In this paper, a class of impulsive bidirectional associative memory (BAM) fuzzy cellular neural networks (FCNNs) with time delays in the leakage terms and distributed delays is formulated and investigated. By establishing an integro-differential inequality with impulsive initial conditions and employing M-matrix theory, some sufficient conditions ensuring the existence, uniqueness and global exponential stability of equilibrium point for impulsive BAM FCNNs with time delays in the leakage terms and distributed delays are obtained. In particular, the estimate of the exponential convergence rate is also provided, which depends on the delay kernel functions and system parameters. It is believed that these results are significant and useful for the design and applications of BAM FCNNs. An example is given to show the effectiveness of the results obtained here.
The online version of this article (doi:10.1186/1029-242X-2011-43) contains supplementary material, which is available to authorized users.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
ZX designed and performed all the steps of proof in this research and also wrote the paper. KL participated in the design of the study and helped to draft and revise manuscript. All authors read and approved the final manuscript.
1 Introduction
The bidirectional associative memory (BAM) neural network models were first introduced by Kosko [1]. It is a special class of recurrent neural networks that can store bipolar vector pairs. The BAM neural network is composed of neurons arranged in two layers, the X-layer and Y-layer. The neurons in one layer are fully interconnected to the neurons in the other layer. Through iterations of forward and backward information flows between the two layer, it performs a two-way associative search for stored bipolar vector pairs and generalize the single-layer autoassociative Hebbian correlation to a two-layer pattern-matched heteroassociative circuits. Therefore, this class of networks possesses good application prospects in some fields such as pattern recognition, signal and image process, and artificial intelligence [2]. In such applications, the stability of networks plays an important role; it is of significance and necessary to investigate the stability. It is well known, in both biological and artificial neural networks, the delays arise because of the processing of information. Time delays may lead to oscillation, divergence or instability which may be harmful to a system. Therefore, study of neural dynamics with consideration of the delayed problem becomes extremely important to manufacture high-quality neural networks. In recent years, there have been many analytical results for BAM neural networks with various axonal signal transmission delays, for example, see [3‐11] and references therein. In addition, except various axonal signal transmission delays, time delay in the leakage term also has great impact on the dynamics of neural networks. As pointed out by Gopalsamy [12, 13], time delay in the stabilizing negative feedback term has a tendency to destabilize a system. Recently, some authors have paid attention to stability analysis of neural networks with time delays in the leakage (or "forgetting") terms [12‐18].
Since FCNNs were introduced by Yang et. al [19, 20], many researchers have done extensive works on this subject due to their extensive applications in classification of image processing and pattern recognition. Specially, in the past few years, the stability analysis on FCNNs with various delays and fuzzy BAM neural networks with transmission delays has been the highlight in the neural network field, for example, see [21‐27] and references therein. On the other hand, in respect of the complexity, besides delay effect, impulsive effect likewise exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time, involving such fields as medicine and biology, economics, mechanics, electronics and telecommunications. Many interesting results on impulsive effect have been gained, e.g., Refs. [28‐37]. As artificial electronic systems, neural networks such as CNNs, bidirectional neural networks and recurrent neural networks often are subject to impulsive perturbations, which can affect dynamical behaviors of the systems just as time delays. Therefore, it is necessary to consider both impulsive effect and delay effect on the stability of neural networks. To the best of our knowledge, few authors have considered impulsive BAM FCNNs with time delays in the leakage terms and distributed delays.
Anzeige
Motivated by the above discussions, the objective of this paper is to formulate and study impulsive BAM FCNNs with time delays in the leakage terms and distributed delays. Under quite general conditions, some sufficient conditions ensuring the existence, uniqueness and global exponential stability of equilibrium point are obtained by the topological degree theory, properties of M-matrix, the integro-differential inequality with impulsive initial conditions and analysis technique.
The paper is organized as follows. In Section 2, the new neural network model is formulated, and the necessary knowledge is provided. The existence and uniqueness of equilibrium point are presented in Section 3. In Section 4, we give some sufficient conditions of exponential stability of the impulsive BAM FCNNs with time delays in the leakage terms and distributed delays. An example is given to show the effectiveness of the results obtained here in Section 5. Finally, in Section 6, we give the conclusion.
2 Model description and preliminaries
In this section, we will consider the model of impulsive BAM FCNNs with time delays in the leakage terms and distributed delays, it is described by the following functional differential equation:
(1)
for i = 1, 2, ..., n, j = 1, 2,..., m, t > 0, where xi (t) and yj (t) are the states of the i th neuron and the j th neuron at time t, respectively; δi ≥ 0 and θj ≥ 0 denote the leakage delays, respectively; fi and gj denote the signal functions of the i th neuron and the j th neuron at time t, respectively; ui , vj and Ii , Jj denote inputs and bias of the i th neuron and the j th neuron, respectively; ai > 0, bj > 0, are constants, ai and bj represent the rate with which the i th neuron and the j th neuron will reset their potential to the resting state in isolation when disconnected from the networks and external inputs, respectively; aij , bji and denote connection weights of feedback template and feedforward template, respectively; αij, βji and denote connection weights of the distributed fuzzy feedback MIN template and the distributed fuzzy feedback MAX template, respectively; and are elements of fuzzy feedforward MIN template and fuzzy feedforward MAX template, respectively; ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operations, respectively; Kij (s) and correspond to the delay kernel functions, respectively. tk is called impulsive moment and satisfies 0 < t1 < t2 < ⋯, ; and denote the left-hand and right-hand limits at tk , respectively; Pik and Qjk show impulsive perturbations of the i th neuron and j th neuron at time tk , respectively.
Anzeige
We always assume and , k ∈ N . The initial conditions are given by
where ϕi (t), φj (t) (i = 1, 2, ..., n; j = 1, 2, ..., m) are bounded and continuous on (-∞, 0], respectively.
If the impulsive operators Pik (xi ) = 0, Qjk (yj ) = 0, i = 1, 2, ..., n, j = 1, 2, ..., m, k ∈ N, then system (1) may reduce to the following model:
(2)
System (2) is called the continuous system of model (1).
Throughout this paper, we make the following assumptions:
(H1) For neuron activation functions fi and gj (i = 1, 2, ..., n; j = 1, 2, ..., m), there exist two positive diagonal matrices F = diag(F1, F2, ..., Fn ) and G = diag(G1, G2, ..., Gm ) such that
for all x, y ∈ R (x ≠ y).
(H2) The delay kernels Kij : [0, +∞) → R and are real-valued piecewise continuous, and there exists δ > 0 such that
Are continuous for λ ∈ [0,δ), i = 1,2, ..., n, j = 1,2, ..., m.
(H3) Let and be Lipschitz continuous in Rn and Rm , respectively, that is, there exist nonnegative diagnose matrices Γk = diag(γ1k, γ2k, ..., γnk ) and such that
where
To begin with, we introduce some notation and recall some basic definitions.
PC[J, Rl ] = {z(t): J → Rl |z(t) is continuous at t ≠ tk , , and exists for t, tk ∈ J, k ∈ N}, where J ⊂ R is an interval, l ∈ N.
PC = {ψ: (-∞, 0] → Rl | ψ(s) is bounded, and ψ(s+) = ψ(s) for s ∈ (-∞, 0), ψ(s-) exists for s ∈ (-∞, 0], ϕ(s-) = ϕ(s) for all but at most a finite number of points s ∈ (-∞, 0]}.
For an m × n matrix A, |A| denotes the absolute value matrix given by |A| = (|aij |)m ×n. For A = (aij )m × n, B = (bij )m × n∈ Rm × n, A ≥ B (A > B) means that each pair of corresponding elements of A and B such that the inequality aij ≥ bij (aij> bij ).
Definition 1A function (x, y) T : (-∞, +∞) → Rn+mis said to be the special solution of system (1) with initial conditions
if the following two conditions are satisfied
(i)
(x, y) Tis piecewise continuous with first kind discontinuity at the points tk , k ∈ K. Moreover, (x, y) Tis right continuous at each discontinuity point.
(ii)
(x, y) Tsatisfies model (1) for t ≥ 0, and x(s) = ϕ(s), y(s) = φ(s) for s ∈ (-∞, 0].
Especially, a point (x*, y*) T ∈ Rn+mis called an equilibrium point of model (1), if (x(t), y(t)) T = (x*, y*) Tis a solution of (1).
Throughout this paper, we always assume that the impulsive jumps Pk and Qk satisfy (referring to [28‐37])
i.e.,
(3)
where (x*, y*) T is the equilibrium point of continuous systems (2). That is, if (x*, y*) T is an equilibrium point of continuous system (2), then (x*, y*) T is also the equilibrium of impulsive system (1).
Definition 2The equilibrium point (x*, y*) Tof model (1) is said to be globally exponentially stable, if there exist constants λ > 0 and M ≥ 1 such that
for all t ≥ 0, where (x(t), y(t)) Tis any solution of system (1) with initial value (ϕ(s), φ(s)) Tand
Definition 3A real matrix D = (dij )n × nis said to be a nonsingular M-matrix if dij ≤ 0, i, j = 1, 2, ..., n, i ≠ j, and all successive principal minors of D are positive.
Lemma 1[38]Let D = (dij )n × nwith dij ≤ 0 (i ≠ j), then the following statements are true:
(i)
D is a nonsingular M-matrix if and only if D is inverse-positive, that is, D-1exists and D-1is a nonnegative matrix.
(ii)
D is a nonsingular M-matrix if and only if there exists a positive vector ξ = (ξ1, ξ2, ..., ξn ) Tsuch that Dξ > 0.
Lemma 2[20]For any positive integer n, let hj : R → R be a function (j = 1, 2, ..., n), then we have
for all α = (α1, α2, ..., αn ) T , u = (u1, u2, ..., un ) T , v = (v1, v2, ..., vn ) T ∈ Rn .
3 Existence and uniqueness of equilibrium point
In this section, we will proof the existence and uniqueness of equilibrium point of model (1). For the sake of simplification, let
then model (2) is reduced to
(4)
It is evident that the dynamical characteristics of model (2) are as same as of model (4).
Theorem 1Under assumptions (H1) and (H2), system (1) has one unique equilibrium point, if the following condition holds,
(C1) there exist vectors ξ = (ξ1, ξ2, ..., ξn ) T > 0, η = (η1, η2, ..., ηm ) T > 0 and positive number λ > 0 such that
Proof. Let , where
for i = 1, 2, ..., n; j = 1, 2, ..., m. Obviously, from assumption (H2), the equilibrium points of model (4) are the solutions of system of equations:
(5)
Define the following homotopic mapping:
H(x1, ..., xn , y1, ..., ym ) = θh(x1, ..., xn , y1, ..., ym ) + (1 - θ)(x1, ..., xn , y1, ..., ym ) T , where θ ∈ [0, 1]. Let Hk (k = 1, 2, ..., n + m) denote the k th component of H(x1, ..., xn , y1, ..., ym ), then from assumption (H1) and Lemma 2, we have
(6)
for i = 1, 2, ..., n, j = 1, 2, ..., m. Denote
Then, the matrix form of (6) is
Since condition (C1) holds, and kij (λ), are continuous on [0, δ ), when λ = 0 in (C1), we obtain
or in matrix form,
(7)
From Lemma 1, we know that C - TL is a nonsingular M-matrix, so (C - TL)-1 is a nonnegative matrix. Let
then Γ is nonempty, and from (6), for any z = (x1,..., xn , y1,..., ym ) T ∈∂Γ, we have
Therefore, for any (x1, ..., xn , y1, ..., ym ) T ∈ ∂Γ and θ ∈ [0, 1], we have H ≠ 0. From homotopy invariance theorem [39], we get
by topological degree theory, we know that (5) has at least one solution in Γ. That is, model (4) has at least an equilibrium point.
Now, we show that the solution of the system of Equations (5) is unique. Assume that and are two solutions of the system of Equations (5), then
it follows that
By using of Lemma 2 and hypothesis (H1), we have
(8)
Let , then the matrix form of (8) is (C -TL)Z ≤ 0. Since C - TL is a nonsingular M-matrix, (C - TL) -1 ≥ 0, thus Z ≤ 0, accordingly, Z = 0, i.e., , . This shows that model (4) has one unique equilibrium point. According to (3), this implies that system (1) has one unique equilibrium point. The proof is completed.
Corollary 1Under assumptions (H1) and (H2), system (1) has one unique equilibrium point if C - TL is a nonsingular M-matrix.
Proof. Since that C - TL is a nonsingular M-matrix, from Lemma 1, there exists a vector ω = (ξ1, ... ξn , η1, ..., ηm ) T > 0 such that (C TL) ω > 0, or (-C + TL) ω < 0. It follows that
From the continuity of kij (λ) and , it is easy to know that there exists λ > 0 such that
That is, condition (C1) holds. This completes the proof.
4 Exponential stability and exponential convergence rate
In this section, we will discuss the global exponential stability of system (1) and give an estimation of exponential convergence rate.
Lemma 3Let a < b ≤ +∞, and u(t) = (u1(t), ..., un (t)) T ∈ PC[[a, b), Rn ] and v(t) = (v1(t), ..., vm (t)) T ∈ PC[[a, b), Rm ] satisfy the following integro-differential inequalities with the initial conditions u(s) ∈ PC[(-∞, 0], Rn ] and v(s) ∈ PC[(-∞, 0], Rm ]:
(9)
for i = 1, 2, ..., n, j = 1, 2, ..., m, where ri > 0, pij > 0, qij > 0, , , , i = 1, 2,...,n, j = 1, 2, ..., m. If the initial conditions satisfy
(10)
in which λ > 0, ξ = (ξ1, ξ2, ..., ξn ) T > 0 and η = (η1, η2, ..., ηm ) T > 0 satisfy
(11)
Then
Proof. For i ∈ {1, 2, ..., n}, j ∈ {1, 2, ..., m} and arbitrary ε > 0, set zi (t) = (κ + ε) ξie-λ(t - a), , we prove that
(12)
If this is not true, no loss of generality, suppose that there exist i0 and t* ∈ [a, b) such that
(13)
for t ∈ [a, t*], i = 1, 2,..., n, j = 1, 2,..., m.
However, from (9) and (12), we get
Since (11) holds, it follows that . Therefore, we have
which contradicts the inequality in (13). Thus (12) holds for all t ∈ [a, b). Letting ε → 0, we have
The proof is completed.
Remark 1. Lemma 3 is a generalization of the famous Halanay inequality.
Theorem 2Under assumptions (H1)-(H3), if the following conditions hold,
(C1) there exist vectors ξ = (ξ1, ξ2, ..., ξn ) T > 0, η = (η1, η2, ..., ηm ) T > 0 and positive number λ > 0 such that
(C2) , where, k ∈ N,
then system (1) has exactly one globally exponentially stable equilibrium point, and its exponential convergence rate equals λ - μ.
Proof. Since (C1) holds, from Theorem 1, we know that system (1) has one unique equilibrium point . Now, we assume that (x1(t), ..., xn (t), y1(t), ..., ym (t)) T is any solution of system (1), let , i = 1, 2, ..., n, , j = 1, 2, ..., m. It is easy to see that system (1) can be transformed into the following system
(14)
where , , and the initial conditions of (14) are
From (H1) and Lemma 2, we calculate the upper right derivative along the solutions of first equation and third equation of (14), we can obtain
for i = 1, 2,..., n, j = 1, 2,..., m.
Let , , ri = ai , pij = |aij |Gj , , , , , then we have
(15)
for i = 1, 2, ..., n, j = 1, 2, ..., m, and from (C1), there exist vectors ξ = (ξ1, ξ2, ..., ξn ) T > 0, η = (η1, η2, ..., ηm ) T > 0 and positive number λ > 0 such that
(16)
Taking , it is easy to prove that
(17)
From Lemma 3, we obtain that
(18)
Suppose that for l ≤ k, the inequalities
(19)
hold, where μ0 = 1. When l = k + 1, we note that
(20)
and
(21)
From (20), (21) and μk ≥ 1, we have
(22)
Combining (15),(16),(22) and Lemma 3, we obtain that
(23)
Applying the mathematical induction, we can obtain the following inequalities
(24)
According to (C2), we have , so we have
and
That is
(25)
It follows that
where , then we have
The proof is completed.
Remark 2. In Theorem 2, the parameters μk and μ depend on the impulsive disturbance of system (1), and λ is actually an estimate of exponential convergence rate of continuous system (2), which depends on the delay kernel functions and system parameters. In order to obtain more precise estimate of the exponential convergence rate of system (1) (or system (2)), we suggest the following optimization problem:
Obviously, for continuous system (2), we can immediately obtain the following corollaries.
Corollary 2Under assumptions (H1) and (H2), if condition (C1) holds, then system (2) has exactly one globally exponentially stable equilibrium point, and its exponential convergence rate equals λ.
Corollary 3Under assumptions (H1) and (H2), system (2) has exactly one globally exponentially stable equilibrium point if C - TL is a nonsingular M-matrix.
Remark 3. Note that Lemma 2 transforms the fuzzy AND (⋀) and the fuzzy OR (⋁) operation into the SUM operation (∑). So above results can be applied to the following classical impulsive BAM neural networks with time delays in the leakage terms and distributed delays:
(26)
for i = 1, 2,..., n; j = 1, 2,..., m.
For model (26), it is easy to obtain the following result:
Theorem 3Under assumptions (H1)-(H3), if the following conditions hold,
(C1') there exist vectors ξ = (ξ1, ξ2, ..., ξn ) T > 0, η = (η1, η2, ..., ηm ) T > 0 and positive number λ > 0 such that
(C2) , where, k ∈ N ,
then system (26) has exactly one globally exponentially stable equilibrium point, and its exponential convergence rate equals λ - μ.
5 An illustrative example
In order to illustrate the feasibility of our above-established criteria in the preceding sections, we provide a concrete example. Although the selection of the coefficients and functions in the example is somewhat artificial, the possible application of our theoretical theory is clearly expressed.
Example. Consider the following impulsive BAM FCNNs with time delays in the leakage terms and distributed delays:
(27)
for k ∈ N, i = 1, 2, j = 1, 2, t > 0, t0 = 0, tk = tk-1+ 0.5k, k ∈ N, where
From above parameters, we have F1 = F2 = 1, G1 = G2 = 1, and .
Solving the following optimization problem
We obtain that λ ≈ 0.3868 > 0, ξ = (1082041, 1327618) T > 0 and η = (716212, 1050021) T > 0, so (C1) holds. From Theorem 1, we know system (27) has a unique equilibrium point, this equilibrium point is (1, 1, 1, 1) T . Also,
That is, (C2) holds. From Theorem 2, the unique equilibrium point (1, 1, 1, 1) T of system (27) is globally exponentially stable, and its exponential convergence rate is about 0.1368. The numerical simulation is shown in Figure 1 and 2.
×
×
6 Conclusions
In this paper, a class of impulsive BAM FCNNs with time delays in the leakage terms and distributed delays has been formulated and investigated. Some new criteria on the existence, uniqueness and global exponential stability of equilibrium point for the networks have been derived by using M-matrix theory and the impulsive delay integro-differential inequality. Our stability criteria are delay-dependent and impulse-dependent. The neuronal output activation functions and the impulsive operators only need to are Lipschitz continuous, but need not to be bounded and monotonically increasing. Some restrictions of delay kernel functions are also removed. It is worthwhile to mention that our technical methods are practical, in the sense that all new stability conditions are stated in simple algebraic forms and provided a more precise estimate of the exponential convergence rate, so their verification and applications are straightforward and convenient. The effectiveness of our results has been demonstrated by the convenient numerical example.
Acknowledgements
This work is supported by the Scientific Research Fund of Sichuan Provincial Education Department under Grant 09ZC057.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
ZX designed and performed all the steps of proof in this research and also wrote the paper. KL participated in the design of the study and helped to draft and revise manuscript. All authors read and approved the final manuscript.