The paper introduces a unified criterion for pth moment input-to-state stability (p-ISS), pth moment integral input-to-state stability (p-iISS), and eσt-weighted pth moment integral input-to-state stability (eσt-p-ISS) of impulsive stochastic function differential system with Markovian switching under the perturbation of the stabilizing impulse and destabilizing impulse. The Lyapunov function approach, comparison principle, and impulsive average dwell-time method are applied in this paper. The linear coefficients of the upper bound of Lyapunov functional differential operators are time-varying functions, including the case of constants, which advances and improves the existing results. In addition, the same results were obtained by applying impulse differential inequality. At the end of this paper, we use a numerical example to verify the validity of the results.
BainovDSimeonovP (1993) Impulsive Differential Equations: Periodic Solutions and Applications. Harlow: Longman Scientific and Technical.
2.
ChenWZhangW (2009) Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays. Automatica45(6): 1481–1488.
3.
DashkovskiySMironchenkoA (2013) Input-to-state stability of nonlinear impulsive system. SIAM Journal on Control and Optimization51(3): 1962–1987.
4.
FuXZhuQ (2018) Stability of nonlinear impulsive stochastic systems with Markovian switching under generalized average dwell time condition. Science China Information Sciences61(11): 112211.
5.
FuXZhuQGuoY (2019) Stabilization of stochastic functional differential systems with delayed impulses. Applied Mathematics and Computation346(C): 776–789.
6.
GaoLWangS (2019) Pth moment input-to-state stability of stochastic impulsive switched delayed systems. Transactions of the Institute of Measurement and Control41(6): 3468–3476.
7.
GuoYZhuQWangF (2020) Stability analysis of impulsive stochastic functional differential equations. Communications in Nonlinear Science and Numerical Simulation82(2): 105013.
8.
HespanhaJLiberzonDTeelA (2008) Lyapunov conditions for input-to-state stability of impulsive systems. Automatica44(11): 2735–2744.
9.
JobertARogersL (2006) Option pricing with Markov-modulated dynamic. SIAM Journal on Control and Optimization44(6): 2063–2078.
10.
KaoYZhuQQiW (2015) Exponential stability and instability of impulsive stochastic functional differential equations with Markovian switching. Applied Mathematics and Computation271: 795–804.
11.
KarafyllisJiangZ (2013) Global stabilization of nonlinear systems based on vector control Lyapunov functions. IEEE Transactions on Automatic Control58(10): 2550–2562.
12.
KhasminskiiR (2012) Stochastic Stability of Differential Equation. Berlin; Heidelberg: Springer-Verlag.
13.
LeiJMackeyM (2007) Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system. SIAM Journal on Control and Optimization67(2): 387–407.
14.
LiMDengF (2018) Moment exponential input-to-state stability of nonlinear switched stochastic systems with Levy noise. IET Control Theory and Applications12(9): 1208–1215.
15.
LiuJLiuXXieW (2011) Input-to-state stability of impulsive and switching hybrid systems with time-delay. Automatica47(5): 899–908.
16.
LuJHoDCaoJ (2010) A unified synchronization criterion for impulsive dynamical networks. Automatica46(7): 1215–1221.
17.
LuoQMaoX (2007) Stochastic population dynamics under regime switching. Journal of Mathematical Analysis and Applications334(1): 69–84.
18.
MaoXYuanCZhouJ (2005) Stochastic differential delay equations of population dynamics. Journal of Mathematical Analysis and Applications304(1): 296–320.
19.
NingCHeYWuM, et al. (2012) Input-to-state stability of nonlinear systems based on an indefinite Lyapunov function. Systems and Control Letters61(12): 1254–1259.
20.
PengSDengF (2017) New criteria on pth moment input-to-state stability of impulsive stochastic delayed differential systems. IEEE Transactions on Automatic Control62(7): 3573–3579.
21.
PengSZhangY (2010) Some new criteria on pth moment stability of stochastic functional differential equations with Markovian switching. IEEE Transactions on Automatic Control55(12): 2886–2890.
22.
RenWXiongJ (2019) Vector Lyapunov function based input-to-state stability of stochastic impulsive switched time-delay system. IEEE Transactions on Automatic Control64(2): 654–669.
SontagE (1998) Comments on integral variants of ISS. Systems and Control Letters34(1-2): 93–100.
25.
WangYWangRZhaoJ (2013) Input-to-state stability of nonlinear impulsive and switched delay systems. IET Control Theory and Applications7(8): 1179–1185.
26.
WuXShiPTangY, et al. (2017) Input-to-state stability of nonlinear stochastic time-varying systems with impulsive effects. International Journal of Robust and Nonlinear Control27(10): 1792–1809.
27.
WuXTangYZhangW (2016) Input-to state stability of impulsive stochastic delayed system under linear assumptions. Automatica66: 195–204.
28.
WuXZhangWTangY (2013) Pth Moment stability of impulsive stochastic delay differential systems with Markovian switching. Communications in Nonlinear Science and Numerical Simulation18(7): 1870–1879.
29.
ZhangB (2016) On asymptotic stability of linear time-varying systems. Automatica68(9): 266–276.
30.
ZhuQ (2014) Pth Moment exponential stability of impulsive stochastic functional differential equations with Markovian switching. Journal of the Franklin Institute-Engineering and Applied Mathematics351(7): 3965–3986.
