American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A backscattering model based on corrector theory of homogenization for the random Helmholtz equation
  • DCDS-B Home
  • This Issue
  • Next Article
    Numerical solution of partial differential equations with stochastic Neumann boundary conditions
October  2019, 24(10): 5355-5375. doi: 10.3934/dcdsb.2019062

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

Tian Zhang 1, , Huabin Chen 1, , Chenggui Yuan 2, and  Tomás Caraballo 3,

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.

Keywords: Stochastic differential equations, time-varying delay, asymptotic behavior, stability, Markov switching.
Mathematics Subject Classification: 60H10, 34K20, 34K50.
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, 2019, 24 (10) : 5355-5375. 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.  Google Scholar

[2]

J. Bao, X. Huang and C. Yuan, Convergence Rate of Euler-Maruyama Scheme for SDEs with Rough Coefficients, arXiv: 1609.06080. Google Scholar

[3]

J. Bao, X. Huang and C. Yuan, Approximation of SPDEs with Hölder Continuous Drifts, arXiv: 1706.05638. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. S. Lipster and A. N. Shiryayev, Theory and Martingale, Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-009-2438-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, IEEE Trans. Automatic Control, 54 (2009), 147-152.   Google Scholar

[16] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London U. K., 2006.  doi: 10.1142/p473.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

H.-L. Ngo and D. Taguchi, On the Euler-Maruyama approximation for one dimensional stochastic differential equations with irregular coefficients, arXiv: 1509.06532. Google Scholar

[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.  Google Scholar

[24]

B.-L. Nikolaos and M. Krstić, Nonlinear Control under Nonconstant Delays, SIAM, U.S., 2013. doi: 10.1137/1.9781611972856.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. Bao, X. Huang and C. Yuan, Convergence Rate of Euler-Maruyama Scheme for SDEs with Rough Coefficients, arXiv: 1609.06080. Google Scholar

[3]

J. Bao, X. Huang and C. Yuan, Approximation of SPDEs with Hölder Continuous Drifts, arXiv: 1706.05638. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. S. Lipster and A. N. Shiryayev, Theory and Martingale, Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-009-2438-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, IEEE Trans. Automatic Control, 54 (2009), 147-152.   Google Scholar

[16] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London U. K., 2006.  doi: 10.1142/p473.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

H.-L. Ngo and D. Taguchi, On the Euler-Maruyama approximation for one dimensional stochastic differential equations with irregular coefficients, arXiv: 1509.06532. Google Scholar

[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.  Google Scholar

[24]

B.-L. Nikolaos and M. Krstić, Nonlinear Control under Nonconstant Delays, SIAM, U.S., 2013. doi: 10.1137/1.9781611972856.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Asymptotic behavior in mean square of the global solution for Eq. (4.1)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Asymptotic behavior in almost sure sense of the global solution for Eq. (4.1)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Asymptotic behavior in mean square of the global solution for Eq. (4.4)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Asymptotic behavior in almost sure sense of the global solution for Eq. (4.4)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Yangzi Hu, Fuke Wu. The improved results on the stochastic Kolmogorov system with time-varying delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1481-1497. doi: 10.3934/dcdsb.2015.20.1481
[2]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693
[3]

Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559
[4]

Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272
[5]

Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263
[6]

Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure & Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667
[7]

Xin-Guang Yang, Jing Zhang, Shu Wang. Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1493-1515. doi: 10.3934/dcds.2020084
[8]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[9]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[10]

Di Wu, Yanqin Bai, Fusheng Xie. Time-scaling transformation for optimal control problem with time-varying delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1683-1695. doi: 10.3934/dcdss.2020098
[11]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[12]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
[13]

Yaozhong Hu, David Nualart, Xiaobin Sun, Yingchao Xie. Smoothness of density for stochastic differential equations with Markovian switching. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3615-3631. doi: 10.3934/dcdsb.2018307
[14]

Dinh Cong Huong, Mai Viet Thuan. State transformations of time-varying delay systems and their applications to state observer design. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 413-444. doi: 10.3934/dcdss.2017020
[15]

Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024
[16]

K. Aruna Sakthi, A. Vinodkumar. Stabilization on input time-varying delay for linear switched systems with truncated predictor control. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 237-247. doi: 10.3934/naco.2019050
[17]

Ruoxia Li, Huaiqin Wu, Xiaowei Zhang, Rong Yao. Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation. Mathematical Control & Related Fields, 2015, 5 (4) : 827-844. doi: 10.3934/mcrf.2015.5.827
[18]

Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043
[19]

Wei Feng, Xin Lu. Global stability in a class of reaction-diffusion systems with time-varying delays. Conference Publications, 1998, 1998 (Special) : 253-261. doi: 10.3934/proc.1998.1998.253
[20]

Lizhao Yan, Fei Xu, Yongzeng Lai, Mingyong Lai. Stability strategies of manufacturing-inventory systems with unknown time-varying demand. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2033-2047. doi: 10.3934/jimo.2017030

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (164)
  • HTML views (345)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences