# American Institute of Mathematical Sciences

doi: 10.3934/eect.2021040
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## New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay

 Vietnam National University-HCMC, Department of Mathematics, International University, Thu Duc city, Ho Chi Minh city, Vietnam

Corresponding author: Pham Huu Anh Ngoc

Received  November 2020 Revised  May 2021 Early access August 2021

Fund Project: This work was supported by the Vietnam National University, Ho Chi Minh city under Grant B 2021-28-01/HD-KHCN

Stochastic functional differential equations with infinite delay are considered. A novel approach to exponential stability of such equations is proposed. New criteria for the mean square exponential stability of general stochastic functional differential equations with infinite delay are presented. Illustrative examples are given.

Citation: Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations & Control Theory, doi: 10.3934/eect.2021040
##### References:
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Ngoc, Explicit criteria for mean square exponential stability of stochastic differential equations, Applied Mathematics Letters, 93 (2019), 22-28.  doi: 10.1016/j.aml.2019.01.014.  Google Scholar [10] P. H. A. Ngoc and T. T. Anh, Stability of nonlinear Volterra equations and applications, Applied Mathematics and Computation, 341 (2019), 1-14.  doi: 10.1016/j.amc.2018.07.027.  Google Scholar [11] P. H. A. Ngoc, Novel criteria for exponential stability in mean square of stochastic functional differential equations, Proceedings of the American Mathematical Society, 148 (2020), 3427-3436.  doi: 10.1090/proc/14994.  Google Scholar [12] P. H. A. Ngoc and L. T. Hieu, A novel approach to mean square exponential stability of stochastic delay differential equations, IEEE Transactions on Automatic Control, 66 (2021), 2351-2356.  doi: 10.1109/TAC.2020.3005587.  Google Scholar [13] G. Pavlovic and S. 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Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 331 (2007), 516-531.  doi: 10.1016/j.jmaa.2006.09.020.  Google Scholar [18] F. Wu and S. Hu, Razumikhin-type theorems on general decay stability and robustness for stochastic functional differential equations, International Journal of Robust and Nonlinear Control, 22 (2012), 763-777.  doi: 10.1002/rnc.1726.  Google Scholar [19] X. Zhao and F. Deng, New-type stability theorem for stochastic functional differential equations with application to SFDSs with distributed delays, International Journal of Systems Science, 45 (2014), 1156-1169.  doi: 10.1080/00207721.2012.745028.  Google Scholar [20] X. Zhao and F. Deng, New type of stability criteria for stochastic functional differential equations via Lyapunov functions, SIAM J. Control Optim., 52 (2014), 2319-2347.  doi: 10.1137/130948203.  Google Scholar [21] X. Zong, G. Yin, L. Y. Wang, T. Li and J. Zhang, Stability of stochastic functional differential systems using degenerate Lyapunov functionals and applications, Automatica J. IFAC, 91 (2018), 197-207.  doi: 10.1016/j.automatica.2018.01.038.  Google Scholar [22] S. Zhou, Z. Wang and D. Feng, Stochastic functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 357 (2009), 416-426.  doi: 10.1016/j.jmaa.2009.04.015.  Google Scholar

show all references

##### References:
 [1] P. Guo and C. J. Li, Razumikhin-type theorems on the moment stability of the exact and numerical solutions for the stochastic pantograph differential equations, Journal of Computational and Applied Mathematics, 355 (2019), 77-909.  doi: 10.1016/j.cam.2019.01.011.  Google Scholar [2] J. Kato, Stability problem in functional differential equations with infinite delay, Funkcialaj Ekvacioj, 21 (1978), 63-80.   Google Scholar [3] R. Khasminskii, Stochastic Stability of Differential Equations, With contributions by G. N. Milstein and M. B. Nevelson. Completely revised and enlarged second edition. Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar [4] J. Lei and M. C. Mackey, Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system, SIAM Journal on Applied Mathematics, 67 (2006/07), 387-407.  doi: 10.1137/060650234.  Google Scholar [5] X. Li and X. Fu, Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks, Journal of Computational and Applied Mathematics, 234 (2010), 407-417.  doi: 10.1016/j.cam.2009.12.033.  Google Scholar [6] Q. Li, Q. Zhang and B. Cao, Mean-square stability of stochastic age-dependent delay population systems with jumps, Acta Mathematicae Applicatae Sinica, English Series, 34 (2018), 145-154.  doi: 10.1007/s10255-018-0732-3.  Google Scholar [7] X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Process. Appl., 65 (1996), 233-250.  doi: 10.1016/S0304-4149(96)00109-3.  Google Scholar [8] X. Mao, Stochastic Differential Equations and Applications, Elsevier, New York, 2007. Google Scholar [9] P. H. A. Ngoc, Explicit criteria for mean square exponential stability of stochastic differential equations, Applied Mathematics Letters, 93 (2019), 22-28.  doi: 10.1016/j.aml.2019.01.014.  Google Scholar [10] P. H. A. Ngoc and T. T. Anh, Stability of nonlinear Volterra equations and applications, Applied Mathematics and Computation, 341 (2019), 1-14.  doi: 10.1016/j.amc.2018.07.027.  Google Scholar [11] P. H. A. Ngoc, Novel criteria for exponential stability in mean square of stochastic functional differential equations, Proceedings of the American Mathematical Society, 148 (2020), 3427-3436.  doi: 10.1090/proc/14994.  Google Scholar [12] P. H. A. Ngoc and L. T. Hieu, A novel approach to mean square exponential stability of stochastic delay differential equations, IEEE Transactions on Automatic Control, 66 (2021), 2351-2356.  doi: 10.1109/TAC.2020.3005587.  Google Scholar [13] G. Pavlovic and S. Jankovi, Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay, Journal of Computational and Applied Mathematics, 236 (2012), 1679-1690.  doi: 10.1016/j.cam.2011.09.045.  Google Scholar [14] W. Rudin, Real and Complex Analysis, 2$^nd$ edition. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.  Google Scholar [15] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Cham, 2013. doi: 10.1007/978-3-319-00101-2.  Google Scholar [16] Z. Yang, E. Zhu, Y. Xu and Y. Tan, Razumikhin-type theorems on exponential stability of stochastic functional differential equations with infinite delay, Acta Appl. Math., 111 (2010), 219-231.  doi: 10.1007/s10440-009-9542-1.  Google Scholar [17] F. Wei and K. Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 331 (2007), 516-531.  doi: 10.1016/j.jmaa.2006.09.020.  Google Scholar [18] F. Wu and S. Hu, Razumikhin-type theorems on general decay stability and robustness for stochastic functional differential equations, International Journal of Robust and Nonlinear Control, 22 (2012), 763-777.  doi: 10.1002/rnc.1726.  Google Scholar [19] X. Zhao and F. Deng, New-type stability theorem for stochastic functional differential equations with application to SFDSs with distributed delays, International Journal of Systems Science, 45 (2014), 1156-1169.  doi: 10.1080/00207721.2012.745028.  Google Scholar [20] X. Zhao and F. Deng, New type of stability criteria for stochastic functional differential equations via Lyapunov functions, SIAM J. Control Optim., 52 (2014), 2319-2347.  doi: 10.1137/130948203.  Google Scholar [21] X. Zong, G. Yin, L. Y. Wang, T. Li and J. Zhang, Stability of stochastic functional differential systems using degenerate Lyapunov functionals and applications, Automatica J. IFAC, 91 (2018), 197-207.  doi: 10.1016/j.automatica.2018.01.038.  Google Scholar [22] S. Zhou, Z. Wang and D. Feng, Stochastic functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 357 (2009), 416-426.  doi: 10.1016/j.jmaa.2009.04.015.  Google Scholar
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