Finite-time decentralized event-triggered feedback control for generalized neural networks with mixed interval time-varying delays and cyber-attacks

: This article investigates the finite-time decentralized event-triggered feedback control problem for generalized neural networks (GNNs) with mixed interval time-varying delays and cyber-attacks. A decentralized event-triggered method reduces the network transmission load and decides whether sensor measurements should be sent out. The cyber-attacks that occur at random are described employing Bernoulli distributed variables. By the Lyapunov-Krasovskii stability theory, we apply an integral inequality with an exponential function to estimate the derivative of the Lyapunov-Krasovskii functionals (LKFs). We present new su ffi cient conditions in the form of linear matrix inequalities. The main objective of this research is to investigate the stochastic finite-time boundedness of GNNs with mixed interval time-varying delays and cyber-attacks by providing a decentralized event-triggered method and feedback controller. Finally, a numerical example is constructed to demonstrate the e ff ectiveness and advantages of the provided control scheme.


Introduction
Neural networks (NNs) are widely used in various fields, such as pattern recognition, combinatorial optimization, image processing, associate memory, signal processing, and fixed-point computations [1][2][3][4], due to their enormous capacity for information processing. The literatures [5][6][7][8][9][10][11][12] classifies neural networks into two categories. The first category consists of static neural networks (SNNs), which rely on the external states of neurons (neural states of neurons). The second category consists of local field neural networks (LFNNs), which depend on the internal states of neurons (local The primary contributions of this article can be summarized as follows: (1) We propose a novel decentralized event-triggered method and feedback controller for GNNs with mixed interval time-varying delays and cyber-attacks. This approach ensures finite-time boundedness while estimating the derivative of the Lyapunov-Krasovskii functionals using an integral inequality with an exponential function. By utilizing this method, we address the challenges associated with system stability and effectiveness in the presence of time-varying delays and cyber-attacks. (2) The event-triggered approach introduced in this work leads to a reduction in network resource utilization. This reduction alleviates the transmission burden on the network by enabling each sensor to autonomously determine the optimal time for signal transmission. By minimizing unnecessary transmissions, the proposed approach enhances the overall efficiency and scalability of the system. (3) We describe random cyber-attacks by utilizing Bernoulli-distributed variables and represent them through a nonlinear function that satisfies a specific condition. This modeling approach allows us to capture the realistic nature of cyber-attacks and effectively incorporate them into the control strategy. (4) We provide an illustrative example accompanied by simulations to demonstrate the feasibility and effectiveness of the proposed control strategy. The results showcase the improved effectiveness achieved in terms of system stability, finite-time boundedness, and resilience against cyber-attacks.
The remainder of this article is structured as follows. In Section 2, we introduce GNNs and provide preliminaries. Section 3 uses a state feedback controller to examine the stochastic finite-time bounded conditions for delayed GNNs with cyber-attacks. Section 4 presents a numerical example to illustrate the effectiveness of the proposed methods. In Section 5, we conclude and discuss our article.
Notations: This article utilizes the following notations: I denotes the identity matrix; ∥·∥ represents the Euclidean vector norm of a matrix; R n indicates the n-dimensional Euclidean space; Prob{X} represents the probability of event X to occur; diag{· · · } refers a block-diagonal matrix; the notation P T and P −1 stand for the transpose and inverse of matrix P, respectively; the expression P < 0 (or P ≤ 0) signifies that the real symmetric matrix P is negative definite (or negative semi-definite); λ min (P) (or λ max (P)) denotes the minimum (or maximum) eigenvalue of real symmetric matrix P; the term L 2 [0, ∞) denotes a function space consisting of quadratically integrable functions over the interval [0, ∞); the notation Sym{P} represents the sum of P and its transpose, i.e., P + P T ; the symbol * indicates the elements below the main diagonal in a symmetric matrix.

Problem formulation and preliminaries
In this article, we introduce a problem involving GNNs with mixed interval time-varying delays. The problem statement is described as follows: where x(t) ∈ R n represents the state vector at time t; n is the number of neurals; z(t) ∈ R n represents the output of the system;Ā = diag{a 1 , a 2 , ..., a n } indicates a diagonal matrix; W, B 0 , B 1 and B 2 refer connection weight matrices; the matrices B w and B u are real constant matrices with known values; ω(t) refers the external disturbance input; u(t) ∈ R m denotes the control input; and f (W x(t)) = [ f 1 (W x 1 (t)), ..., f n (W x n (t))] T and h(W x(t)) = [h 1 (W x 1 (t)), ..., h n (W x n (t))] T indicate the activation functions. The time delays in the system are represented by η i (t)(i = 1, 2) and τ(t), which correspond to interval distributed time-varying delays and interval time-varying delays, respectively.
Additionally, we assume that the neuron activation function and communication network delays satisfy the following assumptions. Assumption (A1). Each of the activation functions, f i (t) and h i (t), where i = 1, 2, ..., n, is assumed to be continuous and bounded, satisfies the following conditions: there exist constants Assumption (A2). Let τ t i k denote the communication network delay at the sampled instant t i k in the ith sensor. It is assumed that 0 < τ t i k <τ i , whereτ = max i∈1,2,...,nτ i .
The diagram of the decentralized event-triggered control for GNNs with cyber-attacks is shown in Figure 1. This structure involves multiple sensors and controllers exchanging information over a communication network, which may experience time delays and potential cyber-attacks. To minimize network transmissions, event generators are employed at each sensor. When a new signal is sampled, it is promptly sent to the corresponding event generator. Additionally, the signal includes the disturbance input vector ω(t), representing either external factors or disturbances that influence the controlled system. The controlled output z(t) represents the desired or targeted system output. Only signals that violate the event-triggered condition are transmitted over the network, thereby reducing communication bandwidth requirements. The primary objective of this decentralized event-triggered structure is to ensure the stability of the neural network while minimizing the impact of cyber-attacks. Furthermore, a three-line table summarizing the algorithm of the proposed method is provided below in Table 1.  Step Description 1 Initialize network parameters, event-triggering thresholds and cyber-attack detection mechanisms 2 Sample input signals, check event-triggering conditions and detect potential cyber-attacks 3 Update network states, transmit relevant information among nodes and implement countermeasures against cyber-attacks The following expression gives the predefined event-triggered criterion for the ith sensor: where a weighting matrix is denoted by Following a similar approach to [35], we establish a sequence of buffers on the actuator side, each with a unique timestamp to hold the controller outputs. This enables the actuators to update the controlled inputs by selecting the corresponding controller output from the buffers. Thus, the input update time set for the actuators is defined as t k+1 h = t k h + jh, where jh = argmini i∈{1,2,...,n} { j i h} can be obtained from (2.2). Define Based on (2.2), the condition for n channels can be derived as follows: where e(t) = x(t k h) − x(t k+1 h + jh), Ω = diag{Ω 1 , ..., Ω n } and σ = diag{σ 1 , ..., σ n }.
The following is a description of the proposed method for designing the controller model, which takes into account both the decentralized event-triggered scheme and cyber-attacks: the controller gain is denoted by K, and the function of cyber-attacks is represented by g(x(t − d(t))), where g(x(t)) = [g 1 (x 1 (t)), g 2 (x 2 (t)), ..., g n (x n (t))] T , d(t) ∈ (0,d] andd is a positive constant. Remark 2.2. The dynamic event-triggered system (ETS) utilizes periodic sampling and a waiting period of h seconds to evaluate the event-triggering condition, allowing for ample decision-making time and preventing the occurrence of Zeno behavior. Zeno behavior refers to the destabilizing effect of rapid and continuous event triggering. The introduction of this temporal constraint ensures a balanced operation, averting undesirable system behavior and instability by triggering events at appropriate intervals.
Remark 2.3. This article considers the probability distribution of cyber-attacks, which is assumed to follow the Bernoulli distribution. The received sensor measurements are represented by ρ(t k ), with ρ(t k ) = 1 indicating actual sensor measurements and ρ(t k ) = 0 indicating that the sensor measurements accessible through the communication network have been attacked.
Combining (2.1) and (2.4), we can derive thaṫ Taking the definition of β(t) and the features of ρ(t k ) into account, we can express (2.5) as the following: We present the following assumptions and lemmas instrumental to deriving our main results. Assumption (A3). For each g i (t), i = 1, 2, ..., n represents a cyber-attack function that is bounded. There exist constants G − i and G + i such that Remark 2.4. Assumption (A3) allows for the analysis of the system's response to cyber-attacks, even in the absence of detailed information about the attack signals. It facilitates the design of resilient control strategies that can handle various attack scenarios and maintain system stability, despite bounded cyber-attacks.
[32] Given positive constants c 1 , c 2 and T with 0 < c 1 < c 2 and X is a symmetric positive definite matrix. The GNNs (2.1) is finite-time bounded with respect to (c 1 , Lemma 2.2. (Jensen's inequality [36]) For a symmetric positive-definite matrix, M ∈ R m×m , and any given scalars d 1 and d 2 , the following inequality holds: , for any matrices R ∈ R and L that satisfies R L L T R ≥ 0, the inequality holds as follows: Lemma 2.4. [37] For any positive scalars a, b > a and α, and any symmetric matrix M = M T > 0 with dimension n × n, the following inequality holds: Remark 2.5. When α = 0, then the specific values of Φ 0 , Φ 1 , Σ 0 , and Σ 1 are given by du . This implies that Lemma 2.4 reduces to the well-known Wirtinger's inequality.
Lemma 2.5. [38] For a full column rank matrix L ∈ R n×m , the singular decomposition is L = U 1 ΣU T 2 , where U 1 and U 2 are orthogonal matrices, and Σ ∈ R n×m is a rectangular diagonal matrix with positive real numbers. Let M be a matrix of the form M = U 1 Lemma 2.6. (Schur complement [39]) If matrices X, Y, and Z have appropriate dimensions and satisfy X = X T and Y = Y T > 0, then the inequality X + Z T Y −1 Z < 0 holds if and only if

Analysis of stochastic finite-time boundedness
In this section, we introduce new sufficient conditions for delayed GNNs that build on the main theorems. To start, we define the parameters as follows: and we define vectors as follows: First, we obtain new sufficient conditions of the finite-time decentralized event-triggered feedback control problem for GNNs with mixed interval time-varying delays and cyber-attacks as follows.

Combining (3.4)-(3.19) with (2.3), we obtain
Applying the Schur complement [39], we can derive that (3.1) is equal to: Then, multiplying (3.21) by e −αt , we can be written as Considering V(x 0 , 0), we can derive that Furthermore, by referring Eq (3.3), we obtain the following: Subsequently, by utilizing Eqs (3.23) to (3.24), we obtain the following: Thus, if the relation in (3.2) holds true, then it implies that E{x Consequently, it can be concluded that the delayed GNNs (2.6) with cyber-attacks are stochastically finite-time bounded with respect to (c 1 , c 2 , T, X, d w ). This completes the proof. □ Remark 3.1. The activation functions play a crucial role in determining the existence and uniqueness of solutions in GNNs. Specifically, the activation functions in GNNs need to satisfy the Lipschitz condition, which ensures that the functions possess certain properties that guarantee the existence and uniqueness of solutions. The Lipschitz condition requires the functions to have a bounded derivative, thereby controlling the rate of change of the functions. In this article, the activation function is assumed to satisfy Assumptions (A1) and (A3). In this case, it is important to note that the activation function does not necessarily need to be non-monotonic and differentiable. The constants . . , n, can take either positive, zero, or negative values. Assumptions (A1) and (A3), as considered in Eqs (3.14)-(3.18) of this article, ensuring not only Therefore, this assumption is weaker and more general than the usual Lipschitz condition. In conclusion, the activation functions in GNNs play a critical role in ensuring the existence and uniqueness of solutions. While the assumption considered in this article relaxes the requirement for non-monotonicity and differentiability, it provides a more general condition that still guarantees the desired properties.
Based on the results of Theorem 3.1, we can now propose a controller design approach for the delayed GNNs (2.6) as follows. Assumption (A5) To handle the nonlinear terms in Theorem 3.1, we adopt a similar assumption to [38], B is assumed to be full column rank, and the singular decomposition for B as B = L 1 B u 0 0 L 2 , where B u 0 is the first u 0 columns of B, L 1 and L 2 are appropriate matrices with compatible dimensions.

26)
where . The other matrices are given in Theorem 3.1. The gain of the controller can be determined as follows: 27) in which B u 0 and L 2 are defined in Assumption (A5).
Proof. Let P and B u be defined as follows based on Assumption (A5): By applying Lemma 2.5, we can find a new variable P k that satisfies PB u = B u P k , from From PB u = B u P k , then we substitute PB u K = B u P k K and define Y = P k K, it follows from (3.1) that (3.28) In terms of the following inequality: One has that −PR −1 j P ≤ −2κ j P + κ 2 j R j .
By replacing −PR −1 j P with −2κ j P + κ 2 j R j in (3.28), it becomes evident that (3.25) is obtained. Thus, the proof is complete. □ The delayed GNNs (2.6) take on the following forms under the decentralized event-triggered scheme without cyber-attacks: (3.29) Following the same approach as the proof for Theorem 3.2, we can derive the following corollary. Furthermore, we also define the vectors as follows: T , e i = 0 n×(i−1)n I n 0 n×(16−i)n , i = 1, 2, ..., 16.
Applying (3.25), (3.26) and (3.27), we derive the corresponding minimum allowable lower bounds (MALBs) of c 2 = 5.1788 and control gain as follows: We show the effectiveness of our results in Case I. Figure 2 illustrates the state responses of x(t) for the delayed GNNs (2.6) without u(t). The responses of the state variable x(t) and the time history of x T (t)Xx(t) in Case I of the delayed GNNs with cyber-attacks (2.6) are depicted in Figures 3 and 4, respectively. Furthermore, the computed MALBs of c 2 for different c 1 are presented in Table 2. From the table, we can see that the MALBs of c 2 from our results increase as c 1 increases. Based on the simulation results presented above, it has been determined that the event-triggered state feedback control scheme for delayed GNNs (2.6) is stochastically finite-time bounded within a specified time interval. This conclusion remains valid even in the presence of cyber-attacks and gains fluctuations.     Case II: This case focuses on delayed GNNs (3.29) under the event-triggered scheme without considering cyber-attacks.
Assuming that the following parameter values are used: 1, 2, 3, ..., 5), τ m = 0.25, τ M = 0.9, τ(t) = 0.7| sin(t)| + 0.25, η 1 = 0.1, η 2 = 0.7, η 1 (t) = 0.4 + 0.3 sin(t), η 2 (t) = 0.5 + 0.2 sin(t), and an external disturbance given by 0.5e −0.5t sin(t). We solve the LMIs in Corollary 3.3 to obtain a feasible solution that guarantees finite-time boundedness with respect to (c 1 , c 2 , T, X, d w ). Applying We present the effectiveness of our findings in Case II. The response of the state variable x(t) and the time history of x T (t)Xx(t) in Case II of the delayed GNNs without cyber-attacks (3.29) are illustrated in Figures 5 and 6, respectively. Additionally, we provide the computed MALBs of c 2 for different c 1 in Table 3. The table reveals that as c 1 increases, the MALB of c 2 also increases according to our findings. The simulation results above confirm that the event-triggered state feedback control scheme for delayed GNNs without cyber-attacks (3.29) is stochastically finite-time bounded within a specified time interval.  x T (t)Xx(t) x T (t)Xx(t) c 2 c 2 =3.2378 Figure 6. Time history of x T (t)Xx(t) for the GNNs without cyber-attacks (3.29) in Case II. Remark 4.1. Figure 5 illustrates the state responses of the GNNs, represented by the variable x(t), in the absence of cyber-attacks. The graph displays three distinct curves, each corresponding to a specific experimental condition. The first curve represents the state response of the GNNs under normal operating conditions. The second curve represents the state response of the GNNs when subjected to external disturbances. The third curve demonstrates the state response of the GNNs when variations are introduced into the system. By examining these three curves, we can gain a comprehensive understanding of the GNNs' behavior under different experimental conditions without the presence of cyber-attacks.

Conclusions
This article proposes the decentralized event-triggered method and feedback controller to ensure the finite-time boundedness of GNNs with mixed interval time-varying delays and stochastic cyber-attacks. By the Lyapunov-Krasovskii stability theory, we apply the integral inequality with the exponential function to estimate the derivative of the LKFs. We also present new sufficient conditions in the form of linear matrix inequalities. The event-triggered approach reduces the network's resource utilization and transmission burden, while the random cyber-attacks are described applying Bernoulli distributed variables. A numerical example is provided to demonstrate the effectiveness and advantages of the proposed control scheme. Additionally, this research can be expanded in the future to include various dynamic systems, such as uncertain NNs [40], complex networks [41], neutral high-order Hopfield NNs [42], neutral-type NNs [43], quaternion-valued neural networks [44], spiking NNs [45], memristive NNs [46], stochastic memristive NNs [47] and synchronization of Lur'e Systems [48]. By focusing on these research directions, the proposed method can be further enhanced to achieve improved performance such as passivity [8], dissipativity [9], H ∞ [5,22,25,26], and extended dissipative performances [12,15,16], even in the presence of cyber-attacks. These advancements will greatly contribute to the development of more robust and high-performing control systems specifically designed for networked applications.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.