","year":"2016","journal-title":"Neural Netw."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.automatica.2016.04.031","article-title":"Delay-independent stabilization of a class of time-delay systems via periodically intermittent control","volume":"71","author":"Chen","year":"2016","journal-title":"Automatica"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.nahs.2018.03.006","article-title":"Stabilization of stochastic complex-valued coupled delayed systems with Markovian switching via periodically intermittent control","volume":"29","author":"Wang","year":"2018","journal-title":"Nonlinear Anal. Hybrid Syst."},{"key":"ref_29","first-page":"341","article-title":"Graph-theoretic approach to synchronizing stochastic coupled systems with time-varying delays on networks via periodically intermittent control","volume":"331","author":"Guo","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s11431-016-6006-5","article-title":"Exponential H\u221e filtering analysis for discrete-time switched neural networks with random delays using sojourn probabilities","volume":"59","author":"Cao","year":"2016","journal-title":"Sci. China Technol. Sci."},{"key":"ref_31","first-page":"3152","article-title":"pth moment exponential input-to-state stability of delayed recurrent neural networks with Markovian switching via vector Lyapunov function","volume":"29","author":"Liu","year":"2017","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1109\/TNNLS.2015.2511196","article-title":"Extended dissipative state estimation for Markov jump neural networks with unreliable links","volume":"28","author":"Shen","year":"2017","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2232","DOI":"10.1109\/TCYB.2017.2729581","article-title":"A flexible terminal approach to sampled-data exponentially synchronization of Markovian neural networks with time-varying delayed signals","volume":"48","author":"Cheng","year":"2018","journal-title":"IEEE Trans. Cybern."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2235","DOI":"10.1088\/0305-4470\/36\/9\/303","article-title":"Stochastic Hopfield neural networks","volume":"36","author":"Hu","year":"2003","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_35","first-page":"84","article-title":"Exponential stability of stochastic delayed Hopfield neural networks","volume":"199","author":"Zhou","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1109\/TSMCB.2010.2053354","article-title":"Exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays","volume":"41","author":"Zhu","year":"2011","journal-title":"IEEE Trans. Syst. Man Cybern. Part B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1016\/j.nonrwa.2011.09.001","article-title":"Ultimate boundedness and an attractor for stochastic Hopfield neural networks with time-varying delays","volume":"13","author":"Wan","year":"2012","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.matcom.2010.11.014","article-title":"Noise suppress exponential growth for hybrid Hopfield neural networks","volume":"86","author":"Zhu","year":"2012","journal-title":"Math. Comput. Simul."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s11063-018-9912-7","article-title":"Stability analysis in a class of Markov switched stochastic Hopfield neural networks","volume":"50","author":"Feng","year":"2019","journal-title":"Neural Process. 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