{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T15:46:15Z","timestamp":1769010375252,"version":"3.49.0"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T00:00:00Z","timestamp":1757289600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Hunan Province, China","award":["2025JJ60047"],"award-info":[{"award-number":["2025JJ60047"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the pth moment exponential stability of random impulsive delayed nonlinear systems (RIDNS) with multiple periodic delayed impulses. Moreover, the continuous dynamics are described by random delay differential equations whose random disturbances are driven by second-order moment processes. Using the periodic impulsive intensity (PII), average delay time (ADT), average impulsive delay (AID), as well as the Lyapunov method, we present some pth exponential stability criteria for impulsive random delayed nonlinear systems with multiple delayed impulses. Furthermore, the criterion is unified, which is not only applicable to stable or unstable original systems but also takes into account the coexistence of stabilizing and destabilizing impulses. The periodic structure of impulses and their intensities introduces an intrinsic temporal symmetry, which plays a critical role in determining the stability behavior of the system. This symmetry-based perspective highlights the fundamental impact of regularly recurring impulsive actions on system dynamics. Several illustrated examples are given to verify the effectiveness of our results.<\/jats:p>","DOI":"10.3390\/sym17091481","type":"journal-article","created":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T10:29:07Z","timestamp":1757327347000},"page":"1481","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Symmetric Analysis of Stability Criteria for Nonlinear Systems with Multi-Delayed Periodic Impulses: Intensity Periodicity and Averaged Delay"],"prefix":"10.3390","volume":"17","author":[{"given":"Yao","family":"Lu","sequence":"first","affiliation":[{"name":"School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China"},{"name":"MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6789-1405","authenticated-orcid":false,"given":"Dehao","family":"Ruan","sequence":"additional","affiliation":[{"name":"School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China"},{"name":"MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3130-4923","authenticated-orcid":false,"given":"Quanxin","family":"Zhu","sequence":"additional","affiliation":[{"name":"MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1650","DOI":"10.1109\/TAC.2004.835360","article-title":"Input-output stability properties of networked control systems","volume":"49","author":"Nesic","year":"2004","journal-title":"IEEE Trans. Automat. Control"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1109\/MCS.2008.931718","article-title":"Hybrid dynamical systems","volume":"29","author":"Goebel","year":"2009","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_3","unstructured":"Yang, T. (2001). Impulsive Control Theory, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"107406","DOI":"10.1016\/j.jfranklin.2024.107406","article-title":"Exponential stability of fractional-order asynchronous switched impulsive systems with time delay and mode-dependent parameter uncertainty","volume":"362","author":"Zhang","year":"2025","journal-title":"J. Franklin. Inst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"107205","DOI":"10.1016\/j.jfranklin.2024.107205","article-title":"Finite-time input-to-state stability and settling-time estimation of impulsive switched systems with multiple impulses","volume":"361","author":"Zhang","year":"2024","journal-title":"J. Franklin. Inst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107152","DOI":"10.1016\/j.jfranklin.2024.107152","article-title":"Finite-time stability via event-triggered delayed impulse control for time-varying nonlinear impulsive systems","volume":"361","author":"Cheng","year":"2024","journal-title":"J. Franklin. Inst."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Aldosari, F., and Ebaid, A. (2024). Analytical and Numerical Investigation for the Inhomogeneous Pantograph Equation. Axioms, 13.","DOI":"10.3390\/axioms13060377"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Albidah, A., Kanaan, N., Ebaid, A., and AlJeaid, H. (2023). Exact and Numerical Analysis of the Pantograph Delay Differential Equation via the Homotopy Perturbation Method. Mathematics, 11.","DOI":"10.3390\/math11040944"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6681","DOI":"10.1002\/rnc.5632","article-title":"Stability criteria of random delay differential systems subject to random impulses","volume":"31","author":"Zhang","year":"2021","journal-title":"Internat. J. Robust Nonlinear Control"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"106194","DOI":"10.1016\/j.sysconle.2025.106194","article-title":"Generalized exponential stability of stochastic delayed systems with delayed impulses and infinite delays","volume":"204","author":"Ruan","year":"2025","journal-title":"Syst. Control Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1109\/TIT.1959.1057532","article-title":"Stability of circuits with randomly time-varying parameters","volume":"5","author":"Bertram","year":"1959","journal-title":"IRE Trans. Circuit Theory"},{"key":"ref_12","unstructured":"Soong, T.T. (1973). Random Differential Equations in Science and Engineering, Academic Press."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Khasminskii, R. (1980). Stochastic Stability of Differential Equations, Springer.","DOI":"10.1007\/978-94-009-9121-7"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1038","DOI":"10.1109\/TAC.2014.2365684","article-title":"Stability criteria of random nonlinear systems and their applications","volume":"60","author":"Wu","year":"2015","journal-title":"IEEE Trans. Automat. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1016\/j.automatica.2019.04.006","article-title":"Practical trajectory tracking of random Lagrange systems","volume":"105","author":"Wu","year":"2019","journal-title":"Automatica"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3164","DOI":"10.1109\/TAC.2015.2504723","article-title":"Noise-to-state stability for a class of random systems with state-dependent switching","volume":"61","author":"Zhang","year":"2015","journal-title":"IEEE Trans. Automat. Control"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1109\/TAC.2015.2476175","article-title":"Noise-to-state stability of random switched systems and its applications","volume":"61","author":"Zhang","year":"2015","journal-title":"IEEE Trans. Automat. Control"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3833","DOI":"10.1002\/rnc.3767","article-title":"Adaptive tracking control for random nonlinear system","volume":"27","author":"Yao","year":"2017","journal-title":"Internat. J. Robust Nonlinear Control"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"35743","DOI":"10.3934\/math.20241696","article-title":"A novel analytical treatment for the Ambartsumian delay differential equation with a variable coefficient","volume":"9","author":"Alyoubi","year":"2024","journal-title":"AIMS Math."},{"key":"ref_20","unstructured":"Tunc, O., Berezansky, L., Tunc, C., and Yao, J. (2020). Enhanced impulsive stabilization results of differential and integro-differential equations of second order. Discret. Contin. Dyn. Syst. Ser. B."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Pinelas, S., Tunc, O., Korkmaz, E., and Tunc, C. (2024). Existence and stabilization for impulsive differential equations of second order with multiple delays. Electron. J. Differ. Equ., 1\u201318.","DOI":"10.58997\/ejde.2024.07"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1049\/cth2.12030","article-title":"Improved noise-to-state stability criteria of random nonlinear systems with stochastic impulses","volume":"15","author":"Feng","year":"2021","journal-title":"IET Control Theory Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"5426","DOI":"10.1016\/j.jfranklin.2021.04.039","article-title":"Further stability results for random nonlinear systems with stochastic impulses","volume":"358","author":"Feng","year":"2021","journal-title":"J. Franklin. Inst."},{"key":"ref_24","first-page":"127517","article-title":"Noise-to-state stability of random impulsive delay systems with multiple random impulses","volume":"436","author":"Feng","year":"2023","journal-title":"Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"106813","DOI":"10.1016\/j.jfranklin.2024.106813","article-title":"Exponential stability of impulsive random delayed nonlinear systems with average-delay impulses","volume":"361","author":"Lu","year":"2024","journal-title":"J. Franklin. Inst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1109\/TAC.2020.2982156","article-title":"Input-to-state stability of impulsive delay systems with multiple impulses","volume":"66","author":"Li","year":"2021","journal-title":"IEEE Trans. Automat. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2621","DOI":"10.1109\/TAC.2023.3335005","article-title":"Exponential stability of stochastic nonlinear delay systems subject to multiple periodic impulses","volume":"69","author":"Xu","year":"2024","journal-title":"IEEE Trans. Automat. Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1016\/j.automatica.2010.04.005","article-title":"A unified synchronization criterion for impulsive dynamical networks","volume":"46","author":"Lu","year":"2010","journal-title":"Automatica"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3763","DOI":"10.1137\/20M1317037","article-title":"Exponential stability of delayed systems with average-delay impulses","volume":"58","author":"Jiang","year":"2020","journal-title":"SIAM J. Control Optim."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1016\/j.automatica.2015.12.030","article-title":"On asymptotic stability of linear time-varying systems","volume":"68","author":"Zhou","year":"2016","journal-title":"Automatica"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1481\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:41:48Z","timestamp":1760035308000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1481"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,8]]},"references-count":30,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["sym17091481"],"URL":"https:\/\/doi.org\/10.3390\/sym17091481","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,8]]}}}