Almost Sure ISS and iISS for Randomly Switched Time-Varying Systems with Time-Delays

This paper delves into almost sure input-to-state stability (ISS) and almost sure integral input-to-state stability (iISS) of randomly switched time-varying systems with time-delays. We provide definitions for almost sure ISS and almost sure iISS, underlining their necessity with an example that shows the disparity between pth moment ISS and almost sure ISS. Following this, we derive criteria for almost sure ISS and almost sure iISS based on indefinite multiple Lyapunov functions and Razumikhin technique. Notably, these criteria give time-varying stability estimators which do not require subsystems to maintain ISS or iISS throughout the entire time interval. Finally, the effectiveness and advantages of our results are demonstrated through a numerical example.