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The p th moment exponential stability of stochastic cellular neural networks with impulses
Advances in Difference Equations volume 2013, Article number: 6 (2013)
This paper studies the p th moment exponential stability of stochastic cellular neural networks with time-varying delays under impulsive perturbations. Based on the Lyapunov function, Razumikhin theory, stochastic analysis and differential inequality technique, criteria on the p th moment exponential stability of this model are derived. These results generalize and improve some of the existing ones. A numerical example illustrates the effectiveness and improvements of our results.
Since Chua and Yang [1, 2] introduced a cellular neural network in 1988, it has received great attention because of its various applications such as classification of patterns, associative memories and optimization, etc. It should be pointed out that time delays are commonly encountered in real systems, which are the source of oscillation and instability both in biological and artificial neural networks, hence it is necessary and important to discuss the delayed cellular neural networks models. Up to now, many results on the stability of delayed neural networks have been developed [3–5]. In fact, in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. Therefore, noise is unavoidable and should be taken into consideration in modeling. Some recent results of stochastic cellular neural networks with delays can be found in [6–15].
On the other hand, it is noteworthy that the state of electronic networks is often subjected to some phenomenon or other sudden noises. On that account, the electronic networks will experience some abrupt changes at certain instants that in turn affect dynamical behaviors of the systems. Therefore, it is necessary to take both stochastic effects and impulsive perturbations into account on dynamical behaviors of delayed neural networks. In recent years, the dynamic analysis of neural networks with impulsive and stochastic effects has been an attractive topic for many researchers, and a large number of stability criteria of these systems have been reported; see [3, 4, 10–12, 16, 17].
In , Sun et al. investigated the following stochastic cellular neural networks model with time-varying delays:
where denotes the potential (or voltage) of a cell i at time t; , n corresponds to the number of units in a neural network; , are activation functions; denotes the rate at which a cell i resets its potential to the resting state when isolated from other cells and inputs; and denote the strengths of connectivity between cells i and j, respectively; denotes the external bias on the i th unit, satisfies and it is a transmission delay. is the diffusion coefficient matrix and is the i th row vector of . is an m-dimensional Brownian motion defined on a complete probability space with a natural filtration .
They investigated the p th moment exponential stability with the help of the method of variation parameter and inequality technique, where denotes a positive constant. More precisely, they established the following fundamental assumptions:
For each , is a differentiable function, namely, there exists ζ such that
(H1) Functions and are Lipschitz-continuous on ℝ with Lipschitz constants , . That is, for all , ,
(H2) There exist nonnegative constants , , such that for all , ,
Huang et al.  studied (1.1) and obtained the p th moment exponential stability by using Dini-derivative and Halanay-type inequality without assumption (H). When , the equilibrium point of the system (1.1) is p th moment exponentially stable, where
where () are positive constants.
Very recently, Li  generalized (1.1); he considered a stochastic cellular neural network under impulsive perturbations. The condition is also needed to ensure exponential stability in mean square.
In this paper, we solve this question and obtain the improved version of the p th moment exponential stability by applying Lyapunov functions, Razumikhin theory and inequality technique. An example is also provided to illustrate the effectiveness of the new results.
In this paper, we study stochastic cellular neural networks with impulses described by the delayed differential equations
where is the time sequence and satisfies , ; , . For , represents the abrupt change of the state at the impulsive moments .
System (2.1) is supplemented with the initial condition given by
where is -measurable and continuous everywhere except at a finite number of points , at which and exist and .
Let denote the family of all nonnegative functions on which are continuous once differentiable in t and twice differentiable in x. If , define an operator associated with (2.1) as
Throughout this paper, the following standard hypothesis is needed:
(H3) , where is the equilibrium point of (2.1) with the initial condition (2.2), satisfies , is a positive constant.
Let , then (2.1) can be written by
In the following, for further study, we first give the following definitions and lemmas.
denotes a vector norm defined by
Definition 2.1 (Mao )
The equilibrium point of the system (2.1) is said to be p th moment exponentially stable if there exist and such that
In such a case,
The right-hand side of (2.5) is commonly known as the p th moment Lyapunov exponent of this solution.
When , it is usually said to be exponentially stable in mean square.
Lemma 2.2 If () denote p nonnegative real numbers, then
where denotes an integer.
A particular form of (2.6), namely
3 Main result
Theorem 3.1 Assume that (H1)-(H3) hold; furthermore, let
there exist , such that ;
, , then the equilibrium point of (2.1) is pth moment exponentially stable.
Proof We define a Lyapunov function . Let and , then we can get the operator associated with the system (2.4) of the form as follows:
Let , there exist , such that
Hence, we can choose such that
For convenience, we denote that for .
It is obvious that
Now, we should prove
Firstly, we prove when ,
If (3.7) is not true, there exists such that
Since is continuous on , which implies that there exists such that
then there exists some satisfying
Hence, for any , ,
By (3.1) and (3.2), we get
which is a contradiction. Hence, (3.7) holds.
Next, we will show
Assuming (3.9) holds for , we shall show that it holds for , i.e.,
Suppose (3.10) is not true. Then we define such that
From (H3) we get
which implies . Let
then for , we can get .
Hence, there exists such that
On the other hand, for any , , either or .
If , we can obtain
If , we can get
Then for any , we get
From the condition (ii), it is obvious that
which is a contradiction. Hence, (3.10) holds.
By induction, we can obtain that (3.9) holds for any , i.e.,
which implies that the equilibrium point of the impulsive system (2.1) is p th moment exponentially stable. This completes the proof of the theorem. □
Theorem 3.2 Assume that (H1)-(H3) hold, ,
if there exist , such that ;
then the equilibrium point of the system (2.1) is pth moment exponentially stable.
Proof Let , the proof of the theorem is similar to that of Theorem 3.1 hence it is omitted. □
Corollary 3.3 Assume that (H1)-(H3) hold, ,
if there exist , such that ;
then the equilibrium point of the system (2.1) is exponentially stable in mean square.
Remark 3.4 In many stability results for stochastic cellular neural networks, is an important condition for their conclusions [13–15], which means that the origin systems without impulses need to be stable. However, by constructing the impulses, we do not need this condition to ensure the equilibrium point of the impulsive system (2.1) is p th moment exponentially stable. Our results show that impulses play an important role in the p th moment exponential stability for the stochastic cellular neural network with time delay, even if the corresponding systems may be unstable themselves. It should be mentioned that our results develop an effective impulse control strategy to stabilize underlying retarded cellular neural networks. And it is particularly meaningful for some practical applications.
Remark 3.5 It is important to emphasize that, in contrast to some existing exponential stability results, see [6, 11, 12, 19], the condition is needed to ensure the equilibrium point of the system (2.1) is p th moment exponentially stable, while in our paper we omit it and obtain the results.
4 Illustrative example
In the following, we will give an example to illustrate the advantages of our results.
Example 1 Consider the following model:
where , , , , . , , .
Obviously, , , (), , .
Let , , , . Then for , we can get , , , .
All conditions of Corollary 3.3 are satisfied, then the equilibrium point is exponentially stable in mean square.
Remark 4.1 If , we have computed , . If , set , then , . If , then , we can compute and ; if , then , and . Hence, in either case, we always have , so the exponential stability in mean square of the system (4.1) cannot be derived by applying the corresponding exponential stability result for cellular neural networks given in the literature [6, 11, 12, 19], since is not satisfied.
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The authors would like to thank the editor and the anonymous referees for their detailed comments and valuable suggestions which considerably improved the presentation of this paper. This work was supported in part by the National Natural Science Foundation of China under Grant No. 11101054, the Hunan Provincial Natural Science Foundation of China under Grant No. 12jj4005, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grant No. 11FEFM11 and the Scientific Research Funds of Hunan Provincial Science and Technology Department of China under Grant No. 2012SK3096.
The authors declare that they have no competing interests.
XL completed the proof and wrote the initial draft. JZ gave some suggestions on the amendment. XL then finalized the manuscript. Correspondence was mainly done by EZ. All authors read and approved the final manuscript.
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Li, X., Zou, J. & Zhu, E. The p th moment exponential stability of stochastic cellular neural networks with impulses. Adv Differ Equ 2013, 6 (2013). https://doi.org/10.1186/1687-1847-2013-6
- p th moment exponentially stable
- stochastic cellular neural network
- Razumikhin theory