Stability Analysis of Impulsive Stochastic Systems With Poisson Jumps and Regime Switching

This paper is devoted to discussing the $p$th-moment input-to-state stability ( $p$-ISS), $p$th-moment integral-ISS ( $p$-iISS), and $p$th-moment $e^{\sigma t}$-ISS ( $e^{\sigma t}$- $p$-ISS) for impulsive stochastic delayed differential system via the Lyapunov–Krasovskii functional and some analytical skills. For the fixed moment impulse, the results show that if the continuous dynamics is $p$-ISS, then the system is $p$-ISS with respect to a lower bound of the average impulsive interval. For the random moment impulse, the sufficient conditions are given to make the system $p$-ISS when continuous dynamics is $p$-ISS, and the intensity and density of random impulses are constrained at the same time. Furthermore, they qualitatively reveal the impact of delay, regime switching and random impulses on the system is $p$-ISS. Finally, examples and numerical simulations are adopted to verify the validity of theoretical analysis.