FurtherResults onStabilityAnalysis forUncertainDelayedNeural Networks with Reliable Memory Feedback Control

School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China School of Electronic Information and Electrical Engineering, Chengdu University, Chengdu, Sichuan 610106, China School of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China


Introduction
As we all know, because the structure of the NN model is similar to the synapse structure of the human brain, it can be described by a variety of differential equations [1][2][3][4][5][6][7]. e wide application of NNs in various fields has received widespread attention, such as signal processing [8], fault diagnosis [9], optimization problem solving [10], pattern recognition [11], image processing [12], and other fields [13][14][15][16]. However, artificial NNs always need to be maintained in practical engineering applications, so the stability of NNs has also been extensively studied by scholars at home and abroad [15,[17][18][19]. In the engineering application of NNs, the signal transmission between synapses has a time delay, and this delay may lead to instability of the NNs, increased oscillation, or performance degradation [20][21][22]. erefore, the stability research of time-varying neural networks (DNNs) has also received extensive attention [23][24][25][26]. Compared with NNs, the latter requires more technical means and engineering requirements to maintain stability in technical analysis [27][28][29]. erefore, the research on DNNs is obviously more important.
us, in order to effectively solve this problem, Lyapunov-Krasovskii functional (LKF) method is proposed [18,[30][31][32][33]. So far, many researchers have made a lot of contributions of how to establish a suitable LKF in order to better study the delayed systems [13,[34][35][36][37]. In [30], the authors proposed a novel LKF, which contains a common double-integral term, an augmented double-integral term, and two delay-product-type terms, was constructed to analyze the exponential stability. In [31,32], the reciprocally convex matrix inequality was an important technique to develop stability criteria for the systems with a time-varying delay which was studied. In [18,33,35], based on some effective integral inequalities, the integral term in LKF was reasonably scaled. In [34], the authors first proposed robust H ∞ control for T-S fuzzy systems with state and input timevarying delays via DTPF. In [13], the authors proposed novel weighting-delay-based stability criteria for system research. In [36,37], the authors mainly studied the sampled-data control and gave the analysis and proof of related stability.
However, there are many methods to construct a reasonable LKF, but only increasing the cross-sectional area will hardly improve, and it will cause a heavy calculation burden [23,38,39]. erefore, the method of constructing LKF from a new perspective has become a hot issue in current research [34,40]. rough in-depth study of existing work, this paper proposes an improved DTPF strategy to construct a new LKF, which fully considers information concerning time delays and the derivative information of both states and time delays, and the conservativeness of the guidelines can be further reduced. e issues discussed above have inspired the purpose of this study.
Based on the above discussions, we establish some new stability criteria for UDNNs, and a reliable memory feedback controller is designed to ensure that the considered system is asymptotically stable. Compared with the existing results [22][23][24][25], the main contributions of this paper are as follows: (1) A novel quadratic function V 1 (x (t)) is constructed via developing an improved TDPF approach, which can fully excavate some intrinsic relationships between the delay derivative information and the time delay (2) Based on this construction method of LKF, the information storage performance of the function is strengthened, an appropriate integral inequality and linear convex combination method are adopted, and a more conservative stability criterion is obtained (3) Different from the earlier work, this paper designs a new RMFC, which fully considers the effective transmission of the three state signals of the controller while enhancing the performance of the controller
Here, A > 0 is a symmetric matrix; H, E are known real constant matrices of appropriate dimensions. F(t) ∈ R l 1 ×l 2 is an unknown time-varying matrix function satisfying F T (t)F(t) ≤ I. W 0 indicates the connection weight matrix, and W 1 expresses the delayed connection weight matrix; a � [a 1 , . . . , a n ] T ∈ R n is a vector. h(t) is time varying and satisfies (2) Based on Assumption 1 in [25], suppose that z * is the balance point of UDNN (1), which can be transferred to the origin by conversion, x(·) � z(·) − z * . en, system (1) can be expressed as where x(·) � [x 1 (·), x 2 (·), . . . , x n (·)] T ∈ R n is the state vector of the transformed system and being the activation function of the converted system. Based on (3), it holds that Remark 1. Compared with the current design methods of the reliability controller [26][27][28][29], this paper introduces the RMFC design with effective lowering of the brake. e reliability control design considered in (10) is more comprehensive than the general reliability control design, which has a wider range of applications. en, the RMFC is as follows: From the above discussion, consider combining (10) and (11) to get the reliability controller design as follows: (3) with controller (12) can be represented as Lemma 1 (see [38]). Given a symmetric positive definite matrix R, scalar α, scalar β, and α < β, and e in [α, β] ⟶ R n . e following are the inequalities under the given conditions: where

Main Results
In this section, we will provide a novel RMFC design scheme for (12). In the following theorem, the asymptotic condition for system (13) is provided under the designed gain matrices K i (i � 1, 2, 3). For simplicity, some relevant notations are defined as in Appendix A. Theorem 1. Given positive scalars h, μ, and ε. System (13) is asymptotically stable if there exist symmetric positive definite matrices P 1 , M 2 , M 3 , G 1 , G 2 , D 1 , and D 2 , any symmetric matrices P 2 , M 1 , M 4 , and Q i (i � 1, 2, . . . , 5), any matrices U 1 , U 2 , N 1 and N 2 with appropriate dimensions such that the following LMIs hold: where other symbols and related equations are listed in Appendix B.
Proof. Construct an augmented LKF as follows: where Mathematical Problems in Engineering e time derivative of V(x(t)) along the trajectory of system (12) is given.
en, the derivative of V i (x(t)) is derived: Combining (24) and (25), we can get the derivative of _ V T (x(t)) as follows: Based on Lemma 1, we can get en, _ V T (x(t)) can be expressed as follows: , and Ω(h(t)) are given in Appendix B.
Mathematical Problems in Engineering Based on system (13), the following zero formula holds: en, based on Lemma 3 in [34], we can get Based on the convex combination technique, Based on (15)- (19), it is easy to come to the conclusion that system (13) is asymptotically stable. is concludes the proof.

Remark
2. In this paper, we consider P h(t) � P 1 − (h − h(t))P 2 . (I) When h − h(t) � 0, P h(t) will be degenerated to the constant matrix P. (II) Compared with the existing methods [25], this paper only needs to consider that P 2 and P 2 are arbitrary symmetric matrices. Furthermore, as long as (25) and (26) are guaranteed, this constraint helps reduce the strength of positive definite conditions. (III) In addition, this construction method makes full use of the delay information and the delay derivative information, thereby increasing the amount of LKF information storage, which helps to construct a more general LKF to further reduce the conservativeness of the criteria. At present, the method used in this paper is more general in constructing the LKF and includes a wider range of usage background and research significance.
Remark 3. Compared with existing research [24], this paper fully considers the relaxation of the requirements for matrix positive definiteness. In V 3 (x(t)), by using M 1,h(t) and M 2,h(t) to replace the constant matrices, the LCCM is used to make constraints. erefore, this method can obtain less conservative criteria through more relaxed positive definite conditions and increase the time-delay information contained in the LKF. (24) and

Remark 4. In order to better solve integral terms
x (s)ds, in this paper, considers Lemma 1. Compared with the Wirtinger-based integral inequality, Jensen's inequality, and other existing inequalities, Lemma 1 has a tighter bound in order to obtain a less conservative criterion. (13) is asymptotically stable if there exist symmetric positive definite matrices P 1 , M 2 , M 3 , G 1 , G 2 , D 1 , and D 2 . Any symmetric matrices P 2 , M 1 , M 4 , and Q i (i � 1, 2, . . . , 5), any matrices U 1 , U 2 , Y 1 , Y 2 , Y 3 and V with appropriate dimensions such that the following LMIs hold:

Theorem 2. Given positive scalars h, μ, ε, and κ. System
where other symbols and related equations are listed in Appendix C.
Proof. Define

Illustrative Example
In this section, two simulation examples are exhibited to express the effectiveness of the established results. Example 1. Consider DNN (13) with the following matrix parameters, and these matrices are based on [13][14][15][16][17][18]: e MAUBs are obtained by eorem 1, and the other results are listed in Table 1. When μ 2 � 0.8, from Table 2, we can clearly see that our result is markedly better than [13][14][15][16][17][18]25]. In addition, it is worth noting that the result obtained in this paper is improved by 21.8325% compared to [25]. For the reason that the TDPF method is employed, the conservativeness of the obtained criterion is reduced and the performance is improved in this paper.
Setting μ � 1, h(t) ≤ 6.0917, h(t) � (60917/20000)sin ((20000 /60917)t) + (60917/20000), and f(x(t)) � diag (0.4, 0.8)tanh(x(t)). In order to verify the stability of the system, we randomly select different initial values to simulate the dynamic response of the system as shown in Figure 1. From Figure 1, we can clearly see that when different initial values are selected, the system tends to a stable state as time increases.
Based on eorem 2, setting κ � 1, we can get the controller gains as follows: Here, the control input trajectory of DNNs is presented in Figure 2.

Mathematical Problems in Engineering
Inverse of matrix A X T + X Stands for a block diagonal matrix By employing eorem 1, for different μ, Table 3 lists the MAUBs based on eorem 1. From Table 3, we can see that when μ takes different values, the MAUBs obtained in this paper are always the largest. For example, when μ � 0.5, the existing result shows h � 3.0587, but the result obtained by applying eorem 1 in this paper is h � 4.0916. Compared with results in [25], the criterion proposed in this paper improves MAUBs by up to 25.2444%.

Conclusion
is study has proposed further results for the stability analysis issue of UDNNs based on the RMFC scheme. First, an improved quadratic function method has been introduced for constructing a novel V 1 (x(t)), which can fully excavate some intrinsic relationships between the delay derivative information and the time delay. Based on the TDPF and LCCM, the information storage has been further improved for obtaining new theoretical results. Second, by using resultful integral inequalities and correlation analysis approaches, several relaxed criteria have been established with respect to the asymptotical stability of the considered UDNNs. ird, a new RMFC has been designed, which can ensure the system stability of UDNNs. Lastly, two numerical experiments have been given to illustrate the significance of the theoretical results. In the future research, we need to further study the UDNNs based on quantized measurements  T im e 1 20 -0. 5 10 -1 0 T im e 1 20 -0. 5 10 -1 0 in order to improve the entire system [41]. In addition, the results obtained in this paper will be extended to the chaotic Lurie system [42,43], quaternion-valued or memristor-based neural networks [44][45][46], T-S fuzzy NNs [47,48], Markov jump systems [49,50], and complex dynamical networks [51][52][53]. ese will occur in the near future. , , e i � 0 n×(i− 1)n I n×n 0 n×(9− i) , i � 1, 2, . . . , 12,
Disclosure e authors declare that the work described is original research that has not been published previously and not under consideration for publication elsewhere, in whole or in part.