Abstract
Impulsive Cohen-Grossberg neural networks with bounded and unbounded delays (i.e., mixed delays) are investigated. By using the Leray-Schauder fixed point theorem, differential inequality techniques, and constructing suitable Lyapunov functional, several new sufficient conditions on the existence and global exponential stability of periodic solution for the system are obtained, which improves some of the known results. An example and its numerical simulations are employed to illustrate our feasible results.
1. Introduction
In the recent years, dynamics of the Cohen-Grossberg neural networks (CGNNs) [1] has been extensively studied because of their immense potentials of application perspective in different areas such as pattern recognition, parallel computing, associative memory, combinational optimization, and signal and image processing [2–6]. The authors of [7–11] have studied the stability of equilibrium point or periodic solution of CGNNs with time-varying delays due to the transmission delays during the communication between neurons which will affect the dynamical behavior of neural networks. Considering the distant past also has influence on the recent behavior of the state, the authors of [12] investigated the stability of equilibrium point for CGNNs with continuously distributed delays. Impulsive effects are also likely to exist in the neural networks, that is, the state of the networks is subject to instantaneous perturbations and experiences abrupt change at certain moments. Authors of [13–16] have studied the stability of equilibrium point for impulsive delay CGNNs. However, the activation functions of CGNNs in [12–16] are bounded, and the restrictions on impulses in [13, 17] are very strong, which limit CGNNs' applications [7].
In theory and applications, global stability of periodic solution of CGNNs is of great importance since the global stability of equilibrium points can be considered as a special case of periodic solution with random period. Moreover, CGNNs model is one of the most popular and typical neural network models. Some other models, such as Hopfield-type neural networks, cellular neural networks, and bidirectional associative memory neural networks, are special cases of the model [14]. To our best knowledge, few authors have considered the existence and global exponential stability of periodic solutions for CGNNs with mixed delays and impulses. Therefore, it is necessary to consider the existence and global exponential stability of periodic solution for impulsive CGNNs with mixed delays.
The main methods used in this paper are Leray-Schauder's fixed point theorem, differential inequality techniques, and Lyapunov functional. Several new sufficient conditions are obtained for the existence and global exponential stability of periodic solution for impulsive CGNNs with mixed delays. Moreover, we discharge some restrictive conditions on the activation functions of the neurons, such as boundedness, monotonicity, and differentiable, we offer the precise convergence index, and give the weak conditions on impulses.
The rest of this paper is organized as follows. In Section 2, we introduce some notations and definitions and state some preliminary results needed in later sections. We then study, in Section 3, the existence of periodic solutions of impulsive CGNNs with mixed delays by using Leray-Schauder's fixed point theorem. In Section 4, with the help of Lyapunov functional, we will derive sufficient conditions for the global exponential stability of the periodic solution. At last, an example and its numerical simulations are employed to illustrate the feasible results of this paper.
2. Preliminaries
Consider the following CGNNs with mixed delays and impulses: where are the impulses at moments , is left continuous at time , and the right limit exists at , that is, and exist and is a strictly increasing sequence such that ; is the state of neuron; and represent an amplification function at time and an appropriately behaved function at time , respectively; and are connection matrices; is the input function; and are the activation functions of the neurons.
Throughout this paper, we assume that (H1), are continuous -periodic functions, is a constant, ;(H2) is continuous and there exist positive constants and such that , , ;(H3)there is a positive constant such that , denotes the derivative of , and , ;(H4) and are -periodic sequence, that is, there exists a positive constant such that , , ;(H5), there are positive constants , , such that , ;(H6)the delay kernels are continuous, integrable and there exist positive constants such that(H7) where ;(H8)there exists a constant such that
From , the antiderivative of exists. We choose an antiderivative of that satisfies . Obviously, . By , we obtain that is strictly monotone increasing about . In view of derivative theorem for inverse function, the inverse function of is differential and . By , composition function is differential. Denote it is easy to see that and . Substituting these equalities into system (2.1), we get which can be rewritten as where , denote the derivative of at point , , is between 0 and .
The existence and global exponential stability of periodic solution for system (2.1) are equivalent to the existence and global exponential stability of periodic solution for system (2.5) or (2.6). So, we investigate the the existence and global exponential stability of periodic solution for system (2.6).
From the definition of , using Lagrange mean value theorem, one gets where is between and . Moreover, we have
For convenience, we denote , to be a column vector, in which the symbol denotes the transpose of a vector.
Definition 2.1. A function is said to be a solution of (2.6) if the
following conditions are satisfied.
(i) is absolutely continuous on each , (ii)For each , and exist and .(iii) satisfies (2.6) for almost everywhere and at
impulsive points, may have discontinuity of the first kind.
The initial condition of (2.6) is of the form
where are continuous functions.
Definition 2.2. Let be an -periodic solution of (2.6) with initial value . If there exist constants , such that for every solution of (2.6) with initial value , where . Then, is said to be globally exponentially stable.
Lemma 2.3 (Leray-Schauder). Let be a Banach space, and let the operator be completely continuous. If the set is bounded, then has a fixed point in , where
3. Existence of Periodic Solutions
Lemma 3.1. Suppose that hold and let be an -periodic solution of system (2.6). Then, where
Proof. Let , . From the first expression of (2.6), we haveIntegrating (3.3) on intervals and adding all of them, by the second formula of (2.6), we have Since , from (3.4), we obtain Substituting (3.5) into (3.4), we obtain (3.1). This completes the proof.
In order to use Lemma 2.3, we take as piecewise continuous periodic solution at , and exist at . Then, is a Banach space with the norm
Set a mapping by settingwhere
It is easy to know the fact that the existence of -periodic solution of (2.6) is equivalent to the existence of fixed point of the mapping in .
Lemma 3.2. Suppose that hold. Then, is completely continuous.
Proof. First, we show that is continuous. For any ,
we take ,
where .
Then, for all and , we have
Hence, is continuous.
Next, we show maps bounded set into bounded set. For any with ,
where is some positive constant, we have
where , , .
Equation (3.10) implies that is uniformly bounded for any .
Hence, is a family of uniformly bounded and
equicontinuous functions on .
By using the Arzela-Ascoli theorem, is compact. Therefore, is completely continuous. This completes the
proof.
Theorem 3.3. Suppose that hold. Then, system (2.6) has an -periodic solution.
Proof. Let , . We consider the operator equation If is a solution of (3.11), for , we obtain where this and imply that This shows that of (3.11) is bounded, which is independent of . In view of Lemma 2.3, we obtain that has a fixed point. Hence, system (2.6) has one -periodic solution with . This completes the proof.
4. Global Exponential Stability of Periodic Solution
In this section, we will construct some suitable Lyapunov functionals to derive sufficient conditions which ensure that (2.6) has a unique -periodic solution, and all solutions of (2.6) exponentially converge to its unique -periodic solution.
Theorem 4.1. Assume that hold and (H9) where , is a positive constant, is given in the proof of this theorem, . Then, system (2.6) has exactly one -periodic solution, which is globally exponentially stable with the convergence index .
Proof. By Theorem 3.3, there exists an -periodic solution of (2.6) with initial value . Suppose that is an arbitrary solution of system (2.6) with
initial value .
Set then, from system (2.5), we
have
By ,
we haveSet
Clearly, are continuous functions on .
Since
and ,
hence are strictly monotone increasing functions.
Therefore, for any and ,
there is a unique such that
Let .
Obviously, Now, we will prove that .
Suppose this is not true, from (4.2), there exists a positive constant such that
Pick small ,
then there exists such that
Let us recall the inequality for sufficiently small then, we obtain
which is a contradiction, and
hence, .
Let .
Obviously, for all we have
It is obvious that
where is defined as that in Definition 2.2.
Define the Lyapunov functional by , In view of (4.1), for , we obtain
We claim that
Contrarily, there must
exist and such that
Together with (4.11) and (4.13),
we obtain
Hence,
which contradicts (4.9). Hence,
(4.12) holds. It follows that
When ,
from the second expression of (4.1), we have
Similar to the steps of
(4.9)–(4.16), we can also prove that
When , again, from the second expression of (4.1), we have
By repeating the same procedure,
we can deduce the following general result:
By ,
we have , which implies that
So, in light of (4.20) and
(4.21), we have
In view of Definition 2.2, the -periodic solution of system (2.6) is globally exponentially
stable with the convergence index .
This completes the proof.
Remark 4.2. Note that we have dropped the restriction: the activation functions are bounded and , which is indispensable in [13, 17].
Remark 4.3. To the best of our knowledge, most of the existing research papers only give the existence of the convergence index, while we offer the precise convergence index in Theorem 4.1.
5. Application
In this section, we give an example to illustrate that our results are feasible. Consider the following Cohen-Grossberg neural networks with mixed delays and impulses: where , , , , , , :
We have , , , , : . Submitting into (4.9), we have , , (see Figures 1(a) and 1(b)). This implies that . Hence, all the conditions needed in Theorem 4.1 are satisfied. Therefore, system (5.1) has a unique 4-periodic solution, which is globally exponentially stable (see Figures 1(c)–1(g)).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Remark 5.1. System (5.1) is a simple CGNNs with time-varying coefficients, mixed delays, and impulses. In system (5.1), the activation functions are all unbounded and , . Thus, none of the results in [13–17] can be applied to (5.1). This implies that the results of this paper are new and complement previously known results.
Acknowledgments
The authors would like to thank the reviewers for their valuable comments and constructive suggestions which considerably improve the presentation of this paper. This work was supported in part by the China Postdoctoral Science Foundation (20070410300, 200801336), the Foundation of Chinese Society for Electrical Engineering, the Hunan Provincial Natural Science Foundation of China (07JJ4001), the Scientific Research Fund of Yunnan Provincial Education Department (07Y10085), the Key Scientific Research Fund of Yunnan Provincial Education Department (5Z0071A), and the Scientific Research Fund of Honghe University (XSS07001).