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This paper develops some criteria for a kind of hybrid stochastic systems with time-delay, which improve existing results on hybrid systems without considering noises. The improved results show that the presence of noise is quite involved in the stability analysis of hybrid systems. New results can be used to analyze the stability of a kind of stochastic hybrid impulsive and switching neural networks (SHISNN). Therefore, stability analysis of SHISNN can be turned into solving a linear matrix inequality (LMI).

With the development of social production, many practical systems cannot be modeled by linear time-invariant systems. In this case, hybrid systems are employed to model many practical systems. A hybrid system is a dynamical system with continuous dynamics, discrete dynamics, and the interaction between them (see, e.g., [

Impulsive stabilization of dynamical systems has attracted increasing interests in fields such as population dynamics, automatic control, drug administration, and communication networks (see [

On the other hand, many real systems are subject to stochastic effects, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. In the past decade years, the stability and stabilization of stochastic dynamic systems have been intensively investigated. Hence, the theory of hybrid stochastic systems has attracted increasing attention because the relating problems are not only academically challenging, but also of practical importance in many branches of science and engineering (see, e.g., [

However, among the existing results, up to now, very little is known about the stability of hybrid stochastic systems (HSS) with time-delay. In this paper, the stability of HSS with time-delay has been discussed. Some new results of the stability of HSS with time-delay have been attained in this paper. New results can be used to analyze the stability of a kind of stochastic hybrid impulsive and switching neural networks (SHISNN). Therefore, stability analysis of SHISNN can be turned into solving a linear matrix inequality.

This paper is organized as follows. In Section

Throughout this paper, unless otherwise specified, we will employ the following definitions. Let

Consider the following HSS with time-delay:

HSS (_{1})–(H_{3}) at any bounded interval

there exists nonnegative constant sequence

let

Let

Let

(i)

(ii)

(iii) Asymptotically stable in mean square: the trivial of HSS (

(iv) Exponentially stable in mean square: the trivial of HSS (

Moreover, in order to finish our results, we will introduce the following lemmas.

Let

If

In this section, we will present some criteria for the stability of systems (

If there exist switching Lyapunov functions

For any

It follows from condition (

Using the proof techniques in Theorem

Assume that conditions in Theorem

When

When

Letting

In Theorem

When

Similar to the proof of Theorem

In Theorem

Similar to paper [

To simplify, we always suppose that the solution of (

In order to obtain the according result, we give out Lemmas

Let

Since

Let

If SHISNN (

activation functions are bounded; that is, for

activation functions are globally Lipschitzian, for

for

for SHISNN (

there exist symmetric and positive-definite matrices

Construct the switching Lyapunov function
_{5}) and Lemmas

It follows from (

Via Schur complement, it is easy to show that (

The hybrid impulsive and switching stochastic systems model has been discussed based on the existing models of hybrid impulsive and switching systems. Some general criteria for the asymptotic and exponential stability analysis of the new model have been established by using switching Lyapunov functions and stochastic analysis techniques. The new result can be used to analyze mean square global asymptotical and exponential stability for a kind of stochastic hybrid impulsive and switching neural networks. Finally, this paper points out that this kind of stability analysis problem can turn into solving a linear matrix inequality (LMI).

The authors declare that there is no conflict of interests regarding the publication of this paper.