Delay-Dependent Stability Criterion of Caputo Fractional Neural Networks with Distributed Delay

Abstract

This paper is concerned with the finite-time stability of Caputo fractional neural networks with distributed delay. The factors of such systems including Caputo’s fractional derivative and distributed delay are taken into account synchronously. For the Caputo fractional neural network model, a finite-time stability criterion is established by using the theory of fractional calculus and generalized Gronwall-Bellman inequality approach. Both the proposed criterion and an illustrative example show that the stability performance of Caputo fractional distributed delay neural networks is dependent on the time delay and the order of Caputo’s fractional derivative over a finite time.

1. Introduction

It is well known that the fractional calculus is a generalization and extension of the traditional integer-order differential and integral calculus. The fractional calculus has gained importance in both theoretical and engineering applications of several branches of science and technology. It draws a great application in nonlinear oscillations of earthquakes and many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic models. Many practical systems in interdisciplinary fields can be described through fractional derivative formulation. For more details on fractional calculus theory, one can see the monographs of Miller and Ross [1], Podlubny [2], Diethelm [3], and Kilbas et al. [4]. In the last few years, there has been a surge in the study of the theory of fractional dynamical systems. Some recent works the theory of fractional differential systems can be seen in [5–10] and references therein. In particular, for the first time, Lazarević [7] investigated the finite-time stability of fractional time-delay systems. In [8], Lazarević and Spasić further introduced the Gronwall’s approach to discuss the finite-time stability of fractional-order dynamic systems.

Compared with the classical integer-order derivatives, fractional-order derivatives provide an excellent approach for the description of memory and hereditary properties of various processes. Therefore, it may be more accurate to model by fractional-order derivatives than integer-order ones. In [11–13], fractional operators were introduced into artificial neural network, and the fractional-order formulations of artificial neural network models were also proposed in research works about biological neurons. Recently, there has been an increasing interest in the investigation of the fractional-order neural networks, and some important and interesting results were obtained [13–19], due to their significance in both theory and applications. In [13], Kaslik and Sivasundaram discussed the stability and multistability, periodic oscillation, bifurcations, and chaos of fractional-order neural networks of Hopfield type. Arena et al. [14] investigated the chaotic behavior in noninteger-order cellular neural networks. Boroomand and Menhaj [15] considered fractional-order Hopfield neural networks and analyzed their stability by means of energy-like function. Huang et al. [16] presented the complex dynamical behaviors of a fractional-order four-cell cellular neural network by means of numerical simulations.

On the other hand, time delay is one of the inevitable problems in practical engineering applications, which has an important effect on the stability and synchronization capability of dynamical systems in the real world. In recent years, there are many important results with respect to integer-order network dynamical systems (see [20–26] and references therein). In [20, 21], the finite-time synchronization problems of various kinds of integer-order dynamical system without delay effect have been investigated. Currently, the dynamical behaviours of integer-order networks dynamical systems with delay [22–26] are discussed by applying the different methods. However, to the best of our knowledge, there are very rare works on the problems for fractional-order delayed neural networks [18, 19]. Zhou et al. [18] discussed numerical simulation of chaotic synchronization of a fractional neuron network system with time-varying delays, while the theoretical result was not established. Chen et al. [19] investigated the uniform stability for a class of fractional-order neural networks with constant delay by the analytical approach.

In this paper, motivated by the works of Lazarević and Spasić [7, 8], we are devoted to establishing the finite-time stability criterion for fractional-order neural networks with distributed delay. Since fractional-order derivatives are nonlocal and have weakly singular kernels, many methods applied to the classical integer-order dynamical systems are not suitable for fractional-order delayed neural networks. Therefore, it is quite interesting and challenging to study the stability problems for fractional-order distributed delay neural networks. In this paper, we will apply the fractional calculus and generalized Gronwall-Bellman inequality (see [9]) to establish the finite-time stability criterion of fractional-order distributed delayed neural networks. The obtained criterion is convenient and feasible to check the considered model’s stability over a finite time.

This paper is organized as follows. In Section 2, we will recall some definitions concerning fractional calculus and elaborate the Caputo fractional distributed delayed neural networks. In Section 3, the finite-time stability criterion of Caputo fractional neural networks with distributed delays is established. An example is given to show the effectiveness and applicability of the proposed result in Section 4. Finally, some concluding remarks are drawn in Section 5.

2. Preliminaries and Model Formulation

In this section, we first recall some definitions of fractional calculus and the well-known results. For more details, one can see [1–4]. Next, we elaborate Caputo fractional neural networks model with distributed delay.

Definition 1. Riemann-Liouville’s fractional integral of order with the lower limit zero for a function is defined as that provided the right side is pointwise defined on , where is the gamma function.

Definition 2. Caputo’s fractional derivative of order for a function is defined as

Definition 3. The Mittag-Leffler function in two parameters is defined as where , , and ; denotes the complex plane. In particular, for , have the interesting property
From [1–4], one knows that the superiority of Caputo’s fractional derivative is that the initial conditions for fractional differential equations under Caputo’s sense take on the similar form as for integer-order differential ones, which also have well understood physical meanings. Furthermore, Caputo’s fractional derivative of a constant is equal to zero. Therefore, we discuss fractional-order neural networks with distributed delay under Caputo’s sense throughout this paper.
In this paper, we are interested in the finite-time stability of Caputo’s fractional-order neural networks with distributed delay by the following state equations: with the initial conditions or in the matrix-vector notation with the initial condition where is an order Caputo’s fractional derivative of ; is a positive constant and satisfies ; is the number of neurons in the indicated neural network; is the state vector of the network at time ; the functions are the activation functions of the neurons at time ; is a diagonal matrix with for ; is the feedback matrix; denotes the maximum possible transmission delay from neuron to another; is the delayed feedback matrix; is an external bias vector.
Throughout this paper, we make the following assumptions.The neurons activation functions and are bounded.The neurons activation functions and are Lipschitz continuous. That is, there exist positive constants such that For , the function is continuous on .
From the assumptions above, we denote , , and .

Note that Caputo’s fractional derivative of a constant is equal to zero [2], and then it follows from Schauder fixed point theorem and assumptions ()–() that the equilibrium points of system (1) exist. We can shift the equilibrium point of system (6) to the origin. Let be an equilibrium point of system (6); ; then system (1) can be written as with the initial conditions where , , and , .

By using the matrix-vector notation, system (12) can be expressed in the form with the initial condition where , and  ,.

From assumption (), we know that the functions and satisfy the following properties: Define the two new functions as follows: It follows from (17) that Thus, system (14) can be further shifted into the following form: where , and . Obviously, one can get , and .

Let denote the Banach space of all continuous functions over a time interval of length , mapping the interval into with the norm defined as follows: for every ,

Definition 4 (see [7, 8], finite-time stability). System (19) with the initial condition , is finite-time stable with respect to , , if and only if implies where is a positive real number and , , denotes the initial time of observation of the system, and denotes time interval .

The following generalized Gronwall-Bellman inequality was derived by Ye et al. [9], which is basic to establish the finite-time stability criterion of system (19).

Lemma 5 (see [9, generalized Gronwall-Bellman inequality]). Let be nonnegative and local integrable on , and let be a nonnegative, nondecreasing continuous function defined on , and let be a real constant, with and then In addition, if is a nondecreasing function , then where is the Mittag-Leffler function with one parameter (see [1–4]).

3. Main Result

In this section, we derive the sufficient conditions for finite-time stability of Caputo fractional neural networks with distributed delays by using the generalized Gronwall-Bellman inequality [9].

Theorem 6. Let system (19) satisfy Assumptions ()–() with the initial condition , , and and then system (19) is finite-time stable with respect to , , where denotes the largest singular value of matrix and is the Mittag-Leffler function with one parameter.

Proof. According to the property of the fractional order , one can obtain that system (19) is equivalent to the following Volterra fractional integral with memory Applying the appropriate properties of the norm on (27), it follows that Let . For , it follows from (28) that where denotes the largest singular value of matrix . Obviously, one can introduce a nondecreasing function as An application of the lemma yields that Hence, using the basic condition of the theorem, inequality (26) yields This completes the proof.

Remark 7. The obtained theorem presents a finite-time stability criterion, which shows that the finite-time stability of Caputo fractional distributed delayed neural networks is dependent on the time delay and the order of Caputo’s fractional derivative.

4. An Illustrative Example

In this section, we give an example to verify the validity and applicability of the given result.

Example 1. Consider the following two-state Caputo fractional neural networks model with distributed delay: with an associated function of the initial state where , , , and .
Now, we apply our theorem to verify that system (33) is finite-time stable. Take Then, it follows from the initial data and system (33) that Using the condition of theorem, we can obtain where is an estimated time of the finite-time stability. Therefore, system (33) is finite-time stable with respect to .

5. Conclusions

In this paper, we have investigated the finite-time stability of Caputo fractional distributed delayed neural networks and have derived the finite-time stability criterion based on the fractional calculus theory and generalized Gronwall-Bellman inequality technique. The proposed criterion with an illustrative example shows that the stability performance of Caputo fractional neural networks with distributed delay is dependent on the time delay and the order of Caputo’s fractional derivative over a finite time. Also, some other dynamical behaviors, such as synchronization and control, of fractional-order network systems will become our future investigative works.

Conflict of Interests

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

Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University (KAU), under Grant 3-130/1434/HiCi. Therefore, the authors acknowledge technical and financial support of KAU.