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This paper mainly tends to utilize Razumikhin-type theorems to investigate p-th moment stability for a class of stochastic switching nonlinear systems with delay. Based on the Lyapunov-Razumik- hin methods, some sufficient conditions are derived to check the stability of stochastic switching nonlinear systems with delay. One numerical example is provided to demonstrate the effectiveness of the results.

Stochastic switching system is an indispensable class of hybrid dynamical systems, which is composed of a family of stochastic subsystems and a rule that orchestrates the switching among them. Yet, there inevitably exists delay phenomenon in the practical systems like physics, biology and economic [

It is well-known that stability is the major issue of control theory. Lyapunov-Razumikhin technique has been a powerful and effective method for investigating stability. Razumikhin developed this technique to study the stability of deterministic systems with delay in [

To the best of our knowledge, there are no results based on the Razumikhin approach referring to the

Consider a family of stochastic switching delay nonlinear systems described by

where

by switching signal

Definition 1.

1) It is continuous, monotone decreasing and differentiable;

2)

3)

4) for any

Definition 2. For

when

Before giving the main results, let us introduce

where

In this section, we shall establish Razumikhin-type theorems on the p-th moment

Assumption 1. Switching signal

Assumption 2. At each switching instant

Then, let us turn our attention to system (1) and give a sufficient result.

Theorem 1. For stochastic switching delay nonlinear systems (1), if there exist a group of Lyapunov functions

for all

where

and at each switching instant

where

Then, for any initial

switching delay nonlinear system (1). Moreover, the system (1) is p-th moment

Proof. Fix the initial data

replaced by

Given switching signal

Let

we will complete this proof. By condition (6), this result follows from

Let

By the continuity of

We claim that (8) holds for all

In order to do so, we first prove that

That is

This can be verified by a contradiction, suppose that inequality (9) is not right, then by the continuity of

as

if

Therefore, for

By condition (4), we can obtain

By the continuity of

By the

By condition (4)

which is a contradiction. Hence, inequality (9) holds for all

Now, let

That is

We will prove that

Suppose that inequality (14) is not right,

By condition (6) and inequality (12), we have

That is

Then by the continuity of

ciently small

if

Therefore, for

By condition (4), we can obtain

By the continuity of

By the

By condition (4)

which is a contradiction. Hence, inequality (14) holds for all

Therefore, by mathematical induction we obtain (8) holds for all

Then,

That is

Thus, the system (1) is p-th moment

In this section, a numerical example is given to illustrate the effectiveness of the main results established in Section 3 as follows.

Consider a family of stochastic switching delay nonlinear systems

where

We choose

When

For the first subsystem, we choose

If

For the second subsystem, we choose

If

By Theorem 1, we can choose

Remark. In the example, a stochastic switching delay nonlinear system is constructed to show the efficiency of the results.

In this paper, p-th moment

zumikhin methods. A numerical example is provided to verify the effectiveness of the main results. Our future research will focus on

The work was supported by the National Natural Science Foundation of China under Grants 11261033 and the Postgraduate Scientific Research Innovation Foundation of Inner Mongolia under Grant 1402020201336.

Haibo Gu,Caixia Gao, (2016) Razumikhin-Type Theorems on p-th Moment Stability for Stochastic Switching Nonlinear Systems with Delay. Journal of Applied Mathematics and Physics,04,1237-1244. doi: 10.4236/jamp.2016.47129