Finite-Time Stability of Fractional-Order BAM Neural Networks with Distributed Delay

Abstract

Based on the theory of fractional calculus, the generalized Gronwall inequality and estimates of mittag-Leffer functions, the finite-time stability of Caputo fractional-order BAM neural networks with distributed delay is investigated in this paper. An illustrative example is also given to demonstrate the effectiveness of the obtained result.

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Yuping Cao 1 and Chuanzhi Bai 2

Academic Editor:Sabri Arik

1, Department of Basic Courses, Lianyungang Technical College, Lianyungang, Jiangsu 222000, China
2, Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, China

Received 8 February 2014; Accepted 1 April 2014; 22 April 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Fractional calculus (integral and differential operations of noninteger order) was firstly introduced 300 years ago. Due to lack of application background and the complexity, it did not attract much attention for a long time. In recent decades fractional calculus is applied to physics, applied mathematics, and engineering [1-6]. Since the fractional-order derivative is nonlocal and has weakly singular kernels, it provides an excellent instrument for the description of memory and hereditary properties of dynamical processes. Nowadays, study on the complex dynamical behaviors of fractional-order systems has become a very hot research topic.

We know that the next state of a system not only depends upon its current state but also upon its history information. Since a model described by fractional-order equations possesses memory, it is precise to describe the states of neurons. Moreover, the superiority of the Caputo's fractional derivative is that the initial conditions for fractional differential equations with Caputo derivatives take on the similar form as those for integer-order differentiation. Therefore, it is necessary and interesting to study fractional-order neural networks both in theory and in applications.

Recently, fractional-order neural networks have been presented and designed to distinguish the classical integer-order models [7-10]. Currently, some excellent results about fractional-order neural networks have been investigated, such as Kaslik and Sivasundaram [11, 12], Zhang et al. [13], Delavari et al. [14], and Li et al. [15, 16]. On the other hand, time delay is one of the inevitable problems on the stability of dynamical systems in the real word [17-20]. But till now, there are few results on the problems for fractional-order delayed neural networks; Chen et al. [21] studied the uniform stability for a class of fractional-order neural networks with constant delay by the analytical approach; Wu et al. [22] investigated the finite-time stability of fractional-order neural networks with delay by the generalized Gronwall inequality and estimates of Mittag-Leffler functions; Alofi et al. [23] discussed the finite-time stability of Caputo fractional-order neural networks with distributed delay.

The integer-order bidirectional associative memory (BAM) model known as an extension of the unidirectional autoassociator of Hopfield [24] was first introduced by Kosko [25]. This neural network has been widely studied due to its promising potential for applications in pattern recognition and automatic control. In recent years, integer-order BAM neural networks have been extensively studied [26-29]. However, to the best of our knowledge, there is no effort being made in the literature to study the finite-time stability of fractional-order BAM neural networks so far.

Motivated by the above-mentioned works, we were devoted to establishing the finite-time stability of Caputo fractional-order BAM neural networks with distributed delay. In this paper, we will apply Laplace transform, generalized Gronwall inequality, and estimates of Mittag-Leffler functions to establish the finite-time stability criterion of fractional-order distributed delayed BAM neural networks.

This paper is organized as follows. In Section 2, some definitions and lemmas of fractional differential and integral calculus are given and the fractional-order BAM neural networks are presented. A criterion for finite-time stability of fractional-order BAM neural networks with distributed delay is obtained in Section 3. Finally, the effectiveness and feasibility of the theoretical result is shown by an example in Section 4.

2. Preliminaries

For the convenience of the reader, we first briefly recall some definitions of fractional calculus; for more details, see [1, 2, 5], for example.

Definition 1.

The Riemann-Liouville fractional integral of order α>0 of a function u:(0,∞)[arrow right]R is given by [figure omitted; refer to PDF] provided that the right side is pointwise defined on (0,∞) , where Γ(·) is the Gamma function.

Definition 2.

The Caputo fractional derivative of order γ>0 of a function u:(0,∞)[arrow right]R can be written as [figure omitted; refer to PDF]

Definition 3.

The Mittag-Leffler function in two parameters is defined as [figure omitted; refer to PDF] where α>0 , β>0 , and z∈... , where ... denotes the complex plane. In particular, for β=1 , one has [figure omitted; refer to PDF]

The Laplace transform of Mittag-Leffler function is [figure omitted; refer to PDF] where t and s are, respectively, the variables in the time domain and Laplace domain and [Lagrangian (script capital L)]{·} stands for the Laplace transform.

In this paper, we are interested in the finite-time stability of fractional-order BAM neural networks with distributed delay by the following state equations: [figure omitted; refer to PDF] or in the matrix-vector notation [figure omitted; refer to PDF] where 1<α , β<2 . The model (6) is made up of two neural fields _Fx and _Fy , where _xi (t) and _yj (t) are the activations of the i th neuron in _Fx and the j th neuron in _Fy , respectively; [figure omitted; refer to PDF] is the state vector of the network at time t ; the functions [figure omitted; refer to PDF] are the activation functions of the neurons at time t ; C=diag...(_ci ) is a diagonal matrix; _ci >0 represents the rate with which the i th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs; B=(_bij)n×n and D=(_dji)n×n are the feedback matrix; τ>0 denotes the maximum possible transmission delay from neuron to another; R=(_rij)n×n and P=(_pji)n×n are the delayed feedback matrix; I=(_I1 ,...,_In^)T and I¯=(_I¯1 ,...,_I¯n^)T are two external bias vectors.

Let ^...1 ([-τ,0],^...n ) be the Banach space of all continuously differential functions over a time interval of length τ , mapping the interval [-τ,0] into ^...n with the norm defined as follows: for every [straight phi]∈^...1 ([-τ,0],^...n ) , [figure omitted; refer to PDF]

The initial conditions associated with (6) are given by [figure omitted; refer to PDF] where _{[straight phi]i} ,_ψj ∈^C1 ([-τ,0],...) .

In order to obtain main result, we make the following assumptions.

(H1) For i,j=1,...,n , the functions _rij (·) and _pji (·) are continuous on [0,τ] .

(H2) The neurons activation functions _fi and _gj (i,j=1,...,n) are bounded.

(H3) The neurons activation functions _fi and _gj are Lipschitz continuous; that is, there exist positive constants _hi ,_lj (i,j=1,...,n) such that [figure omitted; refer to PDF]

Since the Caputo's fractional derivative of a constant is equal to zero, the equilibrium point of system (6) is a constant vector (^x* ,^y* )=(^x1* ,^x2* ,...,^xn* ,^y1* ,^y2* ,...,^yn*)T ∈^...2n which satisfies the system [figure omitted; refer to PDF] By using the Schauder fixed point theorem and assumptions (H1)-(H3), it is easy to prove that the equilibrium points of system (6) exist. We can shift the equilibrium point of system (6) to the origin. Denoting [figure omitted; refer to PDF] then system (6) can be written as [figure omitted; refer to PDF] with the initial conditions [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Similarly, by using the matrix-vector notation, system (15) can be expressed as [figure omitted; refer to PDF] with the initial condition [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Define the functions as follows: [figure omitted; refer to PDF] where i,j=1,...,n . From assumption (H3), we can obtain |_hi (t)|...4;_hi , |_lj (t)|...4;_lj . By (21), we have [figure omitted; refer to PDF] Thus, system (18) can be further written as the following form: [figure omitted; refer to PDF] where H(t)=diag...{_hi (t)} , L(t)=diag...{_lj (t)} .

Definition 4.

System (23) with the initial condition (19) is finite-time stable with respect to {δ,[varepsilon],_t0 ,J} , δ<[varepsilon] , if and only if [figure omitted; refer to PDF] implies [figure omitted; refer to PDF] where δ is a positive real number and [varepsilon]>0 , δ<[varepsilon] , _t0 denotes the initial time of observation of the system, and J denotes time interval J=[_t0 ,_t0 +T) .

A technical result about norm upper-bounding function of the matrix function _Eα,β is given in [30] as follows.

Lemma 5.

If α...5;1 , then, for β=1,2,α , one has [figure omitted; refer to PDF] Moreover, if A is a diagonal stability matrix, then [figure omitted; refer to PDF] where -ω (ω>0 ) is the largest eigenvalue of the diagonal stability matrix A .

Lemma 6 (see [31]).

Let u(t),a(t) be nonnegative and local integrable on [0,T) (T...4;+∞) , and let g be a nonnegative, nondecreasing continuous function defined on [0,T) , g(t)...4;M , and let M be a real constant, α>0 , with [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] Moreover, if a(t) is a nondecreasing function on [0,T) , then [figure omitted; refer to PDF]

3. Main Result

We first give a key lemma in the proof of our main result as follows.

Lemma 7.

Let u(t),v(t) be nonnegative and local integrable on [0,T) (T...4;+∞) , and let _a1 (t),_a2 (t) be nonnegative, nondecreasing and local integrable on [0,T) , and let _b1 , _b2 be two positive constants, α,β>1 , with [figure omitted; refer to PDF] Then [figure omitted; refer to PDF]

Proof.

Substituting (32) into (31), we obtain [figure omitted; refer to PDF] Changing the order of integration in the above double integral, we obtain [figure omitted; refer to PDF] Let a(t)=_a1 (t)+_b1^∫0t ...(t-s^)α-1_a2 (s)ds , g(t)=_b1b2 ((Γ(α)Γ(β))/Γ(α+β)) ; then a(t) is a nonnegative, nondecreasing, and local integrable function and g(t) is a nonnegative, nondecreasing continuous function. Thus, by Lemma 6 (30), one has [figure omitted; refer to PDF] Similarly, we get [figure omitted; refer to PDF]

For convenience, let [figure omitted; refer to PDF] where -γ is the largest eigenvalue of the diagonal stability matrix -C and μ(·) denotes the largest singular value of matrix (·) .

In the following, sufficient conditions for finite-time stability of fractional-order BAM neural networks with distributed delay are derived.

Theorem 8.

Let 1<α,β<2 . If system (23) satisfies (H1)-(H3) with the initial condition (19), and [figure omitted; refer to PDF] where t∈J=[0,T) , then system (23) is finite-time stable with respect to {δ,[varepsilon],0,J} , δ<[varepsilon] .

Proof.

By Laplace transform and inverse Laplace transform, system (23) is equivalent to [figure omitted; refer to PDF] From (40), (41), and Lemma 5, we obtain [figure omitted; refer to PDF] Let U(t)=_{sup...θ∈[t-τ,t]} ||u(θ)||^eγθ , and V(t)=_{sup...θ∈[t-τ,t]} ||v(θ)||^eγθ ; then [figure omitted; refer to PDF] Thus, we have by (42) and (44) that [figure omitted; refer to PDF] where μ(B) denotes the largest singular value of matrix B . Similarly, by (43) and (45), we get [figure omitted; refer to PDF] Hence, by (46) and (47), we have [figure omitted; refer to PDF] Set [figure omitted; refer to PDF] By simple computation, we have [figure omitted; refer to PDF] It follows from (48)-(50) and Lemma 7 that [figure omitted; refer to PDF] By (51), we obtain [figure omitted; refer to PDF] Thus, if condition (39) is satisfied and _{||([straight phi]¯,ψ¯)||1} <δ , then ||(u(t),v(t))||<[varepsilon] , t∈J ; that is, system (23) is finite-time stable. This completes the proof.

4. An Illustrative Example

In this section, we give an example to illustrate the effectiveness of our main result.

Consider the following two-state Caputo fractional BAM type neural networks model with distributed delay [figure omitted; refer to PDF] with the initial condition [figure omitted; refer to PDF] where α=1.2,β=1.3 , τ=0.2 , and _fj (_xj )=_gj (_xj )=(1/2)(|_xj +1|-|_xj -1|) , j=1,2 . It is easy to know that (^x1* ,^x2* ,^y1* ,^y2*)T =(0,0,0,0^)T is an equilibrium point of system (53). Since _{||([straight phi],ψ)||1} =1/15<0.07 , we may let δ=0.07 . Take [figure omitted; refer to PDF] It is easy to check that [figure omitted; refer to PDF] From condition (41) of Theorem 8, we can get [figure omitted; refer to PDF] We can obtain that the estimated time of finite-time stability is T[approximate]23.78 . Hence, system (53) is finite-time stable with respect to {0.07,1,0,[0,30)} .

Acknowledgments

This work is supported by the Natural Science Foundation of Jiangsu Province (BK2011407) and the Natural Science Foundation of China (11271364 and 10771212).

Conflict of Interests

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

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