Stability and Finite-Time Synchronization Analysis for Recurrent Neural Networks with Improved Integral-Type Time-Varying Delays

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Introduction
For the past few years, RNNs have been intensively investigated in various research fields such as machine learning, medical diagnosis, natural language processing, and model prediction [1][2][3][4][5][6][7][8]. Due to a certain transmission time of signal or information, time delays inevitably exist in RNNs, which may lead to oscillation, and even degradation for systems performances. Hence, delayed RNNs have become a significant research topic on their stability and synchronization analysis [9,10]. The integrating approach of terminal sliding mode control and RNNs is proposed [11], based on Lyapunov stability theorem, the convergence and stability of deep learning RNNs are proved. The stability and synchronization of Riemann-Liouville fractional-order inertial neural network with time delay are studied [12] and transform the original inertial system into a conventional system through appropriate variable substitution.
The stable neural network is a necessary prerequisite for practical engineering model; LKF is an effective method for stability analysis of delayed system, whose purpose is to obtain maximum upper bound of time delay and to guarantee the globally asymptotically stability; in other words, to find a positively definite LKF whose difference along the system orbit is less than zero. The main difficulty is how to estimate the upper bound of integral terms of LKF derivative, and three primary methods are proposed to solve this problem: the model transformation method, the free weighted matrix method, and integral inequality method [13].
SMC has been proposed for synchronization of delayed RNNs, which is very robust due to sliding mode invariance when the system state reaches the sliding mode surface [14,15]. Nevertheless, delayed RNNs may be typically expected to attain synchronization in short time in practical applications; it is essential to introduce finite-time synchronization with the understanding that the orbits of two neural networks from different initial states under the control strategy eventually converge over time; finite-time synchronization has fast dynamic behavior and high accuracy synchronization compared with asymptotic and exponential synchronization [16].
Researchers also investigated the consistent asymptotic stability and integrability of nonregressive systems and boundedness of nonlinear delay-dependent perturbed systems based on new LKF method [17]. By considering appropriate LKF and feedback controllers, the finite/fixed-time synchronization of Clifford-valued RNNs with timevarying delays is derived by decomposing the proposed Clifford-valued drive-response models into real-valued drive-response models [18]. A finite-time synchronization for class of drive-response bidirectional associative memory (BAM) neural networks with time-varying delays is developed by employing maximum-value approach, and two new inequalities are also proposed [19]. Furthermore, a novel nonsingular integral terminal sliding control strategy for finite-time synchronization of category of hyperchaotic systems is proposed [20]; in addition, the planned controller scheme confirms that master-slave hyperchaotic systems arrive in existence of parameter uncertainty as quickly as possible by using Lyapunov stability theorem. The combination-combination synchronization of systems under multiple stochastic disturbances is studied by utilizing finitetime Lyapunov theory, and nonsingular terminal SMC technique [21], new fractional-order sliding surface, adaptive combination controller, and some parameter updating laws are deduced at the same time.
In terms of practical applications, an appropriate LKF with double and triple integral terms with details on both lower and upper bounds of delay is completely designed [22]; then, a new class of delay-dependent adequate condition is proposed, so that the error system is ðQ, S, RÞ − γ − strict dissipative. The problem of sampled-data synchronization for delayed multiagent networks with fixed topology is investigated and applied to coupling circuit successfully [23]. The finite-time event-triggered control synchronization for delayed stochastic neural networks is studied with passivity and passification approach [24]; then, a nonfragile ETC is proposed to diminish the communication load during networked transmission.
Based on above findings in the existing literature, the problem of global asymptotic stability and finite-time simultaneous SMC of neural networks with integral nonzero lower bound time-varying delays is debated in depth in this paper. In the process of research, inspired by the integraldifferential equation, the original distributed time-varying delay of integral term is replaced by a time-varying delay. The main contributions of paper are summarized as follows: (i) By utilizing the inequality scaling technique and mutually convex combined inequality, a tight upper bound of augmented LKF derivative is estimated; as a result, a sufficient criterion for universal asymptotic stability of RNNs with the combination of distributed time-varying delay and time-varying delay is derived (ii) In order to eliminate the time that error system state trajectory slides along sliding mode flow pattern until convergence at the origin, a sliding mode manifold is defined by synchronization error; as a result, a semiglobal practical finite-time synchronization of time-varying delayed RNNs is obtained The rest of the remaining part is arranged as follows. The stability analysis of delayed RNNs is described, and some lemmas are given in Section 2. In Section 3, a new criterion which can be used to guarantee the stability of improved integral-type time-varying delayed RNNs is obtained. In Section 4, based on stability of drive system, the finite-time synchronization of drive-response system is derived. Simulation results are provided in Section 5, and the conclusion is drawn in Section 6.
For convenience of understanding, the mathematical notations are summarized in Table 1.
Remark 5. Due to the presence of external input J, equilibrium point in (1) is not located at the origin in most cases. Since changing the equilibrium point does not affect the stability, it is shifted to the origin for the convenience of calculation.
Remark 6. If the neuron activation function is derivable, μ − i and μ + i can take the minimum and maximum values of their derivatives by employing the Lagrange's median theorem.

Time Delay-Dependent Stability Criterion
By constructing incremental LKF and applying Lemma 1, Lemma 2, and Lemma 3, the time delay global asymptotic stability criterion for time-varying delayed RNNs (6) can be obtained.

Remark 8.
It showed no information of the delayed neuron state derivatives for seeking the augmented LKF derivative in (30)-(33); at the same time, in order to reduce the computation complexity and conservative of the stability criterion, the constraints of neuron activation function and the information of time-varying delays lower bound are fully considered in (34).

Remark 9.
The upper bound of delay can be calculated by using LMI toolbox in MATLAB; the smaller step, the higher precision, but with constraint of the running time. 6 Advances in Mathematical Physics

Design of Sliding Mode Controller
Based on stability of drive system (1), the finite-time synchronization of drive-response system is further analyzed. Assume that the external input vectors in drive system (1) is equivalent to response system (2), which represents J 1 = J 2 , the synchronization error system can be obtained by defining error signal kðtÞ = xðtÞ − yðtÞ: where ςðkð•ÞÞ = f ðxð•ÞÞ − f ðyð•ÞÞ.
Proof. Choose the following Lyapunov function candidate: Combining Equation (37) with Equation (39), the time derivative of Y is According to Remark 10, by employing the inequality k hðtÞk ≤ khðtÞk 1 , one has Lemma 4 implies that for ∀t ≤ 0, the inequality _ Y ≤ −ξ 2 ðα+1Þ/2 Y ðα+1Þ/2 + j holds, then kðtÞ = hðtÞ is semiglobal practical finite-time stable. After the state trajectory of the error system (37) under the condition of the sliding mode controller (39) reaches the sliding mode manifold hðtÞ = 0 in a finite time, the system state satisfies hðtÞ = kðtÞ = 0, which means that the synchronization error kðtÞ will converge to zero in a finite time; in other words, drive system (1) and response system (2) are globally synchronized in a finite time. This completes the proof of Theorem 11.
Remark 12. In order to reduce the chattering on the switching control action, the sign function sgn ðhðtÞÞ in (39) is replaced by hðtÞ/ðkhðtÞk + 0:01Þ by using the method of [31].
Remark 13. The method proposed in this paper does not require solving for the feedback control gain, and the switching gain term ρðtÞ does not need to apply the neuron activation function term kð·Þ in the driving system Equation (1).

Remark 14.
In this paper, Sigmoid is chosen as the activation function of drive system, due to Sigmoid function is flatter than tanh and its derivative decreases at a slower rate, which effectively alleviate the gradient disappearance problem and make the neural network learn more efficiently.
Remark 15. This paper investigates a class of continuous differentiable time delay for stability analysis of RNNs; its advantage is to ensure the stronger robustness of time-delay systems by taking into consideration that both the differential and the area tend to be zero, which makes the system stable, but for the case that the time delay is discontinuous, the improved delaydependent integral inequality presented in this paper cannot be directly employed; then, it is necessary to construct proper LKFs and new integral inequalities.

Numerical Example
Consider the time-varying delayed RNNs (6) with the parameters given in [32]: For different upper bounds on time-varying delay derivatives ϱ, the maximum upper bound that guarantees the stability of system s can be obtained according to Theorem 7 as shown in Table 2.
Synchronization trajectory of drive and response systems under the influence of controller are simulated in Figures 2  and 3. Figure 2 shows that drive system (1) does not synchronize with response system (2), and it is clearly seen that drive and response systems achieve synchronization in a finite time under controller in Figure 3.
The corresponding synchronization error kðtÞ, which is equivalent to the sliding mode manifold hðtÞ, is shown in Figure 4.
From Figures 2 to 4, it is clearly seen that drive system (1) and response system (2) can achieve finite-time global synchronization with a synchronization time about 0.07 s.

Advances in Mathematical Physics
Simulation results verify the effectiveness and feasibility of synchronization control method proposed in this paper.

Conclusion
Stability and a finite-time synchronization problem for delayed RNNs with integral time-varying delays are studied in this paper. Firstly, an augmented LKF is developed without involving the information of delayed neural states, and a stability criterion of LMI is obtained. Secondly, a sliding mode flow pattern which is equivalent to synchronization error is presented; then, a suitable sliding mode controller is designed. Finally, the simulation results clearly verify the superiority of presented approach to stability analysis of delayed RNNs and effectiveness of developed SMC method to finite-time synchronization. In the future work, the proposed methods will be extended to coupling circuit, dissipation analysis, chaotic network, filter issues, and practical applying for signal processing.

Data Availability
The data used to support the findings of this study are included within the article.  10 Advances in Mathematical Physics