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## On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations

 1 Department of Mathematics, Nanchang University, Nanchang 330031, China 2 Department of Mathematics, Swansea University, Swansea SA2 8PP, UK 3 Depto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Facultad de Matemáticas, c/ Tarfia s/n, 41012-Sevilla, Spain

Received  May 2018 Revised  October 2018 Published  April 2019

Fund Project: Huabin Chen is partially supported by the National Natural Science Foundation of China (61364005, 11401292, 61773401), the Natural Science Foundation of Jiangxi Province of China (20171BAB201007, 20171BCB23001), and the Foundation of Jiangxi Provincial Educations of China (GJJ160061, GJJ14155) and the National Statistical Science Research Foundation of China (2018LY71). Tomás Caraballo is partially supported by the projects MTM2015-63723-P (MINECO/ FEDER, EU) and P12-FQM-1492 (Junta de Andalucía)

In this paper, the existence and uniqueness, the stability analysis for the global solution of highly nonlinear stochastic differential equations with time-varying delay and Markovian switching are analyzed under a locally Lipschitz condition and a monotonicity condition. In order to overcome a difficulty stemming from the existence of the time-varying delay, one integral lemma is established. It should be mentioned that the time-varying delay is a bounded measurable function. By utilizing the integral inequality, the Lyapunov function and some stochastic analysis techniques, some sufficient conditions are proposed to guarantee the stability in both moment and almost sure senses for such equations, which can be also used to yield the almost surely asymptotic behavior. As a by-product, the exponential stability in $p$th$(p\geq 1)$-moment and the almost sure exponential stability are analyzed. Finally, two examples are given to show the usefulness of the results obtained.

Citation: Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019062
##### References:
 [1] A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, J. Math. Anal. Appl., 292 (2004), 364-380. doi: 10.1016/j.jmaa.2003.12.004. [2] J. Bao, X. Huang and C. Yuan, Convergence Rate of Euler-Maruyama Scheme for SDEs with Rough Coefficients, arXiv: 1609.06080. [3] J. Bao, X. Huang and C. Yuan, Approximation of SPDEs with Hölder Continuous Drifts, arXiv: 1706.05638. [4] H. Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Probability Letters, 80 (2010), 50-56. doi: 10.1016/j.spl.2009.09.011. [5] W. C. H. Daniel and J. Sun, Stability of Takagi-Sugeno Fuzzy delay systems with impulses, IEEE Trans. Fuzzy Syst., 15 (2007), 784-790. [6] W. Fei, L. Hu and X. Mao, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 82 (2017), 165-170. doi: 10.1016/j.automatica.2017.04.050. [7] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7. [8] L. Hu, X. Mao and Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Control Letters, 62 (2013), 178-187. doi: 10.1016/j.sysconle.2012.11.009. [9] L. Hu, X. Mao and L. Zhang, Robust Stability and Boundedness of Nonlinear Hybrid Stochastic Differential Delay Equations, IEEE Trans. Automatic Control, 58 (2013), 2319-2332. doi: 10.1109/TAC.2013.2256014. [10] N. Jacob, Y. Wang and C. Yuan, Stochastic differential delay equations with jumps, under nonlinear growth condition, Stochastics An International Journal of Probability and Stochastic Processes, 81 (2009), 571-588. doi: 10.1080/17442500903251832. [11] R. S. Lipster and A. N. Shiryayev, Theory and Martingale, Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-009-2438-3. [12] J. Luo, J. Zou and Z. Hou, Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching, Science In China (Series A), 46 (2003), 129-138. doi: 10.1360/03ys9014. [13] X. Mao, Robustness of exponential stability of stochastic differential delay equations, IEEE Trans. Automatic Control, 41 (1996), 442-447. doi: 10.1109/9.486647. [14] X. Mao, LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 236 (1999), 350-369. doi: 10.1006/jmaa.1999.6435. [15] X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, IEEE Trans. Automatic Control, 54 (2009), 147-152. [16] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London U. K., 2006. doi: 10.1142/p473. [17] X. Mao, J. Lam and L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Control Letters, 57 (2008), 927-935. doi: 10.1016/j.sysconle.2008.05.002. [18] X. Mao, A. Matasov and A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli, 6 (2000), 73-90. doi: 10.2307/3318634. [19] X. Mao, W. Liu, L. Hu, Q. Luo and J. Lu, Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Control Letters, 73 (2014), 88-95. doi: 10.1016/j.sysconle.2014.08.011. [20] H.-L. Ngo and D. T. Luong, Strong Rate of Tamed Euler-Maruyama Approximation for Stochastic Differential Equations with H$\ddot{o}$lder Continuous Diffusion Coefficients, Brazilian Journal of Probability and Statistics, 31 (2017), 24-40. doi: 10.1214/15-BJPS301. [21] H.-L. Ngo and D. Taguchi, Strong rate of convergence for the Euler-Maruyama approximation of stochastic differential equations with irregular coefficients, Math. Comp., 85 (2016), 1793-1819. doi: 10.1090/mcom3042. [22] H.-L. Ngo and D. Taguchi, On the Euler-Maruyama approximation for one dimensional stochastic differential equations with irregular coefficients, arXiv: 1509.06532. [23] H.-L. Ngo and D. Taguchi, Strong convergence for the Euler-Maruyama approximation of stochastic differential equations with discontinuous coefficients, Statistics and Probability Letters, 125 (2017), 55-63. doi: 10.1016/j.spl.2017.01.027. [24] B.-L. Nikolaos and M. Krstić, Nonlinear Control under Nonconstant Delays, SIAM, U.S., 2013. doi: 10.1137/1.9781611972856. [25] F. Wu, G. Yin and H. Mei, Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity, Journal of Differential Equations, 262 (2017), 1226-1252. doi: 10.1016/j.jde.2016.10.006. [26] F. Wu and S. Hu, Khasmiskii-type theorems for stochastic functional differential equations with infinite delay, Statistics & Probability Letters, 81 (2011), 1690-1694. doi: 10.1016/j.spl.2011.05.005. [27] F. Wu and S. Hu, Attraction, stability and robustness for stochastic functional differential equations with infinite delay, Automatica, 47 (2011), 2224-2232. doi: 10.1016/j.automatica.2011.07.001. [28] S. You, W. Liu, J. Lu, X. Mao and Q. Wei, Stabilization of hybrid systems by feedback control based on discrete-time state observations, SIAM J. Control Optim., 53 (2015), 905-925. doi: 10.1137/140985779. [29] D. Yue and Q. L. Han, Delay-dependent exponential stability of stochastic systems with time-varyin delay, nonlinearity, and Markovian switching, IEEE Trans. Automatic Control, 50 (2005), 217-222. doi: 10.1109/TAC.2004.841935.

show all references

##### References:
 [1] A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, J. Math. Anal. Appl., 292 (2004), 364-380. doi: 10.1016/j.jmaa.2003.12.004. [2] J. Bao, X. Huang and C. Yuan, Convergence Rate of Euler-Maruyama Scheme for SDEs with Rough Coefficients, arXiv: 1609.06080. [3] J. Bao, X. Huang and C. Yuan, Approximation of SPDEs with Hölder Continuous Drifts, arXiv: 1706.05638. [4] H. Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Probability Letters, 80 (2010), 50-56. doi: 10.1016/j.spl.2009.09.011. [5] W. C. H. Daniel and J. Sun, Stability of Takagi-Sugeno Fuzzy delay systems with impulses, IEEE Trans. Fuzzy Syst., 15 (2007), 784-790. [6] W. Fei, L. Hu and X. Mao, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 82 (2017), 165-170. doi: 10.1016/j.automatica.2017.04.050. [7] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7. [8] L. Hu, X. Mao and Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Control Letters, 62 (2013), 178-187. doi: 10.1016/j.sysconle.2012.11.009. [9] L. Hu, X. Mao and L. Zhang, Robust Stability and Boundedness of Nonlinear Hybrid Stochastic Differential Delay Equations, IEEE Trans. Automatic Control, 58 (2013), 2319-2332. doi: 10.1109/TAC.2013.2256014. [10] N. Jacob, Y. Wang and C. Yuan, Stochastic differential delay equations with jumps, under nonlinear growth condition, Stochastics An International Journal of Probability and Stochastic Processes, 81 (2009), 571-588. doi: 10.1080/17442500903251832. [11] R. S. Lipster and A. N. Shiryayev, Theory and Martingale, Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-009-2438-3. [12] J. Luo, J. Zou and Z. Hou, Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching, Science In China (Series A), 46 (2003), 129-138. doi: 10.1360/03ys9014. [13] X. Mao, Robustness of exponential stability of stochastic differential delay equations, IEEE Trans. Automatic Control, 41 (1996), 442-447. doi: 10.1109/9.486647. [14] X. Mao, LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 236 (1999), 350-369. doi: 10.1006/jmaa.1999.6435. [15] X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, IEEE Trans. Automatic Control, 54 (2009), 147-152. [16] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London U. K., 2006. doi: 10.1142/p473. [17] X. Mao, J. Lam and L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Control Letters, 57 (2008), 927-935. doi: 10.1016/j.sysconle.2008.05.002. [18] X. Mao, A. Matasov and A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli, 6 (2000), 73-90. doi: 10.2307/3318634. [19] X. Mao, W. Liu, L. Hu, Q. Luo and J. Lu, Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Control Letters, 73 (2014), 88-95. doi: 10.1016/j.sysconle.2014.08.011. [20] H.-L. Ngo and D. T. Luong, Strong Rate of Tamed Euler-Maruyama Approximation for Stochastic Differential Equations with H$\ddot{o}$lder Continuous Diffusion Coefficients, Brazilian Journal of Probability and Statistics, 31 (2017), 24-40. doi: 10.1214/15-BJPS301. [21] H.-L. Ngo and D. Taguchi, Strong rate of convergence for the Euler-Maruyama approximation of stochastic differential equations with irregular coefficients, Math. Comp., 85 (2016), 1793-1819. doi: 10.1090/mcom3042. [22] H.-L. Ngo and D. Taguchi, On the Euler-Maruyama approximation for one dimensional stochastic differential equations with irregular coefficients, arXiv: 1509.06532. [23] H.-L. Ngo and D. Taguchi, Strong convergence for the Euler-Maruyama approximation of stochastic differential equations with discontinuous coefficients, Statistics and Probability Letters, 125 (2017), 55-63. doi: 10.1016/j.spl.2017.01.027. [24] B.-L. Nikolaos and M. Krstić, Nonlinear Control under Nonconstant Delays, SIAM, U.S., 2013. doi: 10.1137/1.9781611972856. [25] F. Wu, G. Yin and H. Mei, Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity, Journal of Differential Equations, 262 (2017), 1226-1252. doi: 10.1016/j.jde.2016.10.006. [26] F. Wu and S. Hu, Khasmiskii-type theorems for stochastic functional differential equations with infinite delay, Statistics & Probability Letters, 81 (2011), 1690-1694. doi: 10.1016/j.spl.2011.05.005. [27] F. Wu and S. Hu, Attraction, stability and robustness for stochastic functional differential equations with infinite delay, Automatica, 47 (2011), 2224-2232. doi: 10.1016/j.automatica.2011.07.001. [28] S. You, W. Liu, J. Lu, X. Mao and Q. Wei, Stabilization of hybrid systems by feedback control based on discrete-time state observations, SIAM J. Control Optim., 53 (2015), 905-925. doi: 10.1137/140985779. [29] D. Yue and Q. L. Han, Delay-dependent exponential stability of stochastic systems with time-varyin delay, nonlinearity, and Markovian switching, IEEE Trans. Automatic Control, 50 (2005), 217-222. doi: 10.1109/TAC.2004.841935.
Asymptotic behavior in mean square of the global solution for Eq. (4.1)
Asymptotic behavior in almost sure sense of the global solution for Eq. (4.1)
Asymptotic behavior in mean square of the global solution for Eq. (4.4)
Asymptotic behavior in almost sure sense of the global solution for Eq. (4.4)
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