Finite-Time Stability Criteria for a Class of High-Order Fractional Cohen–Grossberg Neural Networks with Delay

*is paper focuses on a class of delayed fractional Cohen–Grossberg neural networks with the fractional order between 1 and 2. Two kinds of criteria are developed to guarantee the finite-time stability of networks based on some analytical techniques. *is method is different from those in some earlier works. Moreover, the obtained criteria are expressed as some algebraic inequalities independent of the Mittag–Leffler functions, and thus, the calculation is relatively simple in both theoretical analysis and practical applications. Finally, the feasibility and validity of obtained results are supported by the analysis of numerical simulations.


Introduction
Neural networks have been paid much attention owing to the powerful applications in diverse fields. With the increasing requirements in practical applications, many researchers have made great efforts to develop various types of neural networks, such as Cohen-Grossberg neural networks (CGNNs) [1], cellular neural networks, bidirectional associative memory neural networks, and recurrent neural networks. As a kind of special recurrent neural network, CGNNs were firstly proposed by Cohen and Grossberg in [1]. CGNN is quite general since it includes some well-known types of neural networks, such as Hopfield neural network, cellular neural network, and shunting neural network. Nowadays, CGNNs have gained more and more interests due to their promising applications in classification, parallel computation and optimization, etc. Many researchers have made great contribution to the research on CGNNs; see [2][3][4][5][6][7][8][9][10][11] and the references therein.
Nowadays, the fractional calculus has achieved significant progress in both theoretical research and practical applications. Compared with the integer-order derivative, the fractional-order derivative has some distinctive features, such as infinite memory and great freedom. Consequently, the fractional-order derivative can better characterize many systems in the real world [12][13][14]. In order to more accurately model the dynamics of neurons, various fractional-order neural networks (FONNs) have been generated based on the integration of the fractional calculus and neural networks. In the recent decades, the research on FONNs has undergone a prosperous development, and there have been numerous works (see [10,[15][16][17][18][19][20] and the references therein). As we all know, the successful applications of FONNs are closely associated to the dynamics of networks, among which stability has been an active topic.
Among various types of stability, finite-time stability (FTS) has aroused more interests in many fields, since many systems always operate over a limited period of time or it is necessary to focus on the behavior of systems within a limited period of time. In the existing works, there are mainly two concepts of FTS. One means that the error of any two state variables tends to zero in a limited time interval, which is also regarded as a special case of Lyapunov asymptotic stability. e other is also called finitetime boundedness, which describes that the quantity related to state does not exceed a prescribed threshold in a limited time interval for a given bound on the initial value [25,26]. It is obvious that "boundedness" is a distinctive feature of this FTS. As revealed in [25,26], it is essentially different from the classical Lyapunov asymptotic stability. In practical applications, this kind of FTS can provide a quantitative bound related to the state, and it has made great contribution for describing the transient performance of system.
In the literature, there have been some interesting results [8-11, 23, 24, 27, 28] on fractional-order Cohen-Grossberg neural networks (FOCGNNs). For example, Ke and Miao [9] investigated the FTS for a class of delayed FOCGNNs by the generalized Bellman-Gronwall inequality. In [23], Zheng et al. studied the FTS and synchronization problem of a class of memristorbased FOCGNNs. Rajivganthi et al. [24] reported the FTS for a class of BAM FOCGNNs with time delays. In these works, the proofs are mainly based on some generalized Gronwall inequalities. In [8], Wan and Wu considered the Mittag-Leffler stability of fuzzy FOCGNNs with deviating argument. Recently, Pratap et al. [10] considered the asymptotic stability and pinning synchronization for a class of delayed FOCGNNs with discontinuous activations, and Huang et al. [11] investigated the Lagrange stability and the asymptotical stability for a class of delayed FOCGNNs. e proofs are mainly based on the Lyapunov theory and some properties related to the fractional calculus. For the FONNs in the aforementioned works, notice that the fractional order α is between 0 and 1. In the real world, the fractional systems with high fractional order can appropriately describe many phenomena and have been successfully applied in physics, biology, and information science (see, for instance, [29][30][31][32]). On the other hand, some classical methods for the case α ∈ (0, 1), such as some Lyapunov methods [10] and LMI method [33], could not be directly extended to the high-order cases. erefore, it is significant to follow through the problems on high-order FONNs.
For FONNs with the fractional order α ∈ (1, 2), there have been many excellent works on the finite-time stability or finite-time synchronization [15][16][17][34][35][36][37]. e analysis is mainly based on the Laplace transform, the inverse Laplace transform, and the generalized Gronwall inequality related to the Mittag-Leffler functions. However, this method can not be directly used to deal with the FTS for FOCGNNs with α ∈ (1, 2) owing to the technical reason. In order to solve this problem, it is desired to investigate a kind of different method.
In this paper, we are devoted to the FTS for a class of delayed FOCGNNs with α ∈ (1, 2). e main contributions are summarized as follows: (i) e fractional order of system considered in this paper is between 1 and 2. A criterion is derived to achieve the FTS of system. Moreover, a criterion is established to ensure the FTS for the equilibrium point of system. (ii) e proofs are based on some analytical techniques, such as the Cauchy-Schwartz inequality, the generalized Gronwall inequality, and some properties of the Caputo derivative. is method is completely different from those in some earlier works [9, 15-17, 24, 34-37]. In particular, the obtained criteria are expressed as some algebraic inequalities and hence, the calculation is relatively easy in practical applications.

Preliminaries
is section starts with recalling some necessary definitions and properties related to the Caputo derivative.
where Γ(·) denotes the Gamma function, i.e., Definition 2 (see [38]). For μ ∈ R + the fractional order integral with order μ of a function θ(t) is defined by Proposition 1 (see [39]). Let μ ∈ R + , m ∈ Z + , and Next, we list two inequalities, which will play a key role in the proofs of main results.

Proposition 2 (generalized Gronwall inequality [40]). Let u(t), v(t) and w(t) be nonnegative L q functions on the in-
Then,

Complexity
Proposition 3 (generalized Bernoulli inequality [41]). Let In what follows, we introduce a class of delayed FOCGNNs, which can be described as where 1 < α < 2 x i (t) ∈ R is the state of the i-th neuron at time t. p i (·) stands for the amplification function and q i (·) corresponds to the behaved function. e constant τ > 0 denotes the time delay. a ij and b ij are the connection weights. f j and g j represent the activation functions. I i stands for the constant external input. For and y(t) stand for two arbitrary solutions for network (7), and let . e initial condition is given as follows: where ‖z(t)‖ � n i�1 |z i (t)|, then network (7) can achieve the finite-time stability w.r.t. δ, ε, T { }.

Main Results
In this section, we are devoted to two kinds of finite-time stability criteria for network (7) based on some properties related to the Caputo derivative and some inequalities. (7). Let us first introduce some further assumptions on the parameters of network (7) and some necessary notation.

Stability Criterion I for Network
(A1) e function p i (i � 1, 2, . . . , n)is a continuous and bounded function such that for x, y ∈ R, where p i , p i , and p * are some positive constants.
(A2) For the functions p i and q i (i � 1, 2, · · · , n), there exists θ i > 0 such that e functions f j and g j (j � 1, 2, · · · , n)are bounded and satisfy the Lipschitz conditions, namely, where F j , G j , ξ j and η j are positive constants. Let where Proof. Let x(t), y(t), and z(t) be defined as in Section 2. By Proposition 1, we obtain Complexity Using the assumptions (A1)-(A2), we have For the term |p i (x i (r)) n j�1 a ij f j (x j (r)) − p i (y i (r)) n j�1 a ij f j (y j (r))|, the assumptions (A1) and (A3) lead to In the same way, we obtain

Stability Criterion II for Network
(H1) e function p i (·) (i � 1, 2, · · · , n) is a continuous and bounded function such that 0 < p i ≤ p i (·) ≤ p i on R , where p i and p i are two positive constants. (H2) e function q i (·) (i � 1, 2, · · · , n) is a monotonic differentiable function such that 0 < q i ≤ q i ′ (·) ≤ q i on R , where q i and q i are two positive constants. (H3) e functions f j and g j (j � 1, 2, · · · , n) satisfy the Lipschitz conditions: where ξ j and η j are two positive constants. (H4) For i, j � 1, 2, · · · , n, a ij , b ij , q i , ξ j and η j satisfy the following condition: where a * � max 1≤j≤n n i�1 |a ij |, b * � max 1≤j≤n n i�1 |b ij |. By an argument similar to that in [9,24], the assumptions (H1)-(H4) can guarantee the existence and uniqueness of equilibrium point for network (7). In what follows, we will concentrate on the finite-time stability for the equilibrium point x * Now, we introduce some notation. Let For network (7), let x(t) represent an arbitrary solution with the initial conditions: Under the assumptions (H1)-(H4), the unique equilibrium point x * of network (7) achieves the FTS w.
Proof. Since x * is the equilibrium point for system (7), we have Based on (7) and (33), we use Proposition 1 to obtain For the term q i (x i (r)) − q i (x * i ), applying Lagrange's mean value theorem, it follows that Following the treatment similar to that of (20), we can obtain inequality (32).

Remark
1. When p i (x i (t)) � 1 and q i (x i (t)) � c i x i (t)(c i > 0), network (7) is reduced to that in [34]. e corresponding results can be easily derived from those in this paper.

Remark 2.
When α � 1, network (7) is reduced to an integer-order one. e corresponding finite-time criteria can be easily obtained by repeating the Proofs of eorems 1 and 2.
Remark 3. For α ∈ (0, 1), Ke and Miao [9] studied the FTS of equilibrium point for a class of delayed FOCGNNs based on the generalized Bellman-Gronwall inequality; Rajivganthi et al. [24] considered the FTS for a class of BAM FOCGNNs with delay by resorting to some inequalities; Zheng et al. [23] reported the FTS for a class of memristorbased FOCGNNs with delay based on a kind of Gronwall's inequality. It seems to us that these methods can not be directly extended to the case of α ∈ (1, 2). Laplace transform, the inverse Laplace transform, and the generalized Gronwall-Bellman inequalities related to the Mittag-Leffler functions. However, this method is not applicable to network (7) owing to the technical reason. In the present paper, a kind of different method was used to discuss the FTS of network (7). More precisely, two kinds of finitetime criteria were obtained based on some properties of the Caputo derivative and some inequalities. Especially, these two criteria are expressed as some algebraic inequalities independent of the Mittag-Leffler functions. erefore, the verification is relatively easy in practical applications.

Numerical Simulations
In this section, two examples are presented to illustrate the effectiveness of two criteria.

Conclusions
In the recent decade, many efforts have been made to the research on FONNs with the fractional order between 1 and 2. e methods are mainly based on the Laplace transform, the inverse Laplace transform, and the generalized Gronwall inequality related to the Mittag-Leffler functions. However, these methods do not work well for the considered FOCGNNs owing to the technical reason. In this paper, the finite-time stability criteria were derived based on the analytic techniques and some inequalities. is kind of method is completely different from the above ones. In particular, the obtained criteria are expressed as some algebraic inequalities independent of the Mittag-Leffler functions and thus, they can be easily verified in practical applications. In the future work, we will investigate the finite-time guaranteed cost control for FONNs with high fractional order. It seems to us that some new techniques would be developed to deal with this problem.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.