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

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.


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][6][7][8][9][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][12][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][14][15][16][17][18][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 fractionalorder 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 fractionalorder 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 integerorder network dynamical systems (see [20][21][22][23][24][25][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][23][24][25][26] are discussed by applying the different methods.However, to the best of our knowledge, there are very rare works on the problems for fractionalorder 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 fractionalorder 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 finitetime 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.

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][2][3][4].Next, we elaborate Caputo fractional neural networks model with distributed delay.Definition 1. Riemann-Liouville's fractional integral of order  > 0 with the lower limit zero for a function  : R + → R  is defined as that provided the right side is pointwise defined on [0, +∞), where Γ(⋅) is the gamma function.
Definition 2. Caputo's fractional derivative of order  for a function  : R + → R  is defined as (2) Definition 3. The Mittag-Leffler function in two parameters is defined as where  > 0,  > 0, and  ∈ C; C denotes the complex plane.
In particular, for  = 1, have the interesting property From [1][2][3][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 0 <  < 1;  is the number of neurons in the indicated neural network; () = ( 1 (), ( 3 ) For ,  = 1, 2, . . ., , the function   (⋅) is continuous on [0, ].
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).
Proof.According to the property of the fractional order 0 <  < 1, one can obtain that system (19) An application of the lemma yields that Hence, using the basic condition of the theorem, inequality 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.

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: where   is an estimated time of the finite-time stability.Therefore, system (33) is finite-time stable with respect to {0.02, 0.1, 0, [0, 2)}.

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.