A New Criterion for Exponential Stability of a Class of Hopfield Neural Network with Time-Varying Delay Based on Gronwall's Inequality

In this paper, we study the problem of exponential stability for the Hopfield neural network with time-varying delays. Different from the existing results, we establish new stability criteria by employing the method of variation of constants and Gronwall's integral inequality. Finally, we give several examples to show the effectiveness and applicability of the obtained criterion.


Introduction
Since Hopfield [1] proposed the Hopfield neural network named after him in 1984, these types of artificial neural networks have been widely applied in many aspects, such as combinative optimization [2][3][4], image processing [5,6], pattern recognition [7], signal processing [8], communication technology [9], and so on. e Hopfield neural network has been extensively investigated in the past decades [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. In the practical application of neural networks, because of the time delay of information transmission between two neurons and the influence of hardware, such as the limited speed of switch, the phenomenon of time delay is inevitable. erefore, the introduction of a time delay in the study of neural networks has widely been of concern [15][16][17][18][19][20][21][22][23]. Because the number of hidden layers and the initial value of connection weights of the neural network are random, the stability of the system is not being guaranteed. If the control system is unstable, the convergence of the network will lose its foundation. erefore, stability is a very important property for neural networks. In the study of the stability of Hopfield neural networks, researchers usually construct Lyapunov functional and combine with linear matrix inequality or integral inequality to analyze the stability of the system. It is no doubt that Lyapunov's method is a powerful tool in the study of the stability of differential equations, but how to construct an appropriate Lyapunov functional is the key to solve these problems. In addition, constructing different Lyapunov functions for the same system will lead to different stability ranges, which is also an uncertain problem. Besides, the operation of the linear matrix inequality is very complicated. Zhang et al. proposed a method based on weight delay to study the stability of a class of recurrent neural networks with time-varying delays [25]. ey obtained a new delay-dependent stability criterion for neural networks with time-varying delays by constructing a Lyapunov-Krasovskii functional and using Jensen's integral inequality. However, the results obtained by the authors are complicated. To describe the complexity of these results, we give another specific example. Wang et al. [27] studied the delay-dependent stability of a class of generalized continuous neural networks with time-varying delays in system (1) (for the meaning of parameters in the formula, please refer to article [27]): _ u(t) � − Au(t) + Bf(u(t)) + Cf(u(t − τ(t))) + I.
(2) ey constructed a new Lyapunov-Krasovskii functional and then used Jensen's integral inequality to obtain the following criterion for system (2). e origin of system (2) is globally asymptotically stable, if for given diagonal matrices Δ 1 and Δ 2 and positive scalars τ m , τ M , ρ m , ρ M , β k , c j , n 1 , and n 2 , there exist symmetric definite matrices P > 0, W s > 0, S s > 0, Q k > 0, and R j > 0, positive definite diagonal matrices V > 0, U > 0, Λ 1 > 0, and Λ 2 > 0, and matrices G r and J such that the following inequalities hold, s � 1, 2, 3: ese symbols are defined in [27]. ere are some problems with this result: (i) Do the matrices P, W s , S s , Q k , and R j exist? (ii) For such complex matrix inequalities, how does one ensure the existence of the unknown matrices? (iii) If they exist, how are they represented?
If one does not solve these problems, the stability of the original equation remains unsolved. In fact, the stability depends only on the coefficient matrices of the system, not on the existence of those unknown matrices.
eir conclusions are also based on the creation of the Lyapunov-Krasovskii functional. ey all assume that some unknown matrices satisfying some matrix inequalities make the system stable, and it is unknown whether these unknown matrices exist.
To solve this problem, in this paper, we will use the technique of integral inequality to construct a new stability criterion, which is only related to the coefficient matrix and independent of those unknown matrices.
In this paper, we define the norms of the n × n matrix M � (a ij ) n×n and the n-dimensional vector as follows: We assume that all activation functions g i (i � 1, 2, . . . , n) satisfy the following conditions: (i) g i : R ⟶ R is continuous and differentiable, and . . , L n Lemma 1. If ‖A − 1 (B + C)‖L < 1 and the activation function g satisfies conditions (i)-(iii), then the equilibrium point of system (4) must exist and be unique.
According to the definition of A, the inverse matrix A − 1 of A exists; therefore, (6) is equivalent to To prove that (8) is true, we create the following mapping: From conditions (i)-(iii), g(u) is a continuous mapping of R n ⟶ R n ; then, H(u) is also a continuous mapping of R n ⟶ R n . According to the definition of the norm of the ndimensional vector and assumption (ii), we have that 2 Computational Intelligence and Neuroscience where set and H(u) is a continuous mapping of Ω ⟶ Ω . According to Brouwer's fixed-point theorem, there must exist u * ∈ Ω such that H(u * ) � u * . As formula (8) holds, there exists an equilibrium point u * in system (4). To prove the uniqueness of the equilibrium point, we We have i.e., According to the condition 1 − ‖A is equation shows that the equilibrium point is unique.
Let the equilibrium point of system (4) be x * and y(t) � x(t) − x * . In this situation, system (4) can be rewritten as _ y(t) � − Ay(t) + Bf(y(t)) + Cf(y(t − τ(t))), where f(y) � g(y + x * ) − g(x * ), f(y(t)) � (f 1 (y 1 (t)), f 2 (y 2 (t)), . . . , f n (y n (t)) T , and the initial state is . e meaning of the other symbols is the same as that of system (6). Let activation function f i (z)(i � 1, 2, . . . , n) be a continuous function that satisfies a Lipschitz condition for all z ∈ R. at is, assume that for some constant L i > 0 and for all z 1 , z 2 ∈ R.
□ Definition 1. System (14) is said to be globally exponentially stable, if there exists a constant M ≥ 1 and α > 0, such that Lemma 2 (Gronwall's inequality [31]). Let K be a nonnegative constant and v(t) and p(t) are nonnegative and continuous functions on the interval α ≤ t ≤ β and satisfy the inequality then

Stability Analysis
In this section, we discuss the global exponential stability condition for the trivial solution of system (14). e linear term in system (14) can be expressed as _ y(t) � − Ay(t).

Theorem 1. Suppose that the activation function f(•) satisfies conditions (i)-(iii) with the Lipschitz constant L; if
then the trivial solution of system (14) is globally exponentially stable.
Proof. For t � 0, the initial value is η � (η 1 , η 2 , . . . , η n ) T ; by using the method of constant variation, we obtain that the solution of system (14) satisfies the following equation: Computational Intelligence and Neuroscience

exp(− A(t − s))(Bf(y(s)) + Cf(y(s − d(s))))ds. (23)
Taking the norm on both sides of the above formula, without loss of generality, for t > τ, we obtain  Computational Intelligence and Neuroscience

‖exp(− A(t − s))‖‖B‖‖f(y(s))‖ds
According to Lemma 1 (Gronwall's inequality), we obtain Since ((‖B‖ + ‖C‖)L(1/1 − τ * )exp(ωτ)) − ω < 0, then the delay system (14) is globally exponentially stable. e proof is completed. □ Remark 1. How to obtain better stability results in time-delay systems has been the concern of many scholars. Some scholars use improved integral inequality techniques and construct better Lyapunov-Krasovskii functional and estimate its derivative to obtain new results. In [29], the authors discuss the exponential stability and generalized dissipative analysis of time-delay generalized neural networks. Based on Lyapunov-Krasovskii functional (LKF) and Wirtinger single integral inequality (WSII) and Wirtinger double integral inequality (WDII) techniques, they establish new criteria for exponential stability of generalized neural networks with delays. However, as we see, their results are still based on the assumption that there are some unknown symmetric matrices. ey only used some examples to verify the validity of the results, but failed to prove the existence of these unknown symmetric matrices theoretically. In this paper, the stability criterion is only related to the coefficient matrix of the system and has nothing to do with other unknown matrices.
Next, we consider several special cases. For the following system without time delay, _ y(t) � − Ay(t) + Bf(y(t)), we have the following corollary.

Corollary 1.
Suppose that the activation function satisfies the Lipschitz condition; if ‖B‖L − ω < 0, the trivial solution of system (27) is globally exponentially stable. For the following system with constant time delay, we can get the following corollary.

Numerical Examples
In this section, we provide four illustrative examples to demonstrate the effectiveness of eorem 1.
Example 3. We consider the following two-dimensional neural network model with variable delay: where the activation function f i (u) � tanh(u), i � 1, 2, satisfies the Lipschitz condition andL � 1.

Remark 2.
In [30], the author also gives a two-dimensional example. According to the criterion of exponential stability obtained in paper [30], it is necessary to find some symmetric matrices that meet the specified matrix inequalities. Although the authors can find these matrices, the results obtained by this method are accidental and uncertain and they cannot guarantee the existence of symmetric matrices that meet the conditions. e stability judgment method used in the example in this paper is according to the data of the coefficient matrix without the unknown parameters or matrix of the third party. Although the result is relatively conservative, this is a sufficient condition and has obvious advantages for judging the stability of the system.

Conclusion
In this work, we have studied the exponential stability for the Hopfield neural network with a time-varying delay. We use the method of variation of constants of ordinary differential equations to obtain an equation satisfied by the state variable of the neural network. en, we used Gronwall's inequality to analyze this system and obtained new criteria for the exponential stability of the neural networks with timevarying delay. Our result is related only to the coefficient matrix of the system and not to the existence of the other unknown matrices. It is easy to test the exponential stability for specific systems by using these criteria.

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.  Computational Intelligence and Neuroscience