Finite-Time Synchronization Analysis for BAM Neural Networks with Time-Varying Delays by Applying the Maximum-Value Approach with New Inequalities

: In this paper, we consider the ﬁnite-time synchronization for drive-response BAM neural networks with time-varying delays. Instead of using the ﬁnite-time stability theorem and integral inequality method, by using the maximum-value method, two new criteria are obtained to ensure the ﬁnite-time synchronization for the considered drive-response systems. The inequalities in our paper, applied to obtaining the maximum-valued and designing the novel controllers, are different from those in existing papers.


Introduction
In 1987, Kosko firstly proposed a two-way associative search for stored bipolar vector pairs and generalized the single-layer auto-associative Hebbian correlation to a two-layer pattern-matched hetero-associative circuit. These are a class of important neural networks, called bidirectional associative memory (BAM) neural networks [1,2]. The dynamic behaviors of BAM neural networks are of significant application prospects in various fields, such as automatic control, signal processing, pattern recognition and associative memory, which has arisen the great interest of researchers. To date, many researchers have analyzed various dynamical behaviors of BAM neural networks, and obtained many dynamical analysis results of BAM neural networks . In recent years, the global asymptotic/exponential synchronization of BAM neural networks has been widely investigated, for example, see [3][4][5][6][7][8][9][10][11][12][13] and references therein. Meanwhile, many scholars have devoted themselves to the study of the finite-time synchronization of BAM neural networks, for example, see [14][15][16][17][18][19][20][21][22]42] and references therein.
In [14], the authors developed a finite-time synchronization for the drive-response fuzzy inertial BAM neural networks by employing integral inequality techniques and the figure analysis approach. Furthermore, the finite-time stochastic synchronization for memristor-based BAM neural networks with time-varying delays and stochastic disturbances was considered in [15]. In addition, the authors built more reasonable switching conditions of the finite-time synchronization by using a stochastic analysis technique. In [16], researchers introduced a different inequality, the fractional-order Gronwall inequality with time delays, and then dealt with the finite-time synchronization problem of fractional-order memristor-based neural networks with time delays. Zhang and Yang [17] studied the finite-time impulsive synchronization issue of fractional-order memristive BAM neural networks involving switch jumps mismatch by designing two impulsive controllers and using the properties of Gamma functions. In [18], the finite-time synchronization of drive-response BAM fuzzy neural networks with time delays and impulsive effects was investigated. The authors' goal was to illustrate the effects of both impulse and time delay in a finite-time control area in terms of novel Lyapunov functionals and special analytical techniques. Motivated by security applications in the image transmission, Wang et al. [19] provided some sufficient conditions to guarantee the finite-time projective synchronization for memristor-based BAM neural networks.
Inspired by [42], we studied the finite-time synchronization of BAM neural networks by applying the maximum-value approach. In this paper, by using different inequality techniques from those in [42] and designing controllers different from those in [14][15][16][17][18][19][20][21][22]42], the finite-time synchronization criteria for drive-response BAM neural networks with time-varying delays are obtained. In proving the main results, one of the two difficulties in our paper was how to construct novel inequalities in the controllers to get V (t) < ψ(t) and the other one was how to establish the new inequalities in the controllers to achieve a finite-time synchronization. Thus, the novelty of this paper is to construct some novel inequalities to obtain the maximum value of the considered functions and to get the finite time t 1 needed to achieve synchronization. As a result, the main contribution of this paper lies in the following three aspects: (1) Some new inequalities which are different from those in [42] are constructed in the process of obtaining the maximum-value of the considered functions; (2) new inequalities described with fractional and integral functions are introduced to design the novel controllers; (3) using the maximum-value approach and designing novel controllers, two different conditions are attained to assure the finite-time synchronization between the drive system and the response system. The rest of this paper is arranged as follows. In Section 2, some necessary preliminaries are given. In Section 3, two novel criteria to realize the finite-time synchronization are put forward for the drive system (1) and the response system (2). In Section 4, we exhibit two examples to validate the effectiveness and feasibility of the derived consequences.

Preliminaries
In this paper, we consider a class of BAM neural networks with time-varying delays described by where i = 1, 2, · · · , n, j = 1, 2, · · · , m, a i > 0 and b j > 0 are constants; x i (t), y j (t) are the states of the ith neuron and jth neuron; constants c ij , d ij , p ji , q ji denote the connection strengths; τ ji (t), σ ij (t) are time delays, 0 < σ ij (t) < σ * , 0 < τ ji (t) < τ * ; functions f j , g i denote the activation functions; constants I i , J j denote the external inputs.
For simplicity, we refer to (1) as the drive system, and consider the response BAM neural networks with time-varying delays described as follows: where u i (t), v j (t) denote the states, the parameters are the same as those in system (1) and P i (t), Q j (t) are the controllers to realize finite-time synchronization between the drive system (1) and the response system (2).
The initial values of system (2) are given as follows: Definition 1. Drive-response systems (1) and (2) are said to be finite-time synchronized if for arbitrary solutions of system (1) and system (2) T , under a suitable designed controller, there exists a time T > 0 which is related to the initial condition, such that for i = 1, 2, · · · , n; j = 1, 2, · · · , m, Lemma 2. If a > 1, b > 1, and 1 a + 1 b = 1, when t > 3 4 , the following inequality holds: Proof. We only need to prove max 1+t t(e t −1) ≤ 1+e

Lemma 4 ([42]).
Assume that z = f(x) defined on (−∞, +∞), and x 0 is a unique local maximum value point. Then, max Proof. The proof is well-known and it is omitted.

Notation 1. In (4), φ(t) is independent of the initial values of the error systems, while in (5), ψ(t)
is dependent on the initial values of the error systems, so the controllers (4) and the controllers (5) are different. Under these two controllers, the finite-time synchronization for the drive system (1) and the response system (2) are achieved under some conditions. Theorem 1. Assume the condition (H 1 ) holds. Then, the drive system (1) and the response system (2) are finite-time synchronized under the controllers (4) in a finite time t 1 , where t 1 = max 3 4 , if there exists a positive constant p ≥ 1 such that the following conditions hold: Proof. Without loss of generalization, we assume that e i (t) = 0, r j (t) = 0 (If e i (t) = 0, r j (t) = 0, then the finite-time synchronization has been proved (if e i (t) = 0 or r j (t) = 0, then the proof is a special case of the following proof).
Namely, lim The proof of Theorem 1 is finished.

Theorem 2.
Assume that (H 1 ) holds. Then, the drive system (1) and the response system (2) are finite-time synchronized under the controllers (5) in a finite time t 2 , where t 2 = max{ 1 3m , 1 e } if there exists a positive constant q ≥ 1 such that the following conditions hold: Proof. Without loss of generalization, we assume that e i (t) = 0, r j (t) = 0 (if e i (t) = 0, r j (t) = 0, then the finite-time synchronization has been proved; if e i (t) = 0 or r j (t) = 0, then the proof is a special case of the following proof).
We construct a Lyapunov function as follows: where, [e i (s)] 2q ds.
Remark 3. The controllers in our paper are different from those in [42]. Firstly, the error items in the controllers in our paper are different from those in [42]; namely, fractional-type functions are designed in the error items of the controllers in our paper, while exponential-and logarithm-type functions are designed in the error items in the controllers from [42]. Secondly, the time t items are different from those in [42]; namely, fractional-and exponential-type functions are designed for the time items of the controllers in our paper, while logarithm-, fractional-and exponential-type functions are designed for the time items in the controllers from [42].
The curves of variables x 1 (t), x 2 (t), u 1 (t) and u 2 (t) are shown in Figure 4, the curves of variables y 1 (t), y 2 (t), v 1 (t) and v 2 (t) are shown in Figure 5 and the error curves of the drive-response system e 1 (t), e 2 (t), r 1 (t) and r 2 (t) are shown in Figure 6.    Figure 6. The curves of e 1 (t), e 2 (t), r 1 (t), r 2 (t) from Example 2.

Conclusions
In this paper, we focused on the finite-time synchronization for a class of driveresponse BAM neural networks with time-varying delays. Furthermore, two novel finitetime synchronization conditions of the above BAM neural networks were derived to assure the finite-time synchronization between the drive system and the response system. In that process, we applied the maximum-value approach and introduced two new inequalities, which were different from those in the existing papers. In the future, we will study the fixed-time synchronization of neural networks.  Data Availability Statement: All data included in this study are available upon request by contacting the corresponding author.