Abstract
A class of neural networks described by nonlinear impulsive neutral nonautonomous differential equations with delays is considered. By means of Lyapunov functionals and differential inequality technique, criteria on global exponential stability of this model are derived. Many adjustable parameters are introduced in criteria to provide flexibility for the design and analysis of the system. The results of this paper are new and they supplement previously known results. An example is given to illustrate the results.
1. Introduction
Many evolution processes in nature exhibit abrupt changes of states at certain moments. That was the reason for the development of the theory of impulsive differential equations and impulsive delay differential equations; see the monographs [1, 2]. But the theory of impulsive neutral differential equations is not well developed due to some theoretical and technical difficulties. For impulsive neutral differential equations, some existence results and oscillation criteria are obtained in [3–5] and some stability conditions are derived in [6]; for neural networks described by impulsive neutral differential equations with delays, the exponential stability results are obtained in [7–11], but their work focuses on the autonomous system. So in this paper, the exponential stability for neural networks described by nonlinear impulsive neutral nonautonomous differential equations with delays is considered.
The purpose of this paper is to study the stability of the following impulsive neural networks with variable coefficients and several time-varying delays:where corresponds to the number of units in a neural network; for , denotes the potential of cell at time ; , , correspond to the transmission delays. (1a) (called continuous part) describes the continuous evolution processes of the neural networks. For , , , and denote the strengths of connectivity between cells and at time , respectively; , , show how the th neuron reacts to the input; is the external bias on the th at time . (1b) (called discrete part) describes that the evolution processes experience abrupt change of states at the moments of (called impulsive moments); for , , the fixed moment satisfies , and ; represents impulsive perturbations of th unit at time ; represents impulsive perturbations of th unit at time , which is caused by the transmission delays; represents the external impulsive input at time .
The theory on linear matrix inequality (LMI) or -Matrix provides effective methods for the analysis of exponential stability of autonomous neural networks. See [7, 9, 10] and the reference therein. But for nonautonomous neural networks, it is invalid. Differential inequalities are important tools for investigating the stability of impulsive differential equations. See [7, 8, 12, 13] and the reference therein. The method in this paper is partially motivated by the work in [7].
In this paper, we will investigate the global exponential stability of the nonautonomous neural networks and focus on the effect of impulse on the dynamic behavior of (1a) and (1b). The results do not require the boundedness of and the differentiability of . So they are new and complement previously known results.
For a continuous function , we denote
Define
For any , , , and , define , , , and as respectively.
For convenience, the following conditions are listed.(H1)For , and ,.(H2)There are positive constants , , such that
for all .(H3)There exist positive constants and , , such that
for all .(H4)There exist positive constants and such that
for .
We assume that (1a) and (1b) are with the following initial conditions: where . According to [13], the initial value problems (1a), (1b), and (8) have the unique solution under assumptions (H2) and (H3).
Definition 1. A function is said to be a solution of (1a) and (1b) on if for ,(i) is absolutely continuous on each interval and , ;(ii)for any , and exist and ;(iii) satisfies (1a) for almost everywhere in and satisfies (1b) for every , .
Obviously, a solution of (1a) and (1b) is continuous at and discontinuous at . Furthermore, has discontinuities of the first kind at the fixed impulsive moments and some moments , . Denote , .
Definition 2. Let and be two solutions of (1a), (1b), and (8) with and , respectively, where and . If there exist and such that then (1a) and (1b) are said to be globally exponentially stable.
2. The Main Result
To study the exponential stability of (1a) and (1b), we need the following lemma.
Lemma 3. Assume that (H1) and (H4) hold and there exist nonnegative vector functions and , where is continuous at (), such thatfor , , . Then for all and , there exists a positive constant such that where and are defined, respectively, as
Proof. By the similar analysis in [14, Lemma 4.1], we can deduce that and exist uniquely and , under the assumption of (H1) and (H4). Consequently, . Choose a positive constant such that . Let
Then for all and , we have
Then
For the sake of contradiction, assume that there exist and such that
From (17), we have
similarly,
Then we have the following cases.(I); then we have the following subcases.(i), . So is continuous at . By (17), we have
From , (17)–(19), and the definition of , we have
which is a contradiction with (20). (ii)There exists a such that . By (17), we have
Noting , we have or . Without loss of generality, we assume that . From (10c) and (22), we get that
Simplifying (23), we obtain , which contradict (12).
If (I) does not hold, then
(II)
Then from (10b) and (17)–(19), we have
which is a contradiction.
From (I) and (II), (16) holds. Letting in (16), we have
So for all , . Let ; then for and , we have
The proof of Lemma 3 is complete.
Theorem 4. Assume that (H1)–(H4) hold. Then systems (1a) and (1b) are globally exponentially stable.
Proof. Let and be solutions of (1a), (1b), and (8) with and , respectively. Let
By (1a) and (1b), for , we have
By (1b) and (H3), we have
By (29)–(31) and Lemma 3, there exists a positive constant such that
where and are defined in (12).
Remark 5. For autonomous system, the exponential stability of the zero solution of (1a) with , is considered in [7]. But the results require that is bounded.
When there is no impulse in systems (1a) and (1b), (1a) and (1b) reduce to the following model which has been studied in [9, 10]:
Corollary 6. Assume that , , and hold. (33) is globally exponentially stable.
Remark 7. For autonomous system, the stability of (33) with , , is considered in [10]. However, the authors assume that , are monotonic, bounded and ,
are constants.
Remark 8. The stability results about the zero solution of are obtained by the fixed-point theory in [15]. But the differentiability of is needed.
3. An Illustrative Example
To show the effectiveness of Theorem 4, consider the following nonautonomous neural networks with impulse:where Obviously, , , and .
Let and , . From the above assumption, the conditions of Theorem 4 are satisfied. Therefore, (34a) and (34b) are globally exponentially stable. and are the solutions of (34a) and (34b) with , and , , respectively. Figures 1(a) and 1(b) depict time response of state variables , without and with impulse effects; Figures 2(a) and 2(b) depict time response of state variables , without and with impulse effects; Figures 3(a) and 3(b) depict the phase plot in the space , without and with impulse effects.
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Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Science Foundation of Shanxi Province (no. 2010021001-1) and the National Natural Science Foundation of China (nos. 11101251 and 11001157).