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This paper focuses on the p th moment exponentially stability for Markov switching and multi-impulse jumps stochastic time-varying delay system, where the switching behavior among subsystems of the target system is determined by Markov chains, and the occurrence of impulsive jumps is decided according to event-triggered impulsive mechanism when certain well-designed conditions are satisfied. By applying the Itô formula, Gronwall inequality and Razumikhin theorem, some novel sufficient criteria are provided to assure the system stability and get rid of Zeno phenomenon. It is worth pointing out that the multi-impulse jumps are our research aim and the range of delays considered is relatively wide, i.e., the daily bounded delay τ (t) ∈ [ 0 , 1) and the unupper bound delay τ (t) ∈ [ 1 , ∞). Subsequently, two diverse event trigger mechanism about impulsive jumps are proposed for such two types of delays, namely the defined event-triggered impulsive mechanism with delay. Finally, the validity and feasibility of the developed theoretical results are verified by two numerical simulations. • Different from the bounded delay system in Wang et al. (2023), Peng et al. (2021),Li and Zhu (2023),Peng et al. (2010), Zhu (2014), Zhua and Cao (2012),Yang and Zhu (2014), Xu and Zhu (2022), we discuss the daily bounded delay τ (t) ∈ [ 0 , 1) and the unupper bound delay τ (t) ∈ [ 1 , ∞). Two categories of event trigger impulsive method named event-triggered impulsive mechanism with delay (ETIMD) are proposed for two types delay, respectively. And in event-triggered condition, it fully consider the impact of delays. Moreover, the upper bound of the delay τ (t) ∈ [ 0 , 1) is required to avoid Zeno phenomenon. While for τ (t) ∈ [ 1 , ∞) , only the lower bound is required to escape Zeno phenomenon, in which is independent of the delayed upper bound. • Compared with Wang et al. (2022), Li et al. (2020), Peng et al. (2021), Zhu (2014), Xu and Zhu (2022), the Markov switching and multi-impulse jumps stochastic time-varying delay system as a benchmark is considered in this paper, where the switching behavior between subsystems is driven by Markov chains, and the occurrence of impulsive jump is decided according to ETIMD strategy. It should be emphasized that the multi-impulse jumps are the research goal. Thus, the system in [27,28,36,43] could be regarded as a special case of the stochastic delay system when the Markov switching is not considered in this study. • By applying the Itô formula, Gronwall inequality and Razumikhin theorem, some novel sufficient criteria of Lyapunov-Razumikhin type for p-ES are provided for unstable subsystems with stable impulses. [ABSTRACT FROM AUTHOR]

In this paper, the global exponential stability and stabilization problems are investigated for memristive neural networks (MNNs) with stochastic disturbances, spatial diffusions and distributed delays. The spatial diffusions are not assumed to be symmetric and distributed delays are relaxed to be unbounded. Then, the presented MNNs are modeled as a class of stochastic partial systems with hybrid delays. Based on nonsmooth analysis, the Lyapunov–Krasovskii functional and inequality approach, some simple algebraic criteria are derived for the global exponential stability as well as the stabilization via two kinds of designed feedback controllers. The derived criteria are easily verified and the obtained results are available for other delayed systems with or without stochastic disturbances and spatial diffusions. Finally, the stability and stabilization results are validated by numerical simulations. • The stochastic model with spatial diffusions and distributed delays is discussed. • The derived results are generalized and they contain previous ones as special cases. • The method is available for other stochastic delayed partial differential systems. • The space set of the spatial diffusions in our model is not symmetrical. • The distributed delays are unbounded. [ABSTRACT FROM AUTHOR]

As the spread of diseases among species becomes more and more common, our society pays increasingly more attention to the harm caused by the disease. Research on eco-epidemiological models will not only promote the development of biology and ecology, but also have important impacts on the economy and health of the society. This paper deals with a stochastic eco-epidemiological model which contains three species and is governed by Ornstein–Uhlenbeck (OU) process. Two common approaches to incorporate the effects of environmental perturbations in stochastic systems are first discussed and compared analytically and computationally. Based on the existence and uniqueness of the global solution to the model, several sharp conditions for the persistence and extinction of the species are established. Then, a criteria for the existence of the stationary distribution to the system is established by constructing suitable Lyapunov functions. Furthermore, the analytical expression of the probability density function of the model around the quasi-equilibrium is obtained. Finally, the impact of the main parameters on the dynamical behaviors and the noise perturbation effect are analysed in the numerical simulations section to illustrate the theoretical results. • A stochastic eco-epidemiological model with Ornstein–Uhlenbeck process is studied. • Some conditions for the persistence and extinction of species are established. • A criteria for the existence of stationary distribution is given. • The probability density function near the quasi-equilibrium is obtained. [ABSTRACT FROM AUTHOR]

• The concepts of average control rate and average control period are proposed to characterize the distribution of work intervals and rest intervals of aperiodically intermittent control. • The constraints on the control rate, control intervals and rest intervals are reduced compared with previous results. • Average control width of AIC instead of the lower bound of all control intervals of AIC, is used to measure the small time delays and large time delays. • For the case of small time delays, the upper bound of time delays can be larger than the lower bound of control widths. This paper focuses on the stabilization problem of complex-valued hybrid stochastic delayed systems via aperiodically intermittent control (AIC). As for AIC in existing literature, strict conditions on the lower bound of control intervals and upper bound of control periods or the maximum proportion of rest intervals are required. In this paper, we relax these constraints by proposing average control ratio and average control period to describe the distribution of control and rest intervals of AIC, i.e., the lower bound of certain control widths can be arbitrarily small, upper bound of certain control periods can be very large and the proportion of rest widths can be any value in (0 , 1). Thus, the conservativeness is reduced compared with the existing related results. Then based on the complex generalized Itô's formula, several novel stabilization criteria are obtained for small time delays and large time delays, respectively, which avoid splitting the real and imaginary parts. Especially, for the case of small time delays, the upper bound of time delays can be larger than the lower bound of all control widths, which has wider applications than previous results. Finally, two examples along with their numerical simulations are given to show the effectiveness and less conservativeness of our results. [ABSTRACT FROM AUTHOR]