Finite-Time Stability of Fractional-Order BAM Neural Networks with Distributed Delay
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.
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. , Delavari et al. , 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.  studied the uniform stability for a class of fractional-order neural networks with constant delay by the analytical approach; Wu et al.  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.  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  was first introduced by Kosko . 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.
Definition 1. The Riemann-Liouville fractional integral of order of a function is given by provided that the right side is pointwise defined on , where is the Gamma function.
Definition 2. The Caputo fractional derivative of order of a function can be written as
Definition 3. The Mittag-Leffler function in two parameters is defined as
where , , and , where denotes the complex plane. In particular, for , one has
The Laplace transform of Mittag-Leffler function is where and are, respectively, the variables in the time domain and Laplace domain and 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: or in the matrix-vector notation where , . The model (6) is made up of two neural fields and , where and are the activations of the th neuron in and the th neuron in , respectively; is the state vector of the network at time ; the functions are the activation functions of the neurons at time ; is a diagonal matrix; represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs; and are the feedback matrix; denotes the maximum possible transmission delay from neuron to another; and are the delayed feedback matrix; and are two external bias vectors.
Let be the Banach space of all continuously differential functions over a time interval of length , mapping the interval into with the norm defined as follows: for every ,
The initial conditions associated with (6) are given by where .
In order to obtain main result, we make the following assumptions.(H1)For , the functions and are continuous on .(H2)The neurons activation functions and are bounded.(H3)The neurons activation functions and are Lipschitz continuous; that is, there exist positive constants such that
Since the Caputo’s fractional derivative of a constant is equal to zero, the equilibrium point of system (6) is a constant vector which satisfies the system 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 then system (6) can be written as with the initial conditions where Similarly, by using the matrix-vector notation, system (15) can be expressed as with the initial condition where
Define the functions as follows: where . From assumption (H3), we can obtain , . By (21), we have Thus, system (18) can be further written as the following form: where , .
Definition 4. System (23) with the initial condition (19) is finite-time stable with respect to , , if and only if
where is a positive real number and , , denotes the initial time of observation of the system, and denotes time interval .
A technical result about norm upper-bounding function of the matrix function is given in  as follows.
Lemma 5. If , then, for , one has Moreover, if is a diagonal stability matrix, then where () is the largest eigenvalue of the diagonal stability matrix .
Lemma 6 (see ). Let be nonnegative and local integrable on , and let be a nonnegative, nondecreasing continuous function defined on , , and let be a real constant, , with Then Moreover, if is a nondecreasing function on , then
3. Main Result
We first give a key lemma in the proof of our main result as follows.
Lemma 7. Let be nonnegative and local integrable on , and let be nonnegative, nondecreasing and local integrable on , and let , be two positive constants, , with Then
Proof. Substituting (32) into (31), we obtain
Changing the order of integration in the above double integral, we obtain
Let , ; then is a nonnegative, nondecreasing, and local integrable function and is a nonnegative, nondecreasing continuous function. Thus, by Lemma 6 (30), one has
Similarly, we get
For convenience, let where is the largest eigenvalue of the diagonal stability matrix and 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.
Proof. By Laplace transform and inverse Laplace transform, system (23) is equivalent to From (40), (41), and Lemma 5, we obtain Let , and ; then Thus, we have by (42) and (44) that where denotes the largest singular value of matrix . Similarly, by (43) and (45), we get Hence, by (46) and (47), we have Set By simple computation, we have It follows from (48)–(50) and Lemma 7 that By (51), we obtain Thus, if condition (39) is satisfied and , then , ; 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 with the initial condition where , , and , . It is easy to know that is an equilibrium point of system (53). Since , we may let . Take It is easy to check that From condition (41) of Theorem 8, we can get We can obtain that the estimated time of finite-time stability is . Hence, system (53) is finite-time stable with respect to .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Natural Science Foundation of Jiangsu Province (BK2011407) and the Natural Science Foundation of China (11271364 and 10771212).
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