A new unified input-to-state stability criterion for impulsive stochastic delay systems with Markovian switching

In this paper, the input-to-state stability (ISS), the integral input-to-state stability (iISS), the stochastic input-to-state stability (SISS) and the eλt (λ > 0)-weighted input-to-state stability (eλt-ISS) for nonlinear time-varying impulsive stochastic delay system with Markovian switching are, respectively, investigated. By using the Lyapunov function approach, we propose one unified criterion for the stabilizing impulse and the destabilizing impulse to guarantee the ISS, iISS, SISS and eλt-ISS for such system. We demonstrate that when the upper bound of the average impulsive interval is given, the stabilizing impulsive effect can stabi-lize a time-varying continuous stochastic system without ISS.We also show that the destabilizing impulsive signal with a given lower bound of the average impulsive interval can preserve the ISS of a time-varying continuous stochastic system. The method used is the Lyapunov function approach incorporating with the proof by contradiction. The coefficient of the estimation term for the system state in the generalized monotonicity condition is not only time-varying, but also allows to be sign-changed. In addition, one criterion for guaranteeing the ISS of nonlinear time-varying stochastic hybrid system under no impulsive effect is derived. Two examples including one coupled dynamic system model subject to external random perturba-tion of the continuous input and impulsive input disturbances are provided to illustrate the efficacy of the findings obtained through theoretical analysis.Huabin Chen, Peng Shi, Cheng‐Chew Li