American Institute of Mathematical Sciences

Advanced
doi: 10.3934/mcrf.2020017

Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations

Ran Dong 1,, and  Xuerong Mao 2,

1. 

Warwick Manufacturing Group, University of Warwick, Coventry, CV4 7AL, UK
2. 

Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, UK

* Corresponding author: Ran Dong

Received  July 2019 Revised  November 2019 Published  December 2019

Fund Project: The first author was partially supported by the PhD studentship of the University of Strathclyde. The second author is partially supported by the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship) and the EPSRC (EP/K503174/1)

In 2013, Mao initiated the study of stabilization of continuous-time hybrid stochastic differential equations (SDEs) by feedback control based on discrete-time state observations. In recent years, this study has been further developed while using a constant observation interval. However, time-varying observation frequencies have not been discussed for this study. Particularly for non-autonomous periodic systems, it's more sensible to consider the time-varying property and observe the system at periodic time-varying frequencies, in terms of control efficiency. This paper introduces a periodic observation interval sequence, and investigates how to stabilize a periodic SDE by feedback control based on periodic observations, in the sense that, the controlled system achieves $ L^p $-stability for $ p>1 $, almost sure asymptotic stability and $ p $th moment asymptotic stability for $ p \ge 2 $. This paper uses the Lyapunov method and inequalities to derive the theory. We also verify the existence of the observation interval sequence and explain how to calculate it. Finally, an illustrative example is given after a useful corollary. By considering the time-varying property of the system, we reduce the observation frequency dramatically and hence reduce the observational cost for control.

Keywords: Stochastic differential equations, periodic stochastic systems, markovian switching, asymptotic stability, feedback control, periodic discrete-time observations.
Mathematics Subject Classification: Primary: 93D15, 93E03; Secondary: 93D20.
Citation: Ran Dong, Xuerong Mao. Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020017
References:
[1]

L. Arnold and C. Tudor, Stationary and almost periodic solutions of almost periodic affine stochastic differential equations, Stochastics and Stochastic Reports, 64 (1998), 177-193.  doi: 10.1080/17442509808834163.  Google Scholar

[2]

G. K. Basak, A. Bisi and M. K. Ghosh, Stability of a random diffusion with linear drift, J. Math. Anal. Appl., 202 (1996), 604-622.  doi: 10.1006/jmaa.1996.0336.  Google Scholar

[3]

P. H. Bezandry and T. Diagana, Almost Periodic Stochastic Processes, Springer, New York, 2011. doi: 10.1007/978-1-4419-9476-9.  Google Scholar

[4]

R. Dong, Stabilization of Stochastic Differential Equations by Feedback Controls Based on Discrete-time Observations, PhD thesis, University of Strathclyde, UK, 2019. Google Scholar

[5]

R. Dong, Almost sure exponential stabilization by stochastic feedback control based on discrete-time observations, Stochastic Analysis and Applications, 36 (2018), 561-583.  doi: 10.1080/07362994.2018.1433046.  Google Scholar

[6]

R. Dong and X. R. Mao, On $p$th moment stabilization of hybrid systems by discrete-time feedback control, Stochastic Analysis and Applications, 35 (2017), 803-822.  doi: 10.1080/07362994.2017.1324798.  Google Scholar

[7]

L. Y. Hu, Y. Ren and T. B. Xu, $p$-Moment stability of solutions to stochastic differential equations driven by $G$-Brownian motion, Applied Mathematics and Computation, 230 (2014), 231-237.  doi: 10.1016/j.amc.2013.12.111.  Google Scholar

[8]

C. X. Huang, Y. G. He, L. H. Huang and W. J. Zhu, $p$th moment stability analysis of stochastic recurrent neural networks with time-varying delays, Information Sciences, 178 (2008), 2194-2203.  doi: 10.1016/j.ins.2008.01.008.  Google Scholar

[9]

Y. D. Ji and H. J. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Transactions on Automatic Control, 35 (1990), 777-788.  doi: 10.1109/9.57016.  Google Scholar

[10]

Y. Y. Li, J. Q. Lu, C. H. Kou, X. R. Mao and J. F. Pan, Robust stabilization of hybrid uncertain stochastic systems by discrete-time feedback control, Optimal Control Applications and Methods, 38 (2017), 847-859.  doi: 10.1002/oca.2293.  Google Scholar

[11]

X. Y. Li and X. R. Mao, A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica J. IFAC, 48 (2012), 2329-2334.  doi: 10.1016/j.automatica.2012.06.045.  Google Scholar

[12]

J. Q. Lu, Y. Y. Li, X. R. Mao and Q. W. Qiu, Stabilization of hybrid systems by feedback control based on discrete-time state and mode observations, Asian Journal of Control, 19 (2017), 1943-1953.  doi: 10.1002/asjc.1515.  Google Scholar

[13]

X. R. Mao, Stability of stochastic differential equations with Markovian switching, Sto. Proc. Their Appl., 79 (1999), 45-67.  doi: 10.1016/S0304-4149(98)00070-2.  Google Scholar

[14]

X. R. Mao, Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Transactions on Automatic Control, 47 (2002), 1604-1612.  doi: 10.1109/TAC.2002.803529.  Google Scholar

[15]

X. R. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[16]

X. R. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.  Google Scholar

[17]

X. R. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Transactions on Automatic Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696.  Google Scholar

[18]

X. R. Mao, G. G. Yin and C. G. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.  Google Scholar

[19] X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473.  Google Scholar
[20]

X. R. Mao, W. Liu, L. J. Hu, Q. Luo and J. Q. Lu, Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Systems Control Lett., 73 (2014), 88-95.  doi: 10.1016/j.sysconle.2014.08.011.  Google Scholar

[21]

Y. G. Niu, D. W. C. Ho and J. Lam, Robust integral sliding mode control for uncertain stochastic systems with time-varying delay, Automatica J. IFAC, 41 (2005), 873-880.  doi: 10.1016/j.automatica.2004.11.035.  Google Scholar

[22]

R. Rifhat, L. Wang and Z. D. Teng, Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients, Physica A: Statistical Mechanics and its Applications, 481 (2017), 176-190.  doi: 10.1016/j.physa.2017.04.016.  Google Scholar

[23]

J. L. Sabo and D. M. Post, Quantifying periodic, stochastic, and catastrophic environmental variation, Ecological Monographs, 78 (2008), 19-40.  doi: 10.1890/06-1340.1.  Google Scholar

[24]

J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, New Jersey, 1991. Google Scholar

[25]

G. F. Song, B.-C. Zheng and X. R. Mao, Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time observations of state and mode, IET Control Theory Appl., 11 (2017), 301-307.  doi: 10.1049/iet-cta.2016.0635.  Google Scholar

[26]

M. H. Sun, J. Lam, S. Y. Xu and Y. Zou, Robust exponential stabilization for Markovian jump systems with mode-dependent input delay, Automatica J. IFAC, 43 (2007), 1799-1807.  doi: 10.1016/j.automatica.2007.03.005.  Google Scholar

[27]

I. Tsiakas, Periodic stochastic volatility and fat tails, Journal of Financial Econometrics, 4 (2006), 90-135.  doi: 10.1093/jjfinec/nbi023.  Google Scholar

[28]

C. Wang and R. P. Agarwal, Almost periodic solution for a new type of neutral impulsive stochastic Lasota-Wazewska timescale model, Applied Mathematics Letters, 70 (2017), 58-65.  doi: 10.1016/j.aml.2017.03.009.  Google Scholar

[29]

C. Wang, R. P. Agarwal and S. Rathinasamy, Almost periodic oscillations for delay impulsive stochastic Nicholson's blowflies timescale model, Computational and Applied Mathematics, 37 (2018), 3005-3026.  doi: 10.1007/s40314-017-0495-0.  Google Scholar

[30]

G. C. Wang, Z. Wu and J. Xiong, A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information, IEEE Transactions on Automatic Control, 60 (2015), 2904-2916.  doi: 10.1109/TAC.2015.2411871.  Google Scholar

[31]

Y. Wang and Z. Liu, Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25 (2012), 2803-2821.  doi: 10.1088/0951-7715/25/10/2803.  Google Scholar

[32]

L. G. Wu, P. Shi and H. J. Gao, State estimation and sliding mode control of Markovian jump singular systems, IEEE Transactions on Automatic Control, 55 (2010), 1213-1219.  doi: 10.1109/TAC.2010.2042234.  Google Scholar

[33]

S. R. You, L. J. Hu, W. Mao and X. R. Mao, Robustly exponential stabilization of hybrid uncertain systems by feedback controls based on discrete-time observations, Statist. Probab. Lett., 102 (2015), 8-16.  doi: 10.1016/j.spl.2015.03.006.  Google Scholar

[34]

S. R. You, W. Liu, J. Q. Lu, X. R. Mao and Q. W. 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.  Google Scholar

show all references

References:
[1]

L. Arnold and C. Tudor, Stationary and almost periodic solutions of almost periodic affine stochastic differential equations, Stochastics and Stochastic Reports, 64 (1998), 177-193.  doi: 10.1080/17442509808834163.  Google Scholar

[2]

G. K. Basak, A. Bisi and M. K. Ghosh, Stability of a random diffusion with linear drift, J. Math. Anal. Appl., 202 (1996), 604-622.  doi: 10.1006/jmaa.1996.0336.  Google Scholar

[3]

P. H. Bezandry and T. Diagana, Almost Periodic Stochastic Processes, Springer, New York, 2011. doi: 10.1007/978-1-4419-9476-9.  Google Scholar

[4]

R. Dong, Stabilization of Stochastic Differential Equations by Feedback Controls Based on Discrete-time Observations, PhD thesis, University of Strathclyde, UK, 2019. Google Scholar

[5]

R. Dong, Almost sure exponential stabilization by stochastic feedback control based on discrete-time observations, Stochastic Analysis and Applications, 36 (2018), 561-583.  doi: 10.1080/07362994.2018.1433046.  Google Scholar

[6]

R. Dong and X. R. Mao, On $p$th moment stabilization of hybrid systems by discrete-time feedback control, Stochastic Analysis and Applications, 35 (2017), 803-822.  doi: 10.1080/07362994.2017.1324798.  Google Scholar

[7]

L. Y. Hu, Y. Ren and T. B. Xu, $p$-Moment stability of solutions to stochastic differential equations driven by $G$-Brownian motion, Applied Mathematics and Computation, 230 (2014), 231-237.  doi: 10.1016/j.amc.2013.12.111.  Google Scholar

[8]

C. X. Huang, Y. G. He, L. H. Huang and W. J. Zhu, $p$th moment stability analysis of stochastic recurrent neural networks with time-varying delays, Information Sciences, 178 (2008), 2194-2203.  doi: 10.1016/j.ins.2008.01.008.  Google Scholar

[9]

Y. D. Ji and H. J. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Transactions on Automatic Control, 35 (1990), 777-788.  doi: 10.1109/9.57016.  Google Scholar

[10]

Y. Y. Li, J. Q. Lu, C. H. Kou, X. R. Mao and J. F. Pan, Robust stabilization of hybrid uncertain stochastic systems by discrete-time feedback control, Optimal Control Applications and Methods, 38 (2017), 847-859.  doi: 10.1002/oca.2293.  Google Scholar

[11]

X. Y. Li and X. R. Mao, A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica J. IFAC, 48 (2012), 2329-2334.  doi: 10.1016/j.automatica.2012.06.045.  Google Scholar

[12]

J. Q. Lu, Y. Y. Li, X. R. Mao and Q. W. Qiu, Stabilization of hybrid systems by feedback control based on discrete-time state and mode observations, Asian Journal of Control, 19 (2017), 1943-1953.  doi: 10.1002/asjc.1515.  Google Scholar

[13]

X. R. Mao, Stability of stochastic differential equations with Markovian switching, Sto. Proc. Their Appl., 79 (1999), 45-67.  doi: 10.1016/S0304-4149(98)00070-2.  Google Scholar

[14]

X. R. Mao, Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Transactions on Automatic Control, 47 (2002), 1604-1612.  doi: 10.1109/TAC.2002.803529.  Google Scholar

[15]

X. R. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[16]

X. R. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.  Google Scholar

[17]

X. R. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Transactions on Automatic Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696.  Google Scholar

[18]

X. R. Mao, G. G. Yin and C. G. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.  Google Scholar

[19] X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473.  Google Scholar
[20]

X. R. Mao, W. Liu, L. J. Hu, Q. Luo and J. Q. Lu, Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Systems Control Lett., 73 (2014), 88-95.  doi: 10.1016/j.sysconle.2014.08.011.  Google Scholar

[21]

Y. G. Niu, D. W. C. Ho and J. Lam, Robust integral sliding mode control for uncertain stochastic systems with time-varying delay, Automatica J. IFAC, 41 (2005), 873-880.  doi: 10.1016/j.automatica.2004.11.035.  Google Scholar

[22]

R. Rifhat, L. Wang and Z. D. Teng, Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients, Physica A: Statistical Mechanics and its Applications, 481 (2017), 176-190.  doi: 10.1016/j.physa.2017.04.016.  Google Scholar

[23]

J. L. Sabo and D. M. Post, Quantifying periodic, stochastic, and catastrophic environmental variation, Ecological Monographs, 78 (2008), 19-40.  doi: 10.1890/06-1340.1.  Google Scholar

[24]

J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, New Jersey, 1991. Google Scholar

[25]

G. F. Song, B.-C. Zheng and X. R. Mao, Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time observations of state and mode, IET Control Theory Appl., 11 (2017), 301-307.  doi: 10.1049/iet-cta.2016.0635.  Google Scholar

[26]

M. H. Sun, J. Lam, S. Y. Xu and Y. Zou, Robust exponential stabilization for Markovian jump systems with mode-dependent input delay, Automatica J. IFAC, 43 (2007), 1799-1807.  doi: 10.1016/j.automatica.2007.03.005.  Google Scholar

[27]

I. Tsiakas, Periodic stochastic volatility and fat tails, Journal of Financial Econometrics, 4 (2006), 90-135.  doi: 10.1093/jjfinec/nbi023.  Google Scholar

[28]

C. Wang and R. P. Agarwal, Almost periodic solution for a new type of neutral impulsive stochastic Lasota-Wazewska timescale model, Applied Mathematics Letters, 70 (2017), 58-65.  doi: 10.1016/j.aml.2017.03.009.  Google Scholar

[29]

C. Wang, R. P. Agarwal and S. Rathinasamy, Almost periodic oscillations for delay impulsive stochastic Nicholson's blowflies timescale model, Computational and Applied Mathematics, 37 (2018), 3005-3026.  doi: 10.1007/s40314-017-0495-0.  Google Scholar

[30]

G. C. Wang, Z. Wu and J. Xiong, A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information, IEEE Transactions on Automatic Control, 60 (2015), 2904-2916.  doi: 10.1109/TAC.2015.2411871.  Google Scholar

[31]

Y. Wang and Z. Liu, Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25 (2012), 2803-2821.  doi: 10.1088/0951-7715/25/10/2803.  Google Scholar

[32]

L. G. Wu, P. Shi and H. J. Gao, State estimation and sliding mode control of Markovian jump singular systems, IEEE Transactions on Automatic Control, 55 (2010), 1213-1219.  doi: 10.1109/TAC.2010.2042234.  Google Scholar

[33]

S. R. You, L. J. Hu, W. Mao and X. R. Mao, Robustly exponential stabilization of hybrid uncertain systems by feedback controls based on discrete-time observations, Statist. Probab. Lett., 102 (2015), 8-16.  doi: 10.1016/j.spl.2015.03.006.  Google Scholar

[34]

S. R. You, W. Liu, J. Q. Lu, X. R. Mao and Q. W. 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.  Google Scholar

Figure 1.  Sample averages of $ |x|^2 $ from $ 500 $ simulated paths by the Euler-Maruyama method with step size $ 1e-5 $ and random initial values. Upper plot shows original system (55); lower plot shows controlled system (56) for mean square asymptotically stabilization with corresponding observation frequencies
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Plot of parameters $ K_1(t) $, $ K_2(t) $, $ K_3(t) $ and $ \lambda(t) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Plot of observation intervals. The dashed blue line shows the auxiliary function and the solid orange line is observation interval sequence
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Period partition, observation interval and observation times in each subinterval
Subinterval Observation interval Observation times
[0, 0.5) 0.05556 9
[0.5, 1) 0.1 5
[1, 2.42) 0.142 10
[2.42, 3) 0.19333 3
[3, 4.27) 0.21167 6
[4.27, 5) 0.10429 7
[5, 5.48) 0.06 8
[5.48, 6.37) 0.01745 51
[6.37, 11.28) 0.00164 2988
[11.28, 12) 0.01714 42
Table Options

Download as excel

Subinterval Observation interval Observation times
[0, 0.5) 0.05556 9
[0.5, 1) 0.1 5
[1, 2.42) 0.142 10
[2.42, 3) 0.19333 3
[3, 4.27) 0.21167 6
[4.27, 5) 0.10429 7
[5, 5.48) 0.06 8
[5.48, 6.37) 0.01745 51
[6.37, 11.28) 0.00164 2988
[11.28, 12) 0.01714 42
[1]

Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011
[2]

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
[3]

Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control & Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359
[4]

Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471
[5]

Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025
[6]

Victor Kozyakin. Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3537-3556. doi: 10.3934/dcdsb.2018277
[7]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[8]

Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control & Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019
[9]

Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo, Rohanin Ahmad. An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 109-125. doi: 10.3934/naco.2013.3.109
[10]

Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207
[11]

Sie Long Kek, Mohd Ismail Abd Aziz. Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 275-288. doi: 10.3934/naco.2015.5.275
[12]

Yong He. Switching controls for linear stochastic differential systems. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020005
[13]

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
[14]

Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697
[15]

Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182
[16]

Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253
[17]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005
[18]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
[19]

Sofian De Clercq, Wouter Rogiest, Bart Steyaert, Herwig Bruneel. Stochastic decomposition in discrete-time queues with generalized vacations and applications. Journal of Industrial & Management Optimization, 2012, 8 (4) : 925-938. doi: 10.3934/jimo.2012.8.925
[20]

Qingling Zhang, Guoliang Wang, Wanquan Liu, Yi Zhang. Stabilization of discrete-time Markovian jump systems with partially unknown transition probabilities. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1197-1211. doi: 10.3934/dcdsb.2011.16.1197

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (18)
  • HTML views (33)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences