Uniform input-to-state stability of time-varying stochastic systems with delay-dependent impulses

Impulsive stochastic systems, as classical hybrid systems, are prevalent across numerous scientific disciplines, attracting significant research interest in their stability characteristics. This paper investigates the input-to-state stability properties of impulsive stochastic systems featuring time varying or state-dependent delayed impulses within a two metrics framework. By applying the Lyapunov method, we establish sufficient criteria for input-to-state stability in impulsive stochastic systems, meeting the non-exponential Lyapunov function candidate and eventually uniformly bounded impulse frequency conditions. Importantly, these criteria do not require strict constraints on the size relationship between delays and impulse intervals; they encompass cases where the delay is shorter than the impulse interval and are also applicable when the delay exceeds the impulse interval. Our theoretical analysis highlights the distinct effects of stable and unstable delayed impulses on system stability, extending existing literature conclusions and theoretically demonstrating the robustness of the input-to-state stability property under the perturbation caused by uncertain impulse moments. Finally, the theoretical findings are applied to examples where existing methods are ineffective, and numerical simulations are presented to demonstrate the advantages of the new approach.