# zbMATH — the first resource for mathematics

Razumikhin-type theorems on polynomial stability of hybrid stochastic systems with pantograph delay. (English) Zbl 1443.60064
Summary: 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.
##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 93E15 Stochastic stability in control theory
Full Text:
##### References:
 [1] J. Appleby; E. Buckwar, Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation, Electron. J. Qual. Theo., 2016, 1-32 (2016) · Zbl 1389.60069 [2] C. T. H. Baker; E. Buckwar, Continuous $$\theta$$-methods for the stochastic pantograph equation, Electron. T. Numer. Anal., 11, 131-151 (2000) · Zbl 0968.65004 [3] W. Fei; L. Hu; X. Mao; M. Shen, Structured robust stability and boundedness of nonlinear hybrid delay systems, SIAM J. Control. Optim., 56, 2662-2689 (2018) · Zbl 1394.93343 [4] Z. Fan; M. Song; M. Liu, The $$\alpha$$ th moment stability for the stochastic pantograph equation, J. Comput. Appl. Math, 233, 109-120 (2009) · Zbl 1206.65021 [5] Z. Fan; M. Liu; W. Cao, Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations, J. Math. Anal. Appl., 325, 1142-1159 (2007) · Zbl 1107.60030 [6] P. Guo; C. Li, Almost sure exponential stability of numerical solutions for stochastic pantograph differential equations, J. Math. Anal. Appl., 460, 411-424 (2018) · Zbl 1382.65018 [7] P. Guo; C. Li, Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations, BIT Numer. Math., 59, 77-96 (2019) · Zbl 1418.60057 [8] L. Hu; X. Mao; L. Zhang, Robust stability and boundedness of nonlinear hybrid stochastic differential delay equations, IEEE Trans. Automa. Control., 58, 2319-2332 (2013) · Zbl 1369.93693 [9] L. Huang; F. Deng, Razumikhin-type theorems on stability of neutral stochastic functional differential equations, IEEE Trans. Automa. Control., 53, 1718-1723 (2008) · Zbl 1367.34096 [10] S. Jankovic; J. Randjelovic; M. Jovanovic, Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations, J. Math. Anal. Appl., 355, 811-820 (2009) · Zbl 1166.60040 [11] Y. Ji; H. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Trans. Automa. Control., 35, 777-788 (1990) · Zbl 0714.93060 [12] K. Liu; A. Chen, Moment decay rates of solutions of stochastic differential equations, Tohoku Math. J., 53, 81-93 (2001) · Zbl 0997.93097 [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, 672-685 (2014) · Zbl 1334.60117 [14] X. Mao; J. Lam; S. Xu; H. Gao, Razumikhin method and exponental stability of hybrid stochastic delay interval systems, J. Math. Anal. Appl., 314, 45-66 (2006) · Zbl 1127.60072 [15] X. Mao; A. Matasov; A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli., 6, 73-90 (2000) · Zbl 0956.60060 [16] X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stoch. Proc. Appl., 65, 233-250 (1996) · Zbl 0889.60062 [17] X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal., 28, 389-401 (1997) · Zbl 0876.60047 [18] X. Mao, Almost sure polynomial stability for a class of stochastic differential equations, Quart. J. Math. Oxford Ser., 43, 339-348 (1992) · Zbl 0765.60058 [19] X. Mao, Polynomial stability for perturbed stochastic differential equations with respect to semimartingales, Stoch. Proc. Appl., 41, 101-116 (1992) · Zbl 0753.60051 [20] X. Mao; J. Lam; L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Syst. Control. Lett., 57, 927-935 (2008) · Zbl 1149.93027 [21] S. Peng; Y. Zhang, Razumikhin-type theorems on pth moment exponential stability of impulsive stochastic delay differential equations, IEEE Trans. Automa. Control., 55, 1917-1922 (2010) · Zbl 1368.93771 [22] G. Pavlovic; S. Jankovic, Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay, J. Comput. Appl. Math., 236, 1679-1690 (2012) · Zbl 1247.34128 [23] B. S. Razumikhin, On the stability of systems with a delay, Prikl. Mat. Mekh., 20, 500-512 (1956) [24] L. Shaikhet, Stability of stochastic hereditary systems with Markov switching, Theory. Stoch. Proc., 2, 180-184 (1996) · Zbl 0939.60049 [25] Y. Shen; X. Liao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, Chinese Science Bulletin., 44, 2225-2228 (1999) · Zbl 1043.60049 [26] M. Shen; W. Fei; X. Mao; S. Deng, Exponential stability of highly nonlinear neutral pantograph stochastic differential equations, Asian. J. Control., 22, 1-13 (2020) [27] F. Wu; G. Yin; L. Wang, Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching, Math. Control. Related. Fields., 5, 697-719 (2015) · Zbl 1323.34091 [28] F. Wu; S. Hu, Razumikhin type theorems on general decay stability and robustness for stochastic functional differential equations, I. J. Robust. Nonlinear. Control., 22, 763-777 (2012) · Zbl 1276.93081 [29] X. Wu; W. Zhang; Y. Tang, pth Moment stability of impulsive stochastic delay differential systems with Markovian switching, Commun. Nonlinear. Sci. Numer. Simulation, 18, 1870-1879 (2013) · Zbl 1291.35432 [30] Y. Xiao; M. Song; 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, 214-225 (2011) · Zbl 1214.65003 [31] C. Yuan; X. Mao, Robust stability and controllability of stochastic differential delay equations with Markovian switching, Automatica., 40, 343-354 (2004) · Zbl 1040.93069 [32] Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with lévy noise and markov switching, Int. J. Control., 90, 1703-1712 (2017) · Zbl 1367.93711 [33] S. Zhou; M. Xue, Exponential stability for nonlinear hybrid stochastic pantograph equations and numerical approximation, Acta. Math. Sci., 34, 1254-1270 (2014) · Zbl 1324.65006 [34] T. Zhang; H. Chen; C. Yuan; T. Caraballo, On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations, Discrete Contin. Dyn. Syst. Ser. B., 24, 5355-5375 (2019) · Zbl 1420.60077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.