{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:29:05Z","timestamp":1760059745145,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T00:00:00Z","timestamp":1751846400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Universities of Anhui","award":["2022AH050988","2022YQQ096","2023qyhz07","12271003","DQKJ202406"],"award-info":[{"award-number":["2022AH050988","2022YQQ096","2023qyhz07","12271003","DQKJ202406"]}]},{"name":"Startup Foundation for Introduction Talent of AHPU","award":["2022AH050988","2022YQQ096","2023qyhz07","12271003","DQKJ202406"],"award-info":[{"award-number":["2022AH050988","2022YQQ096","2023qyhz07","12271003","DQKJ202406"]}]},{"name":"Enterprise Cooperation Project of Anhui Institute of Future Technology","award":["2022AH050988","2022YQQ096","2023qyhz07","12271003","DQKJ202406"],"award-info":[{"award-number":["2022AH050988","2022YQQ096","2023qyhz07","12271003","DQKJ202406"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022AH050988","2022YQQ096","2023qyhz07","12271003","DQKJ202406"],"award-info":[{"award-number":["2022AH050988","2022YQQ096","2023qyhz07","12271003","DQKJ202406"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Laboratory of Electric Drive and Control of Anhui Province, Anhui Polytechnic University","award":["2022AH050988","2022YQQ096","2023qyhz07","12271003","DQKJ202406"],"award-info":[{"award-number":["2022AH050988","2022YQQ096","2023qyhz07","12271003","DQKJ202406"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The stability of hybrid stochastic differential equations (SDEs in short) depends on multiple factors, such as the structures and parameters of subsystems, switching rules, delay, etc. Regarding stability analysis for hybrid stochastic systems incorporating subsystems with diverse structures, existing research results require the system to possess either Markovian properties or finite memory characteristics. However, the stability problem remains unresolved for hybrid stochastic differential equations with infinite memory (hybrid IMSDEs in short), as no systematic theoretical framework currently exists for such systems. To bridge this gap, this paper develops a rigorous stability analysis for a class of hybrid IMSDEs by introducing a suitably chosen phase space and leveraging the theory of fading memory spaces. We establish sufficient conditions for exponential stability, extending the existing results to systems with unbounded memory effects. Finally, a numerical example is provided to illustrate the effectiveness of the proposed criteria.<\/jats:p>","DOI":"10.3390\/sym17071077","type":"journal-article","created":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T04:43:37Z","timestamp":1751863417000},"page":"1077","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Structured Stability of Hybrid Stochastic Differential Equations with Superlinear Coefficients and Infinite Memory"],"prefix":"10.3390","volume":"17","author":[{"given":"Chunhui","family":"Mei","sequence":"first","affiliation":[{"name":"School of Mathematics\u2013Physics and Finance, Anhui Polytechnic University, Wuhu 241000, China"},{"name":"The Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, Wuhu 241000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2197-7289","authenticated-orcid":false,"given":"Mingxuan","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Mathematics\u2013Physics and Finance, Anhui Polytechnic University, Wuhu 241000, China"},{"name":"The Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, Wuhu 241000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4153","DOI":"10.1007\/s00009-016-0737-1","article-title":"Existence and dependence results for semilinear functional stochastic differential equations with infinite delay in a Hilbert space","volume":"13","author":"Guedda","year":"2016","journal-title":"Mediterr. J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"105435","DOI":"10.1016\/j.sysconle.2022.105435","article-title":"Stabilisation with general decay rate by delay feedback control for nonlinear neutral stochastic functional differential equations with infinite delay","volume":"172","author":"Shi","year":"2023","journal-title":"Syst. Control Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1118","DOI":"10.3934\/cpaa.2025026","article-title":"Asymptotic Bismut formula for Lions derivative of McKean-Vlasov neutral stochastic differential equations with infinite memory","volume":"24","author":"Wang","year":"2025","journal-title":"Commun. Pure Appl. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"109541","DOI":"10.1016\/j.aml.2025.109541","article-title":"On the decay rate for a stochastic delay differential equation with an unbounded delay","volume":"166","author":"Yao","year":"2025","journal-title":"Appl. Math. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.matcom.2020.10.002","article-title":"Dynamics of a stochastic Markovian switching predator\u2013prey model with infinite memory and general L\u00e9vy jumps","volume":"181","author":"Lu","year":"2021","journal-title":"Math. Comput. Simul."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Lu, B., Zhu, Q., and He, P. (2022). Exponential stability of highly nonlinear hybrid differently structured neutral stochastic differential equations with unbounded delays. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6070385"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1007\/s10255-024-1078-7","article-title":"On \u03b2-extinction and stability of a stochastic Lotka-Volterra system with infinite delay","volume":"40","author":"Zhao","year":"2024","journal-title":"Acta Math. Appl. Sin."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Liu, L., Zhang, Y., Tian, Y., Wei, D., and Huang, Z. (2025). Sliding mode control for stochastic SIR models with telegraph and L\u00e9vy noise: Theory and applications. Symmetry, 17.","DOI":"10.3390\/sym17060963"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2319","DOI":"10.1109\/TAC.2013.2256014","article-title":"Robust stability and boundedness of nonlinear hybrid stochastic differential delay equations","volume":"58","author":"Hu","year":"2013","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2662","DOI":"10.1137\/17M1146981","article-title":"Structured robust stability and boundedness of nonlinear hybrid delay systems","volume":"56","author":"Fei","year":"2018","journal-title":"SIAM J. Control Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1080\/21642583.2019.1651679","article-title":"Analysis on structured stability of highly nonlinear pantograph stochastic differential equations","volume":"7","author":"Shen","year":"2019","journal-title":"Syst. Sci. Control Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2283","DOI":"10.1016\/j.jfranklin.2022.01.023","article-title":"Exponential stability of highly nonlinear hybrid NSDEs with multiple time-Dependent delays and different structures and the Euler-Maruyama method","volume":"359","author":"Lu","year":"2022","journal-title":"J. Frankl. Inst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"111409","DOI":"10.1016\/j.automatica.2023.111409","article-title":"Structured stabilisation of superlinear delay systems by bounded discrete-time state feedback control","volume":"159","author":"Xu","year":"2024","journal-title":"Automatica"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"101198","DOI":"10.1016\/j.nahs.2022.101198","article-title":"Stabilisation of hybrid system with different structures by feedback control based on discrete-time state observations","volume":"45","author":"Shi","year":"2022","journal-title":"Nonlinear Anal. Hybrid Syst."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hino, Y., Murakami, S., and Naito, T. (1991). Functional Differential Equations with Infinite Delay, Springer.","DOI":"10.1007\/BFb0084432"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"101026","DOI":"10.1016\/j.nahs.2021.101026","article-title":"Exponential stabilization by delay feedback control for highly nonlinear hybrid stochastic functional differential equations with infinite delay","volume":"40","author":"Mei","year":"2021","journal-title":"Nonlinear Anal. Hybrid Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.jde.2023.07.028","article-title":"Hybrid stochastic functional differential equations with infinite delay: Approximations and numerics","volume":"374","author":"Li","year":"2023","journal-title":"J. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1007\/s12555-022-0579-5","article-title":"Delay feedback control based on discrete-time state observations for highly nonlinear hybrid stochastic functional differential equations with infinite delay","volume":"22","author":"Zhao","year":"2024","journal-title":"Int. J. Control. Autom. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1226","DOI":"10.1016\/j.jde.2016.10.006","article-title":"Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity","volume":"262","author":"Wu","year":"2017","journal-title":"J. Differ. Equ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1080\/15326349.2024.2398514","article-title":"Numerical solutions of regime-switching functional diffusions with infinite delay","volume":"40","author":"Zhen","year":"2024","journal-title":"Stoch. Model."},{"key":"ref_21","unstructured":"Kallenberg, O. (2006). Foundations of Modern Probability, Springer."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2577","DOI":"10.1016\/j.automatica.2009.07.007","article-title":"Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay","volume":"45","author":"Hu","year":"2009","journal-title":"Automatica"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","unstructured":"Mao, X. (2007). Stochastic Differential Equations and Applications, Horwood Publishing. [2nd ed.].","DOI":"10.1533\/9780857099402"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Zhang, M., Zhu, Q., and Tian, J. (2024). Convergence rates of partial truncated numerical algorithm for stochastic age-dependent cooperative Lotka\u2013Volterra system. Symmetry, 16.","DOI":"10.3390\/sym16121659"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/7\/1077\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:05:38Z","timestamp":1760033138000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/7\/1077"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,7]]},"references-count":25,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["sym17071077"],"URL":"https:\/\/doi.org\/10.3390\/sym17071077","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,7,7]]}}}