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July  2022, 27(7): 3725-3747. doi: 10.3934/dcdsb.2021204

Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author: Tian Zhang

Received  January 2021 Revised  June 2021 Published  July 2022 Early access  August 2021

Fund Project: This work was supported by the National Natural Science Foundation of China under Grant No. 12071428 and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ20A010002

This paper focuses on the $ p $th moment and almost sure stability with general decay rate (including exponential decay, polynomial decay, and logarithmic decay) of highly nonlinear hybrid neutral stochastic pantograph differential equations driven by L$ \acute{e} $vy noise (NSPDEs-LN). The crucial techniques used are the Lyapunov functions and the nonnegative semi-martingale convergence theorem. Simultaneously, the diffusion operators are permitted to be controlled by several additional functions with time-varying coefficients, which can be applied to a broad class of the non-autonomous hybrid NSPDEs-LN with highly nonlinear coefficients. Besides, $ H_\infty $ stability and the almost sure asymptotic stability are also concerned. Finally, two examples are offered to illustrate the validity of the obtained theory.

Citation: Tian Zhang, Chuanhou Gao. Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3725-3747. doi: 10.3934/dcdsb.2021204
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J. A. D. Appleby and E. Buckwar, Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation, Proceedings of the 10'th Colloquium on the Qualitative Theory of Differential Equations, 2 (2016), 32 pp. doi: 10.14232/ejqtde.2016.8.2.

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T. CaraballoM. J. Garrido-Atinenza and J. Real, Stochastic stabilization of differential systems with general decay rate, Syst. Control Lett., 48 (2003), 397-406.  doi: 10.1016/S0167-6911(02)00293-1.

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H. Chen and C. Yuan, On the asymptotic behavior for neutral stochastic differential delay equations, IEEE Trans. Automat. Control, 64 (2019), 1671-1678.  doi: 10.1109/TAC.2018.2852607.

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Z. FanM. Liu and W. Cao, Existence and uniqueness of the solutions and convergence of semi-implicit euler methods for stochastic pantograph equations, J. Math. Anal. Appl., 325 (2007), 1142-1159.  doi: 10.1016/j.jmaa.2006.02.063.

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Z. FanM. Song and M. Liu, The $\alpha$th moment stability for the stochastic pantograph equation, J. Comput. Appl. Math., 233 (2009), 109-120.  doi: 10.1016/j.cam.2009.04.024.

[6]

P. Guo and C.-J. Li, Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations, BIT, 59 (2019), 77-96.  doi: 10.1007/s10543-018-0723-z.

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P. Guo and C.-J. Li, Almost sure stability with general decay rate of exact and numerical solutions for stochastic pantograph differential equations, Numer. Algorithms, 80 (2019), 1391-1411.  doi: 10.1007/s11075-018-0531-1.

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Q. GuoX. Mao and R. Yue, Almost sure exponential stability of stochastic differential delay equations, SIAM J. Control Optim., 54 (2016), 1919-1933.  doi: 10.1137/15M1019465.

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J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7.

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L. HuX. Mao and Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Syst. Control Lett., 62 (2013), 178-187.  doi: 10.1016/j.sysconle.2012.11.009.

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L. Hu, Y. Ren and Q. He, Pantograph stochastic differential equations driven by $G$-Brownian motion, J. Math. Anal. Appl., 480 (2019), 123381, 11 pp. doi: 10.1016/j.jmaa.2019.123381.

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B. LiD. Li and D. Xu, Stability analysis for impulsive stochastic delay differential equations with Markovian switching, J. Frankl. Institut., 350 (2013), 1848-1864.  doi: 10.1016/j.jfranklin.2013.05.009.

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M. Li and F. Deng, Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with L$\acute{e}$vy noise, Nonlinear Anal. Hybrid Syst., 24 (2017), 171-185.  doi: 10.1016/j.nahs.2017.01.001.

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J. Liu, Z.-Y. Li and F. Deng, Asymptotic behavior analysis of Markovian switching neutral-type stochastic time-delay systems, Appl. Math. Comput., 404 (2021), 126205, 14 pp. doi: 10.1016/j.amc.2021.126205.

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L. Liu and F. Deng, $p$th moment exponential stability of highly nonlinear neutral pantograph stochastic differential equations driven by L$\acute{e}$vy noise, Appl. Math. Lett., 86 (2018), 313-319.  doi: 10.1016/j.aml.2018.07.003.

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[22]

J. Luo, A note on exponential stability in $p$th mean of solutions of stochastic delay differential equations, J. Computat. Appl. Math., 198 (2007), 143-148.  doi: 10.1016/j.cam.2005.11.019.

[23]

W. Mao, L. Hu and X. Mao, Almost sure stability with general decay rate of neutral stochastic pantograph equations with Markovian switching, Electron. J. Qual. Theory Differ. Equ., (2019), Paper No. 52, 17 pp. doi: 10.14232/ejqtde.2019.1.52.

[24]

W. MaoL. Hu and X. Mao, The asymptotic stability of hybrid stochastic systems with pantograph delay and non-Gaussian L$\acute{e}$vy noise, J. Frankl. Inst., 357 (2020), 1174-1198.  doi: 10.1016/j.jfranklin.2019.11.068.

[25]

X. Mao, Robustness of exponential stability of stochastic differential delay equations, IEEE Trans. Automat. Control, 41 (1996), 442-447.  doi: 10.1109/9.486647.

[26]

X. MaoY. Shen and C. Yuan, Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stochastic Process. Appl., 118 (2008), 1385-1406.  doi: 10.1016/j.spa.2007.09.005.

[27] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Swithcing, Imperial College Press, London, UK, 2006.  doi: 10.1142/p473.
[28]

M. Milo$\check{s}$evi$\acute{c}$ and M. Jovanovi$\acute{c}$, A Taylor polynomial approach in approximations of solution to pantograph stochastic differential equations with Markovian switching, Math. Comput. Model., 53 (2011), 280-293.  doi: 10.1016/j.mcm.2010.08.016.

[29]

G. Pavlovi$\acute{c}$ and S. Jankovi$\acute{c}$, The Razumikhin approach on general decay stability for neutral stochastic functional differential equations, J. Frankl. Institut., 350 (2013), 2124-2145.  doi: 10.1016/j.jfranklin.2013.05.025.

[30]

G. Shen, W. Xu and D. Zhu, The stability with general decay rate of neutral stochastic functional hybrid differential equations with L$\acute{e}$vy noise, Syst. Control Lett., 143 (2020), 104742, 9 pp. doi: 10.1016/j.sysconle.2020.104742.

[31]

M. ShenW. FeiX. Mao and S. Deng, Exponential stability of highly nonlinear neutral pantograph stochastic differential equations, Asian J. Control, 22 (2020), 436-448.  doi: 10.1002/asjc.1903.

[32]

A. Wu, S. You, W. Mao, X. Mao and L. Hu, On exponential stability of hybrid neutral stochastic differential delay equations with different structures, Nonlinear Anal. Hybrid Syst., 39 (2021), 100971, 17 pp. doi: 10.1016/j.nahs.2020.100971.

[33]

F. Wu and S. Hu, Razumikhin-type theorems on general decay stability and robustness for stochastic functional differential equations, Int. J. Robust Nonlinear Control, 22 (2012), 763-777.  doi: 10.1002/rnc.1726.

[34]

F. WuS. Hu and C. Huang, Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay, Syst. Control Lett., 59 (2010), 195-202.  doi: 10.1016/j.sysconle.2010.01.004.

[35]

F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1715-1734.  doi: 10.3934/dcdsb.2013.18.1715.

[36]

Y. Xiao and H. Zhang, Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations, Int. J. Comput. Math., 88 (2011), 2955-2968.  doi: 10.1080/00207160.2011.563843.

[37]

L. Xu and H. Hu, Boundedness analysis of stochastic pantograph differential systems, Appl. Math. Lett., 111 (2021), 106630, 7 pp. doi: 10.1016/j.aml.2020.106630.

[38]

H. YangZ. YangP. Wang and D. Han, Mean-square stability analysis for nonlinear stochastic pantograph equations by transformation approach, J. Math. Anal. Appl., 479 (2019), 977-986.  doi: 10.1016/j.jmaa.2019.06.061.

[39]

S. YouW. LiuJ. LuX. Mao and Q. Qiu, 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.

[40]

S. YouW. MaoX. Mao and L. Hu, Analysis on exponential stability of highly pantograph stochastic differential equations with highly nonlinear coefficients, Appl. Math. Comput., 263 (2015), 73-83.  doi: 10.1016/j.amc.2015.04.022.

[41]

H. Yuan and C. Song, Convergence and stability of exponential integrators for semi-linear stochastic pantograph integro-differential equations with jump, Chaos, Solitons and Fractals, 140 (2020), 110172, 18 pp. doi: 10.1016/j.chaos.2020.110172.

[42]

H. ZhangY. Xiao and F. Guo, Convergence and stability of a numerical method for nonlinear stochastic pantograph equations, J. Frankl. Inst., 351 (2014), 3089-3103.  doi: 10.1016/j.jfranklin.2014.02.004.

[43]

T. Zhang and H. Chen, The stability with a general decay of stochastic delay differential equations with Markovian switching, Appl. Math. Comput., 359 (2019), 294-307.  doi: 10.1016/j.amc.2019.04.057.

[44]

T. ZhangH. ChenC. Yuan and T. Caraballo, On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5355-5375.  doi: 10.3934/dcdsb.2019062.

[45]

W. ZhangJ. Ye and H. Li, Stability with general decay rates of stochastic differential delay equations with Poisson jumps and Markovian switching, Statist. Probab. Lett., 92 (2014), 1-11.  doi: 10.1016/j.spl.2014.04.024.

[46]

X. Zhao and F. Deng, Moment stability of nonlinear stochastic systems with time delays based on $\mathcal{H}$-representation technique, IEEE Trans. Automat. Control, 59 (2014), 814-819.  doi: 10.1109/TAC.2013.2279909.

[47]

S. Zhou and Y. Hu, Numerical approximation for nonlinear stochastic pantograph equations with Markovian switching, Appl. Math. Comput., 286 (2016), 126-138.  doi: 10.1016/j.amc.2016.03.040.

[48]

S. Zhou and M. Xue, Exponential stability for nonlinear hybrid stochastic pantograph equations and numerical approximation, Acta Math. Sci. Ser. B (Engl. Ed.), 34 (2014), 1254-1270.  doi: 10.1016/S0252-9602(14)60083-7.

[49]

Q. Zhu and Q. Zhang, $p$th moment exponential stabilisation of hybrid stochastic differential equations by feedback controls based on discrete-time state observations with a time delay, IET Control Theory Appl., 11 (2017), 1992-2003.  doi: 10.1049/iet-cta.2017.0181.

[50]

X. ZongG. YinL. Y. WangT. Li and J.-F. 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.

show all references

References:
[1]

J. A. D. Appleby and E. Buckwar, Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation, Proceedings of the 10'th Colloquium on the Qualitative Theory of Differential Equations, 2 (2016), 32 pp. doi: 10.14232/ejqtde.2016.8.2.

[2]

T. CaraballoM. J. Garrido-Atinenza and J. Real, Stochastic stabilization of differential systems with general decay rate, Syst. Control Lett., 48 (2003), 397-406.  doi: 10.1016/S0167-6911(02)00293-1.

[3]

H. Chen and C. Yuan, On the asymptotic behavior for neutral stochastic differential delay equations, IEEE Trans. Automat. Control, 64 (2019), 1671-1678.  doi: 10.1109/TAC.2018.2852607.

[4]

Z. FanM. Liu and W. Cao, Existence and uniqueness of the solutions and convergence of semi-implicit euler methods for stochastic pantograph equations, J. Math. Anal. Appl., 325 (2007), 1142-1159.  doi: 10.1016/j.jmaa.2006.02.063.

[5]

Z. FanM. Song and M. Liu, The $\alpha$th moment stability for the stochastic pantograph equation, J. Comput. Appl. Math., 233 (2009), 109-120.  doi: 10.1016/j.cam.2009.04.024.

[6]

P. Guo and C.-J. Li, Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations, BIT, 59 (2019), 77-96.  doi: 10.1007/s10543-018-0723-z.

[7]

P. Guo and C.-J. Li, Almost sure stability with general decay rate of exact and numerical solutions for stochastic pantograph differential equations, Numer. Algorithms, 80 (2019), 1391-1411.  doi: 10.1007/s11075-018-0531-1.

[8]

Q. GuoX. Mao and R. Yue, Almost sure exponential stability of stochastic differential delay equations, SIAM J. Control Optim., 54 (2016), 1919-1933.  doi: 10.1137/15M1019465.

[9]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7.

[10]

L. HuX. Mao and Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Syst. Control Lett., 62 (2013), 178-187.  doi: 10.1016/j.sysconle.2012.11.009.

[11]

L. Hu, Y. Ren and Q. He, Pantograph stochastic differential equations driven by $G$-Brownian motion, J. Math. Anal. Appl., 480 (2019), 123381, 11 pp. doi: 10.1016/j.jmaa.2019.123381.

[12]

A. Iserles, On the generalized pantograph functional-differential equation, Eur. J. Appl. Math., 4 (1993), 1-38.  doi: 10.1017/S0956792500000966.

[13]

A. Iserles and J. Terjéki, Stability and asymptotic stability of functional-differential equations, J. Lond. Math. Soc., 51 (1995), 559-572.  doi: 10.1112/jlms/51.3.559.

[14]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer Science & Business Media, 2013.

[15]

B. LiD. Li and D. Xu, Stability analysis for impulsive stochastic delay differential equations with Markovian switching, J. Frankl. Institut., 350 (2013), 1848-1864.  doi: 10.1016/j.jfranklin.2013.05.009.

[16]

M. Li and F. Deng, Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with L$\acute{e}$vy noise, Nonlinear Anal. Hybrid Syst., 24 (2017), 171-185.  doi: 10.1016/j.nahs.2017.01.001.

[17]

R. LiM. Liu and W. Pang, Convergence of numerical solutions to stochastic pantograph equations with Markovian switching, Appl. Math. Comput., 215 (2009), 414-422.  doi: 10.1016/j.amc.2009.05.013.

[18]

J. Liu, Z.-Y. Li and F. Deng, Asymptotic behavior analysis of Markovian switching neutral-type stochastic time-delay systems, Appl. Math. Comput., 404 (2021), 126205, 14 pp. doi: 10.1016/j.amc.2021.126205.

[19]

J. Liu and J. Zhou, Convergence rate of Euler-Maruyama scheme for stochastic pantograph differential equations, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1697-1705.  doi: 10.1016/j.cnsns.2013.10.015.

[20]

L. Liu and F. Deng, $p$th moment exponential stability of highly nonlinear neutral pantograph stochastic differential equations driven by L$\acute{e}$vy noise, Appl. Math. Lett., 86 (2018), 313-319.  doi: 10.1016/j.aml.2018.07.003.

[21]

M. LiuZ. Yang and G. Hu, Asymptotical stability of the numerical methods with the constant step size for the pantograph equation, BIT, 45 (2005), 743-759.  doi: 10.1007/s10543-005-0022-3.

[22]

J. Luo, A note on exponential stability in $p$th mean of solutions of stochastic delay differential equations, J. Computat. Appl. Math., 198 (2007), 143-148.  doi: 10.1016/j.cam.2005.11.019.

[23]

W. Mao, L. Hu and X. Mao, Almost sure stability with general decay rate of neutral stochastic pantograph equations with Markovian switching, Electron. J. Qual. Theory Differ. Equ., (2019), Paper No. 52, 17 pp. doi: 10.14232/ejqtde.2019.1.52.

[24]

W. MaoL. Hu and X. Mao, The asymptotic stability of hybrid stochastic systems with pantograph delay and non-Gaussian L$\acute{e}$vy noise, J. Frankl. Inst., 357 (2020), 1174-1198.  doi: 10.1016/j.jfranklin.2019.11.068.

[25]

X. Mao, Robustness of exponential stability of stochastic differential delay equations, IEEE Trans. Automat. Control, 41 (1996), 442-447.  doi: 10.1109/9.486647.

[26]

X. MaoY. Shen and C. Yuan, Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stochastic Process. Appl., 118 (2008), 1385-1406.  doi: 10.1016/j.spa.2007.09.005.

[27] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Swithcing, Imperial College Press, London, UK, 2006.  doi: 10.1142/p473.
[28]

M. Milo$\check{s}$evi$\acute{c}$ and M. Jovanovi$\acute{c}$, A Taylor polynomial approach in approximations of solution to pantograph stochastic differential equations with Markovian switching, Math. Comput. Model., 53 (2011), 280-293.  doi: 10.1016/j.mcm.2010.08.016.

[29]

G. Pavlovi$\acute{c}$ and S. Jankovi$\acute{c}$, The Razumikhin approach on general decay stability for neutral stochastic functional differential equations, J. Frankl. Institut., 350 (2013), 2124-2145.  doi: 10.1016/j.jfranklin.2013.05.025.

[30]

G. Shen, W. Xu and D. Zhu, The stability with general decay rate of neutral stochastic functional hybrid differential equations with L$\acute{e}$vy noise, Syst. Control Lett., 143 (2020), 104742, 9 pp. doi: 10.1016/j.sysconle.2020.104742.

[31]

M. ShenW. FeiX. Mao and S. Deng, Exponential stability of highly nonlinear neutral pantograph stochastic differential equations, Asian J. Control, 22 (2020), 436-448.  doi: 10.1002/asjc.1903.

[32]

A. Wu, S. You, W. Mao, X. Mao and L. Hu, On exponential stability of hybrid neutral stochastic differential delay equations with different structures, Nonlinear Anal. Hybrid Syst., 39 (2021), 100971, 17 pp. doi: 10.1016/j.nahs.2020.100971.

[33]

F. Wu and S. Hu, Razumikhin-type theorems on general decay stability and robustness for stochastic functional differential equations, Int. J. Robust Nonlinear Control, 22 (2012), 763-777.  doi: 10.1002/rnc.1726.

[34]

F. WuS. Hu and C. Huang, Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay, Syst. Control Lett., 59 (2010), 195-202.  doi: 10.1016/j.sysconle.2010.01.004.

[35]

F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1715-1734.  doi: 10.3934/dcdsb.2013.18.1715.

[36]

Y. Xiao and H. Zhang, Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations, Int. J. Comput. Math., 88 (2011), 2955-2968.  doi: 10.1080/00207160.2011.563843.

[37]

L. Xu and H. Hu, Boundedness analysis of stochastic pantograph differential systems, Appl. Math. Lett., 111 (2021), 106630, 7 pp. doi: 10.1016/j.aml.2020.106630.

[38]

H. YangZ. YangP. Wang and D. Han, Mean-square stability analysis for nonlinear stochastic pantograph equations by transformation approach, J. Math. Anal. Appl., 479 (2019), 977-986.  doi: 10.1016/j.jmaa.2019.06.061.

[39]

S. YouW. LiuJ. LuX. Mao and Q. Qiu, 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.

[40]

S. YouW. MaoX. Mao and L. Hu, Analysis on exponential stability of highly pantograph stochastic differential equations with highly nonlinear coefficients, Appl. Math. Comput., 263 (2015), 73-83.  doi: 10.1016/j.amc.2015.04.022.

[41]

H. Yuan and C. Song, Convergence and stability of exponential integrators for semi-linear stochastic pantograph integro-differential equations with jump, Chaos, Solitons and Fractals, 140 (2020), 110172, 18 pp. doi: 10.1016/j.chaos.2020.110172.

[42]

H. ZhangY. Xiao and F. Guo, Convergence and stability of a numerical method for nonlinear stochastic pantograph equations, J. Frankl. Inst., 351 (2014), 3089-3103.  doi: 10.1016/j.jfranklin.2014.02.004.

[43]

T. Zhang and H. Chen, The stability with a general decay of stochastic delay differential equations with Markovian switching, Appl. Math. Comput., 359 (2019), 294-307.  doi: 10.1016/j.amc.2019.04.057.

[44]

T. ZhangH. ChenC. Yuan and T. Caraballo, On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5355-5375.  doi: 10.3934/dcdsb.2019062.

[45]

W. ZhangJ. Ye and H. Li, Stability with general decay rates of stochastic differential delay equations with Poisson jumps and Markovian switching, Statist. Probab. Lett., 92 (2014), 1-11.  doi: 10.1016/j.spl.2014.04.024.

[46]

X. Zhao and F. Deng, Moment stability of nonlinear stochastic systems with time delays based on $\mathcal{H}$-representation technique, IEEE Trans. Automat. Control, 59 (2014), 814-819.  doi: 10.1109/TAC.2013.2279909.

[47]

S. Zhou and Y. Hu, Numerical approximation for nonlinear stochastic pantograph equations with Markovian switching, Appl. Math. Comput., 286 (2016), 126-138.  doi: 10.1016/j.amc.2016.03.040.

[48]

S. Zhou and M. Xue, Exponential stability for nonlinear hybrid stochastic pantograph equations and numerical approximation, Acta Math. Sci. Ser. B (Engl. Ed.), 34 (2014), 1254-1270.  doi: 10.1016/S0252-9602(14)60083-7.

[49]

Q. Zhu and Q. Zhang, $p$th moment exponential stabilisation of hybrid stochastic differential equations by feedback controls based on discrete-time state observations with a time delay, IET Control Theory Appl., 11 (2017), 1992-2003.  doi: 10.1049/iet-cta.2017.0181.

[50]

X. ZongG. YinL. Y. WangT. Li and J.-F. 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.

Figure 1.  Poisson jump process
Figure 2.  State trajectories of two subsystems
Figure 3.  State trajectory of whole system
Figure 4.  Poisson jump process
Figure 5.  State trajectories of two subsystems
Figure 6.  State trajectory of whole system
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