# American Institute of Mathematical Sciences

August  2020, 25(8): 3217-3232. doi: 10.3934/dcdsb.2020059

## Razumikhin-type theorems on polynomial stability of hybrid stochastic systems with pantograph delay

 1 School of mathematics and information technology, Jiangsu Second Normal University, Nanjing 210013, China 2 Department of Applied Mathematics, Donghua University, Shanghai 201620, China 3 Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK

* Corresponding author: Liangjian Hu

Received  July 2019 Revised  September 2019 Published  August 2020 Early access  February 2020

Fund Project: The author Wei Mao is supported by the National Natural Science Foundation of China (11401261) and "333 High-level Personnel Training Project" of Jiangsu Province. The author Liangjian Hu is supported by the National Natural Science Foundation of China (11471071). The author Xuerong Mao is supported by the Leverhulme Trust (RF-2015-385), the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the EPSRC (EP/K503174/1)

The main aim of this paper is to investigate the polynomial stability of hybrid stochastic systems with pantograph delay (HSSwPD). By using the Razumikhin technique and Lyapunov functions, we establish several Razumikhin-type theorems on the $p$th moment polynomial stability and almost sure polynomial stability for HSSwPD. For linear HSSwPD, sufficient conditions for polynomial stability are presented.

Citation: Wei Mao, Liangjian Hu, Xuerong Mao. Razumikhin-type theorems on polynomial stability of hybrid stochastic systems with pantograph delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3217-3232. doi: 10.3934/dcdsb.2020059
##### References:
 [1] J. Appleby and E. Buckwar, Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation, Electron. J. Qual. Theo., 2016 (2016), 1-32. [2] C. T. H. Baker and E. Buckwar, Continuous $\theta$-methods for the stochastic pantograph equation, Electron. T. Numer. Anal., 11 (2000), 131-151. [3] W. Fei, L. Hu, X. Mao and M. Shen, Structured robust stability and boundedness of nonlinear hybrid delay systems, SIAM J. Control. Optim., 56 (2018), 2662-2689.  doi: 10.1137/17M1146981. [4] Z. Fan, M. 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. [5] Z. Fan, M. 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. [6] P. Guo and C. Li, Almost sure exponential stability of numerical solutions for stochastic pantograph differential equations, J. Math. Anal. Appl., 460 (2018), 411-424.  doi: 10.1016/j.jmaa.2017.10.002. [7] P. Guo and C. Li, Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations, BIT Numer. Math., 59 (2019), 77-96.  doi: 10.1007/s10543-018-0723-z. [8] L. Hu, X. Mao and L. Zhang, Robust stability and boundedness of nonlinear hybrid stochastic differential delay equations, IEEE Trans. Automa. Control., 58 (2013), 2319-2332.  doi: 10.1109/TAC.2013.2256014. [9] L. Huang and F. Deng, Razumikhin-type theorems on stability of neutral stochastic functional differential equations, IEEE Trans. Automa. Control., 53 (2008), 1718-1723.  doi: 10.1109/TAC.2008.929383. [10] S. Jankovic, J. Randjelovic and M. Jovanovic, Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations, J. Math. Anal. Appl., 355 (2009), 811-820.  doi: 10.1016/j.jmaa.2009.02.011. [11] Y. Ji and H. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Trans. Automa. Control., 35 (1990), 777-788.  doi: 10.1109/9.57016. [12] K. Liu and A. Chen, Moment decay rates of solutions of stochastic differential equations, Tohoku Math. J., 53 (2001), 81-93.  doi: 10.2748/tmj/1178207532. [13] M. Milosevic, Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler-Maruyama approximation, Appl. Math. Comput., 237 (2014), 672-685.  doi: 10.1016/j.amc.2014.03.132. [14] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.  doi: 10.1142/p473. [15] X. Mao, J. Lam, S. Xu and H. Gao, Razumikhin method and exponental stability of hybrid stochastic delay interval systems, J. Math. Anal. Appl., 314 (2006), 45-66.  doi: 10.1016/j.jmaa.2005.03.056. [16] X. Mao, A. Matasov and A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli., 6 (2000), 73-90.  doi: 10.2307/3318634. [17] X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stoch. Proc. Appl., 65 (1996), 233-250.  doi: 10.1016/S0304-4149(96)00109-3. [18] X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal., 28 (1997), 389-401.  doi: 10.1137/S0036141095290835. [19] X. Mao, Almost sure polynomial stability for a class of stochastic differential equations, Quart. J. Math. Oxford Ser., 43 (1992), 339-348.  doi: 10.1093/qmath/43.3.339. [20] X. Mao, Polynomial stability for perturbed stochastic differential equations with respect to semimartingales, Stoch. Proc. Appl., 41 (1992), 101-116.  doi: 10.1016/0304-4149(92)90149-K. [21] X. Mao, J. Lam and L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Syst. Control. Lett., 57 (2008), 927-935.  doi: 10.1016/j.sysconle.2008.05.002. [22] S. Peng and Y. Zhang, Razumikhin-type theorems on pth moment exponential stability of impulsive stochastic delay differential equations, IEEE Trans. Automa. Control., 55 (2010), 1917-1922.  doi: 10.1109/TAC.2010.2049775. [23] G. Pavlovic and S. Jankovic, Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay, J. Comput. Appl. Math., 236 (2012), 1679-1690.  doi: 10.1016/j.cam.2011.09.045. [24] B. S. Razumikhin, On the stability of systems with a delay, Prikl. Mat. Mekh., 20 (1956), 500-512. [25] L. Shaikhet, Stability of stochastic hereditary systems with Markov switching, Theory. Stoch. Proc., 2 (1996), 180-184. [26] Y. Shen and X. Liao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, Chinese Science Bulletin., 44 (1999), 2225-2228.  doi: 10.1007/BF02885926. [27] M. Shen, W. Fei, X. Mao and S. Deng, Exponential stability of highly nonlinear neutral pantograph stochastic differential equations, Asian. J. Control., 22 (2020), 1-13. [28] F. Wu, G. Yin and L. Wang, Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching, Math. Control. Related. Fields., 5 (2015), 697-719.  doi: 10.3934/mcrf.2015.5.697. [29] F. Wu and S. Hu, Razumikhin type theorems on general decay stability and robustness for stochastic functional differential equations, I. J. Robust. Nonlinear. Control., 22 (2012), 763-777.  doi: 10.1002/rnc.1726. [30] X. Wu, W. Zhang and Y. Tang, pth Moment stability of impulsive stochastic delay differential systems with Markovian switching, Commun. Nonlinear. Sci. Numer. Simulation, 18 (2013), 1870-1879.  doi: 10.1016/j.cnsns.2012.12.001. [31] Y. Xiao, M. Song and M. Liu, Convergence and stability of the semi-implicit Euler method with variable step size for a linear stochastic pantograph differential equation, Int. J. Numer. Anal. Model., 8 (2011), 214-225. [32] C. Yuan and X. Mao, Robust stability and controllability of stochastic differential delay equations with Markovian switching, Automatica., 40 (2004), 343-354.  doi: 10.1016/j.automatica.2003.10.012. [33] Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with lévy noise and markov switching, Int. J. Control., 90 (2017), 1703-1712.  doi: 10.1080/00207179.2016.1219069. [34] S. Zhou and M. Xue, Exponential stability for nonlinear hybrid stochastic pantograph equations and numerical approximation, Acta. Math. Sci., 34 (2014), 1254-1270.  doi: 10.1016/S0252-9602(14)60083-7. [35] T. Zhang, H. Chen, C. 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.

show all references

##### References:
 [1] J. Appleby and E. Buckwar, Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation, Electron. J. Qual. Theo., 2016 (2016), 1-32. [2] C. T. H. Baker and E. Buckwar, Continuous $\theta$-methods for the stochastic pantograph equation, Electron. T. Numer. Anal., 11 (2000), 131-151. [3] W. Fei, L. Hu, X. Mao and M. Shen, Structured robust stability and boundedness of nonlinear hybrid delay systems, SIAM J. Control. Optim., 56 (2018), 2662-2689.  doi: 10.1137/17M1146981. [4] Z. Fan, M. 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. [5] Z. Fan, M. 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. [6] P. Guo and C. Li, Almost sure exponential stability of numerical solutions for stochastic pantograph differential equations, J. Math. Anal. Appl., 460 (2018), 411-424.  doi: 10.1016/j.jmaa.2017.10.002. [7] P. Guo and C. Li, Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations, BIT Numer. Math., 59 (2019), 77-96.  doi: 10.1007/s10543-018-0723-z. [8] L. Hu, X. Mao and L. Zhang, Robust stability and boundedness of nonlinear hybrid stochastic differential delay equations, IEEE Trans. Automa. Control., 58 (2013), 2319-2332.  doi: 10.1109/TAC.2013.2256014. [9] L. Huang and F. Deng, Razumikhin-type theorems on stability of neutral stochastic functional differential equations, IEEE Trans. Automa. Control., 53 (2008), 1718-1723.  doi: 10.1109/TAC.2008.929383. [10] S. Jankovic, J. Randjelovic and M. Jovanovic, Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations, J. Math. Anal. Appl., 355 (2009), 811-820.  doi: 10.1016/j.jmaa.2009.02.011. [11] Y. Ji and H. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Trans. Automa. Control., 35 (1990), 777-788.  doi: 10.1109/9.57016. [12] K. Liu and A. Chen, Moment decay rates of solutions of stochastic differential equations, Tohoku Math. J., 53 (2001), 81-93.  doi: 10.2748/tmj/1178207532. [13] M. Milosevic, Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler-Maruyama approximation, Appl. Math. Comput., 237 (2014), 672-685.  doi: 10.1016/j.amc.2014.03.132. [14] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.  doi: 10.1142/p473. [15] X. Mao, J. Lam, S. Xu and H. Gao, Razumikhin method and exponental stability of hybrid stochastic delay interval systems, J. Math. Anal. Appl., 314 (2006), 45-66.  doi: 10.1016/j.jmaa.2005.03.056. [16] X. Mao, A. Matasov and A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli., 6 (2000), 73-90.  doi: 10.2307/3318634. [17] X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stoch. Proc. Appl., 65 (1996), 233-250.  doi: 10.1016/S0304-4149(96)00109-3. [18] X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal., 28 (1997), 389-401.  doi: 10.1137/S0036141095290835. [19] X. Mao, Almost sure polynomial stability for a class of stochastic differential equations, Quart. J. Math. Oxford Ser., 43 (1992), 339-348.  doi: 10.1093/qmath/43.3.339. [20] X. Mao, Polynomial stability for perturbed stochastic differential equations with respect to semimartingales, Stoch. Proc. Appl., 41 (1992), 101-116.  doi: 10.1016/0304-4149(92)90149-K. [21] X. Mao, J. Lam and L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Syst. Control. Lett., 57 (2008), 927-935.  doi: 10.1016/j.sysconle.2008.05.002. [22] S. Peng and Y. Zhang, Razumikhin-type theorems on pth moment exponential stability of impulsive stochastic delay differential equations, IEEE Trans. Automa. Control., 55 (2010), 1917-1922.  doi: 10.1109/TAC.2010.2049775. [23] G. Pavlovic and S. Jankovic, Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay, J. Comput. Appl. Math., 236 (2012), 1679-1690.  doi: 10.1016/j.cam.2011.09.045. [24] B. S. Razumikhin, On the stability of systems with a delay, Prikl. Mat. Mekh., 20 (1956), 500-512. [25] L. Shaikhet, Stability of stochastic hereditary systems with Markov switching, Theory. Stoch. Proc., 2 (1996), 180-184. [26] Y. Shen and X. Liao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, Chinese Science Bulletin., 44 (1999), 2225-2228.  doi: 10.1007/BF02885926. [27] M. Shen, W. Fei, X. Mao and S. Deng, Exponential stability of highly nonlinear neutral pantograph stochastic differential equations, Asian. J. Control., 22 (2020), 1-13. [28] F. Wu, G. Yin and L. Wang, Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching, Math. Control. Related. Fields., 5 (2015), 697-719.  doi: 10.3934/mcrf.2015.5.697. [29] F. Wu and S. Hu, Razumikhin type theorems on general decay stability and robustness for stochastic functional differential equations, I. J. Robust. Nonlinear. Control., 22 (2012), 763-777.  doi: 10.1002/rnc.1726. [30] X. Wu, W. Zhang and Y. Tang, pth Moment stability of impulsive stochastic delay differential systems with Markovian switching, Commun. Nonlinear. Sci. Numer. Simulation, 18 (2013), 1870-1879.  doi: 10.1016/j.cnsns.2012.12.001. [31] Y. Xiao, M. Song and M. Liu, Convergence and stability of the semi-implicit Euler method with variable step size for a linear stochastic pantograph differential equation, Int. J. Numer. Anal. Model., 8 (2011), 214-225. [32] C. Yuan and X. Mao, Robust stability and controllability of stochastic differential delay equations with Markovian switching, Automatica., 40 (2004), 343-354.  doi: 10.1016/j.automatica.2003.10.012. [33] Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with lévy noise and markov switching, Int. J. Control., 90 (2017), 1703-1712.  doi: 10.1080/00207179.2016.1219069. [34] S. Zhou and M. Xue, Exponential stability for nonlinear hybrid stochastic pantograph equations and numerical approximation, Acta. Math. Sci., 34 (2014), 1254-1270.  doi: 10.1016/S0252-9602(14)60083-7. [35] T. Zhang, H. Chen, C. 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.
 [1] 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 and Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697 [2] Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885 [3] Alexandra Rodkina, Henri Schurz, Leonid Shaikhet. Almost sure stability of some stochastic dynamical systems with memory. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 571-593. doi: 10.3934/dcds.2008.21.571 [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 and Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471 [5] Yanqiang Chang, Huabin Chen. Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021301 [6] Xiaojin Huang, Hongfu Yang, Jianhua Huang. Consensus stability analysis for stochastic multi-agent systems with multiplicative measurement noises and Markovian switching topologies. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 601-610. doi: 10.3934/naco.2021024 [7] 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 [8] Karim El Mufti, Rania Yahia. Polynomial stability in viscoelastic network of strings. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1421-1438. doi: 10.3934/dcdss.2022073 [9] Litan Yan, Wenyi Pei, Zhenzhong Zhang. Exponential stability of SDEs driven by fBm with Markovian switching. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6467-6483. doi: 10.3934/dcds.2019280 [10] Henri Schurz. Moment attractivity, stability and contractivity exponents of stochastic dynamical systems. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 487-515. doi: 10.3934/dcds.2001.7.487 [11] Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control and Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359 [12] Azeddine Elmajidi, Elhoussine Elmazoudi, Jamila Elalami, Noureddine Elalami. Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 143-159. doi: 10.3934/dcdss.2021035 [13] 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 and Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011 [14] Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345 [15] Cónall Kelly, Alexandra Rodkina. Constrained stability and instability of polynomial difference equations with state-dependent noise. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 913-933. doi: 10.3934/dcdsb.2009.11.913 [16] Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations and Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1 [17] Reza Kamyar, Matthew M. Peet. Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2383-2417. doi: 10.3934/dcdsb.2015.20.2383 [18] Alaa Hayek, Serge Nicaise, Zaynab Salloum, Ali Wehbe. Exponential and polynomial stability results for networks of elastic and thermo-elastic rods. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1183-1220. doi: 10.3934/dcdss.2021142 [19] Wensheng Yin, Jinde Cao. Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4493-4513. doi: 10.3934/dcdsb.2020109 [20] Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations and Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493

2021 Impact Factor: 1.497