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In this paper, the Razumikhin approach is applied to the study of both p-th moment and almost sure stability on a general decay for a class of impulsive stochastic functional differential systems with Markovian switching. Based on the Lyapunov-Razumikhin methods, some sufficient conditions are derived to check the stability of impulsive stochastic functional differential systems with Markovian switching. One numerical example is provided to demonstrate the effectiveness of the results.

Impulsive stochastic systems with Markovian switching is a class of hybrid dynamical systems, which is composed of both the logical switching rule of continuous-time finite-state Markovian process and the state represented by a stochastic differential system [

It is well-known that stability is the major issue in the study of control theory, one of the most important techniques applied in the investigation of stability for various classes of stochastic differential systems is based on a stochastic version of the Lyapunov direct method. However, the so-called Razumikhin technique combined with Lyapunov functions has also been a powerful and effective method in the study of stability. Recalled that Razumikhin developed this technique to study the stability of deterministic systems with delay in [

Mao incorporated the Razumikhin approach in stochastic functional differential equations [

In the above cited papers, both the p-th moment and almost sure stability on a general decay are investigated, but mostly used in stochastic differential equations. And As far as I know, a little work has been done on the impulsive stochastic differential equations or systems. In this paper, we will close this gap by extending the general decay stability to the impulsive stochastic differential systems. To the best of our knowledge, there are no results based on the general decay stability of impulsive stochastic delay differential systems with Markovian switching. And the main aim of the present paper is attempt to investigate the p-th moment and almost sure stability on a general decay of impulsive stochastic delay differential systems with Markovian switching. Since the delay phenomenon and the Markovian switching exists among impulsive stochastic systems, the whole systems become more complex and may oscillate or be not stable, we introduce Razumikhin-type theorems and Lyapunov methods to give the conditions that make the systems stable. By the aid of Lyapunov-Razumikhin approach, we obtain the p-th moment general decay stability of impulsive stochastic delay differential systems with Markovian. In order to establish the criterion on almost surely general decay stability of impulsive stochastic delay differential systems with Markovian, the Holder inequality, Burkholder-Davis-Gundy inequality and Borel- Cantelli’s lemma are utilized in this paper.

The paper is organized as follows. Firstly, the problem formulations, definitions of general dacay stability and some lemmas are given in Section 2. In Section 3, the main results on p-th moment and almost sure stability on a general decay of impulsive stochastic delay differential systems with Markovian switching are obtained with Lyapunov-Razumikhin methods. An example is presented to illustrate the main results in Section 4. In the last section the conclusions are given.

Throughout this paper, let

Let

Let

where

We assume that the Markov chain

In this paper, we consider the following impulsive stochastic delay differential systems with Markovian switching

where the initial value

represents the impulsive perturbation of x at time

For the existence and uniqueness of the solution we impose a hypothesis:

Assumption (H): For

For all

For all

Definition 1

(1) It is continuous, monotone increasing and differentiable;

(2)

(3)

(4) for any

Definition 2 For

when

Definition 3 impulsive stochastic delay differential systems with Markovian switching (1) is said to be almost surely stable with decay

when

Let

where

Lemma 1 (Burkholder-Davis-Cundy inequality) Let

where

Lemma 2 (Borel-Cantelli’s lemma)

(1) If

That is, there exist a set

(2) If the sequence

That is, there exists a set

In this section, we shall establish some criteria on the p-th moment exponential stability and almost exponential stability for system (1) by using the Razumikhin technique and Lyapunov functions.

Theorem 1 For systems (1), let (H) hold, and

(H_{1}) For all

(H_{2}) For all

For all

where

(H_{3}) For all

where

Then, for any initial

Proof. Fix the initial data

Then it follows from condition (H_{1}) that

In the following, we will use the mathematical induction method to show that

In order to do so, we first prove that

This can be verified by a contradiction. Hence, suppose that inequality (9) is not true, than there exist some

the interval

Define

Consequently, for all

And so

By condition (H_{2}) we have

Consequently,

Applying the

By condition (14), we obtain

On the other hand, a direct computation yields

that is

which is a contradiction. So inequality (9) holds and (8) is true for

Then, we will prove that (8) holds for

Suppose (18) is not true, i.e. there exist some

Then, it follows from the condition (H_{3}) and (17) that

which implies that the

interval

Define

Fix any

If

Therefore,

by condition (H_{2}) we have

Consequently,

Similar to (15), applying the

By condition (25), we obtain

On the other hand, by (20) and (22), we have

that is

which is a contradiction. So inequality (18) holds. Therefore, by mathematical induction, we obtain (8) holds for all_{1}), we have

which implies

i.e., system (1) is pth moment exponentially stable with decay

Theorem 2 For system (1), suppose all of the conditions of Theorem 1 are satisfied. Let

Then, for any initial

Proof. Fix the initial data

where

Choose

is the maximum integer not more than x. Then for any

For each i when

By Theorem 1, we have

By Holder inequality, condition (26) and Theorem 1, we derives that

Similarly, by the Lemma 1 and (32), we obtain

where

Substituting (31), (32) and (33) into (30) yields

Thus, it follows from (29) and (34), we obtain

Using Chebyshev inequality, we have

Since

That is

Thus, the system (1) is almost surely stable with decay

In this section, a numerical example is given to illustrate the effectiveness of the main results established in Section 3 as follows. Consider an impulsive stochastic delay system with Markovian switching as follows

where

And independent of the scalar Brownian motion

Choosing

and

By Theorem 1, we know that

In this paper, p-th moment and almost surely stability on a general decay have been investigated for a class of impulsive stochastic delay systems with Markovian switching. Some sufficient conditions have been derived to check the stability criteria by using the Lyapunov-Razumikhin methods. A numerical example is provided to verify the effectiveness of the main results.

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

Zhiyu Zhan,Caixia Gao, (2016) Razumikhin-Type Theorems on General Decay Stability of Impulsive Stochastic Functional Differential Systems with Markovian Switching. Journal of Applied Mathematics and Physics,04,1617-1629. doi: 10.4236/jamp.2016.48172