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This article extends stochastic input‐to‐state stability (SISS$$ \mathrm{SISS} $$) to stochastic time‐delayed hybrid systems with the properties of edge dependent. By quoting multiple Lyapunov–Krasovskii functional, comparison principle in association with admissible edge‐dependent average dwell time (AED‐ADT$$ \mathrm{AED}\hbox{-} \mathrm{ADT} $$) approach and admissible amputated edge‐dependent average impulsive interval (AAED‐AII$$ \mathrm{AAED}\hbox{-} \mathrm{AII} $$) technology, several novel stability criteria of hybrid time‐delayed stochastic systems are established. We allow impulsive jump and subsystem switching to occur at different instants. Furthermore, a new method named AAED‐AII is recommended for gap between impulses, which is an improvement for mode‐dependent average impulsive interval. Finally, a practical example is given to show the effectiveness of the new design techniques. [ABSTRACT FROM AUTHOR]

This paper studies integral input-to-state stability (iISS) of a class of hybrid delayed systems with impulse and switching behavior, in which the delay-dependent impulse and switching behavior are allowed for asynchronous occurrence. Relative to admissible edge-dependent average dwell time (AED-ADT) concept of the switching signal, a fresh concept of admissible edge-dependent average impulsive interval (AED-AII) is proposed for the impulse signal. Several sufficient criteria for iISS are achieved by using multiple Lyapunov–Krasovskii functional (MLKF) tool, AED-ADT switching, and AED-AII schemes. An improved inequality that describing the relation among Lyapunov function continuous switching signal and discrete impulse is constructed, where the jump amplitude at discrete switching points and impulsive moments including delay-dependent part and delay-independent part are involved. Compared with existing results, our conclusions have the following novelties: (1) the influence of time delay existing in continuous dynamics and discrete impulse on the function part of L-K functional is dug out, which is less conservative; (2) the AED-AII scheme proposed in this paper is combined with the AED-ADT switching method, which contains a larger set of the impulsive and switching signal. Moreover, the stability criterion is suitable for the arbitrary time delay in situation that all the subsystems are destabilizing. Finally, an example is given to verify the validity of the results. [ABSTRACT FROM AUTHOR]

• A more general framework of a hybrid system is developed to characterize the overall system subject to switch-ing signals and delay-dependent impulsive jumps. It relaxes the assumption that the impulsive behavior and switching signal have appeared simultaneously. There are two kinds of time series, impulsive and switching time series. These sequences need to be considered separately, which causes significant difficulties and challenges; • A novel definition called AED-AII is employed to portray the impulse signal. Within the framework of this new concept, when the subsystem switches from different modes to the ith mode, the AII corresponding to the ith mode may also be different. This approach is less conservative in the sense that it relaxes the limitation of MDAII to all-edge-dependence. Thus, the results are more flexible and more applicable for the stability of impulsive switched systems; • The relation of AED-AII, AED-ADT, impulsive gain and the decay rate of the Lyapunov functional for each mode is revealed. Compared with the MDAII and MDADT methods used in [23] and [16] , the results obtained in our paper are less conservative, which may make them applicable a larger class of switching signals and impulsive sequences. Compared with [24] , our results are more relaxed in that the conditions in [24] are designed without accounting for the effect of a delay-dependent impulse emerging in Lyapunov function part and the contribution existing in the functional part V2. When the influence of the switching signal is not considered, the conclusions in this paper return to the corresponding results in [9] , which implies that the results in this paper are more general. Moreover, [25] is not concerned with the delay existing in the impulse, and thus, the method may not be easily applied to our system. In other words, when the impulsive jumps are not affected by the time delay, which implies that ζ 2 i = 0. Theorem 1 in our setup can be reduced to Lemma 1 in [25]. Then, applying our results to impulsive switched delayed systems with delay-independent impulses, a relaxed criterion on the upper bound of dwell time is obtained. In this summary, for a kind of hybrid deterministic system, we emphasize the input-to-state stability (ISS) property. Meanwhile, the delay-dependent impulse and switching behavior are permitted to arise asynchronously. In other words, the impulses can arise many times or not once within a period of switching interval. For impulsive signals, A fresh idea called admissible edge-dependent average impulsive interval (AED-AII) is addressed according to properties of edges. By developing the multiple Lyapunov-Krasovskii functionals (MLKFs), merged with the AED-ADT switching scheme and AED-AII method, some ameliorated criteria in accordance with the ISS of hybrid delayed systems are established with regard to stabilizing and destabilizing impulses. Compared with previous work, our setup enjoys the important features: the candidate functions fully taps the effect of the delay existing in the system dynamics and impulse on the function part of the MLKFs; the AED-AII method combined with the AED-ADT switching scheme in our paper contains a larger group of the impulsive and switching signals; and for the case in which all the subsystems are unstable, the stability criterion is suitable for the arbitrary time delay. Finally, an example shows that the results of this paper are effective. [ABSTRACT FROM AUTHOR]