Robust Exponential Stability of Impulsive Stochastic Neural Networks with Leakage Time-Varying Delay

Abstract

This paper investigates mean-square robust exponential stability of the equilibrium point of stochastic neural networks with leakage time-varying delays and impulsive perturbations. By using Lyapunov functions and Razumikhin techniques, some easy-to-test criteria of the stability are derived. Two examples are provided to illustrate the efficiency of the results.

1. Introduction

In recent years, stability for neural networks with time delay has been extensively studied due to their great applications in some practical engineering problems such as signal processing, associate memory, and combinatorial optimization (see [1–3]). In particular, the leakage delay, which exists in the negative feedback terms (known as forgetting or leakage terms) of the system, has great impact on the dynamical behavior of neural networks (see [4–10]). Gopalsamy [4] initially discussed the problem of bidirectional associative memory neural networks with constant delays in the leakage term by using model transformation technique. Then, many results of stability of neural networks with delays in the leakage terms are obtained (see, [5–12]).

However, besides time delay, neural networks are often subject to impulsive perturbation—the abrupt changes at certain instants, which may be caused by switching phenomenon, frequency change, or other sudden noise (see [13, 14]). The impulsive effect can affect the dynamical behaviors of the system. Now there are many results on stability of the neural networks with time delays in the leakage term and impulsive perturbations under the corresponding delayed neural networks without impulses must be stable themselves (see [5–8]). To best of the authors’ knowledge, this is the first attempt to investigate the stability of the systems under the corresponding delayed neural networks without impulses which are unstable themselves.

On the other hand, in real nervous systems, the synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes [15]. It is well known that a neural network could be stabilized or destabilized by certain stochastic inputs. Therefore, noise disturbances have an important effect on the stability of neural networks. Recently, many interesting results on stochastic effect to the stability of neural networks with delays have been reported (see [11, 16, 17]). Moreover, uncertainties are unavoidable in practical implementation of neural networks due to modeling errors and parameter fluctuation, which also cause instability and poor performance [12]. Hence, we can obtain a more perfect model of this situation if we include parameter uncertainties and stochastic effects in neural networks.

Motivated by the above, it is of practical and theoretical importance to study the stability problem of impulsive neural networks with time-varying delays in the leakage term. In this paper, we will investigate stability for a class of stochastic neural networks with time-varying delay in the leakage term and impulses. By using Razumikhin techniques [18–21], some new robust mean-square exponential stability criteria will be given under the corresponding delayed neural networks without impulses which are stable and unstable, respectively.

2. Problem Formulation

Consider the following uncertain neural networks model with impulses and leakage time-varying delay where is the neuron state vector of the neural networks, , represents the neuron activation function, and are the connection weight matrix. is the time-varying delay and is the leakage time-varying delay satisfying , ,  . is a -dimensional Brownian motion defined on complete probability space .    is impulsive time. represents the jump in the state at with determining the size of the jump and . The initial conditions , where denotes the family of all bounded measurable and valued random variable , satisfying ; denotes the mathematical expectation. .

Throughout this paper, symmetric matrix (resp., ) means that the matrix is positive semidefinite (resp., positive definite). denotes an identity matrix. The notation represents the transpose of the matrix . The symmetric terms in asymmetric matrix are denoted by . and mean the largest and the smallest eigenvalue of , respectively.

In this paper, we assume that is the equilibrium point of the system (1). And we have the following assumptions.(A₁)The neuron activation function is continuous on and satisfies where are some real constants and they may be positive, zero, or negative.(A₂)Consider where , , and .(A₃)Consider − . Let , and system (1) becomes where , .(A₄)We consider the parameter uncertainties expressed as where , , , , and are known real constant matrices; , , , , and are unknown matrices representing the parameter uncertainties, which are assumed to be the following form: where , , , () are known real constant matrices and are unknown real time-varying matrix functions satisfying , .

Remark 1. Assumptions (A₁)–(A₄) imply that system (4) satisfies the local Lipschitz condition and linear growth condition. Thus there exists a unique solution of system (4).

Definition 2. The equilibrium point of system (1) is said to be robustly exponentially stable in the mean square, if there exists scalars and such that for any and initial condition satisfying which implies ,   .

Lemma 3 (see [22]). Given matrices , , and with and a scalar , then

3. Main Results

Theorem 4. Suppose that assumptions (A₁)–(A₄) hold and for prescribed scalar , choose positive scalars , and , , , such that . Then the equilibrium point of system (1) is robustly exponentially stable in the mean square over any impulse time sequences satisfying , if there exist positive definite matrix and definite diagonal matrices , , and positive constants , , such that the following LMI holds:
where , + , , , .

Proof. Since the matrix inequality (9) holds, we can choose small enough scalars , satisfying and , such that
where , , .
Let , ; then , , .
Define Lyapunov function From the condition (8), applying Schur complement [23] and Lemma 3, we have Therefore, when , When , applying the Itô formula, we have Let From assumption (A₁), the following inequalities hold for any diagonal matrices , , Set Combining (15)–(16) together, we have where Using the similar method for uncertain parameters as above and from (10), we can get . Then we have ; that is, Let , . For any , there exists a , such that .
In the following, we will prove that when the initial function satisfies , we have First, for , Then we will prove that If the above inequality does not hold, then there exist , such that . Set ; then and . Hence for all , , which implies that It follows from (20) that for any , and , we have which leads to . This is a contradiction.
Thus (23) holds.
Now we assume that for some , , ; we will prove that From (13), we have Suppose (26) does not hold; then there exists and . Set , and then from (27), and . In the sequel, the proof is very similar with the proof of (23). Therefore (26) holds. By mathematical induction, inequality (21) holds. This together with , we have This completes the proof of Theorem 4.

Remark 5. Theorem 4 shows that robustly exponential stability of system (1) can be achieved by adjusting suitable impulsive control and appropriate impulsive intervals even if the given networks without impulses may be unstable or chaotic themselves.

Remark 6. In [5], the authors investigated the stability of neural networks with delayed leakage term and impulsive perturbations. However, the neural networks without impulses must be stable. Moreover, parameter uncertainties and stochastic effects were not taken into account in the models. Hence, the results in this paper have wider adaptive range.

Theorem 7. Suppose that assumptions (A₁)–(A₄) hold and for prescribed scalars , choose positive scalars , , , , , satisfying and . Then the equilibrium point of system (1) is robustly exponentially stable in the mean square over any impulse time sequences satisfying , if there exist positive definite matrix and positive definite diagonal matrices , and positive constants , , such that (8) and the following LMI hold:
where and , .

Proof. Since the matrix inequality (29) holds, we can choose small enough scalars , satisfying , such that the following matrix inequality holds:
where , , and .
Set , ; then , , .
Define Lyapunov function and function Similar to the proof Theorem 4, if (8) and (29) hold, we have and for , which implies that
For any , there exists , such that . In the following, we will prove that when the initial function satisfies , we have First, for , Then we will prove that If (37) does not hold, there exists . From (36), and . Set ; then and .
So for and any , we have . It follows from (34) that for any , and , Then we have . This is a contradiction. Thus (37) holds.
Suppose that for some , , . We will prove that We first claim that . The proof is very similar to [18, 19], so we omit it here. Next, we show that Suppose not; then we have . Without loss of generality, we assume . There are two cases to be considered.
Case 1. for any . Then we have It follows from (34) that for any , and (38) holds, which leads to This is a contradiction.
Case 2. There exist some , such that .
Set . Then we have and . Hence for , we have It follows from (34) that for any , and (38) holds, which leads to This is a contradiction.
Therefore (40) holds. By the same methods, we can prove . Then from (33), we get Suppose (39) does not hold; then there exist and . If for all , set . Otherwise let . Then for and any , we have . It follows from (34) that for any , , and , (38) holds. Then, .
This is a contradiction. Thus (39) holds.
The next proof is very similar to Theorem 4. This completes the proof.

Remark 8. Theorem 7 shows that the system will remain exponentially stable on the condition that the impulses, which may destabilize the system, do not occur too frequently.

Remark 9. If there is no leakage delay, that is, , , , , , then the system (4) is the one investigated in [19]. If there is no stochastic perturbation either, then system (4) is the one investigated in [18, 20].

4. Examples

In this section, we present some examples to verify the effectiveness of the theoretical results.

Example 1. Consider (4) with two neurons. The uncertain parameters satisfy assumption (A₄), where , , , , , , , , . . , and it is obvious that , . Choose , , , , for ; then the LMIs in Theorem 4 have the following feasible solution via MATLAB LMI toolbox: , , , , , . Thus from Theorem 4, the equilibrium of system (4) is robustly exponentially stable in the mean square.

Example 2. We consider neural network shown in [5] as follows: where . As shown in [5], the system (46) is stable when , and it becomes unstable when . We consider that the system has the following impulsive perturbation at times : where , . It is obvious that , . For , choosing , , , , , by using the LMI toolbox in MATLAB, a feasible solution of Theorem 4 is Thus from Theorem 4, the equilibrium of system (46) is robustly exponentially stable in the mean square. It can be seen that the impulses play an important role in inducing the stability of neural network in the leakage time delay.

5. Conclusion

Robust exponential stability of stochastic neural networks with time-varying delay in the leakage term under impulsive perturbations is investigated. The leakage delay is time varying and the impulsive perturbations depend not only on the current state of neurons at impulse times but also on the state of neurons in its recent history. Based on Lyapunov functions and Razumikhin techniques, some new criteria are derived. Some examples have been given to demonstrate that, even though the corresponding delayed neural networks without impulses are unstable, impulses may compensate the deviating trend.

Conflict of Interests

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

Acknowledgments

The authors wish to thank the referees for their suggestions and comments. This work is supported by the National Science Foundation of China (11171374) and the Key National Science Foundation of Shandong Province (ZR2011AZ001).