Sampling-Based Event-Triggered Control for Neutral-Type Complex-Valued Neural Networks with Partly Unknown Markov Jump and Time-Varying Delay

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ere are two methods usually used in the study of CVNNs: one is to divide the neural networks into the real part and imaginary part, then, the original CVNNs will be changed into real-valued neural networks [8][9][10]. e other method is when the activation function in CVNNs cannot be separated, the stability condition of the system will be sure by the complex-valued LKFs under the condition that the activation function satisfies the complex-valued Lipschitz continuity [11,12], and it would increase the difficulty of the analysis. Considering the network works which are transformed in the first method are real-valued systems, it is easier to understand. Unfortunately, the dimensions of the obtained real-valued neural networks are twice the dimension of the original CVNNs. Also, the partial derivatives of the real and imaginary parts of state variables of the activation function are required not only to exist but also to be continuous and bounded, resulting in problems in our analysis and the application of the obtained conclusions would be limited. According to the state of the art, the LKFs method is often used to deal with neural networks problems because of its simplicity and effectiveness [10,[13][14][15], so it is necessary to construct LKFs with conjugate transpose of state vector to study the CVNNs by the method which do not separate the original system.
Due to the complexity of the reality, certain systems are sometimes difficult to apply to the reality, making the research of uncertain systems more valuable and practical [16]. H∞ control provides a good method of dealing with unstructured uncertain systems, and research related has grown over the past two decades. Besides, it was increasingly used to analyse problems in the field of robotics, aerospace, and power systems [13,[16][17][18][19][20]. Reference [16] was concerned with the stabilization problem for uncertain T-S fuzzy systems with time-varying delays via a robust H∞ state-feedback controller. And more general LKFs method with relaxed conditions was constructed through an improved time-delay interval segmentation method. e state-feedback H∞ control problem of time-delay systems is studied in [13], and the reciprocal convex inequality was used to obtain stability conditions of the system. However, the research we mentioned is all committed to the real systems; there is little related research on the H∞ performance of CVNNs, and it is one of the main tasks in this paper. Furthermore, many scholars have specialized in the dynamic system with Markov jumping on account of the universal of Markov phenomenon, and abundant achievements have been made [14,[21][22][23][24][25][26][27]. In the applications of practical engineering, the analysis and control synthesis of Markov jumping systems are troubled by many dicey factors, such as the partial unknown transmission probability and the uncertainty of transmission rates. Some preliminary results have been obtained in the study of Markov jumping systems with partially unknown transmission probability [21,23,24,27]. To our knowledge, less effort has been made on CVNNs with Markov phenomenon. To sum up, it is necessary to analyze the stability and H∞ performance of Markov jumping CVNNs in the case of partially unknown transmission probability.
e so-called event-triggered control refers to a control of the tasks; whether to be executed is determined by the given eventtriggered conditions in advance rather than according to the time. e control tasks would execute d immediately when the event-triggered mechanism is broken out. Comparing to the controller with time mechanism [32], events-triggered control can save the computing resources, battery energy, and communication resources obviously. Note that the event-triggered was continuous in the inchoate phase and that special hardware is needed to monitor the status continuously. Yue dong et al. proposed the (SBET) control, which is a discrete one [34]. And the monitor only needs to observe the state of the system at discrete instants with a SBET scheme that can effectively reduce the number of control tasks and save the communication resources significantly. In literature [37], the global asymptotic stability of the CVNNs under the framework of event-triggered control was studied by dividing the system into real and imaginary parts. ere is an output-feedback H∞ control under the event-triggered framework with nonuniform sampling used to explain the stability of networked control systems by Peng and Zheng [33]. Unfortunately, there is almost no research on CVNNs with a SBET control. So, how to stabilize the CVNNs by designing a sampling-based event-triggered controller is of our interest.
To date, this paper focuses on the stabilization of CVNNs with partly unknown Markov jump and time-varying delay. Firstly, a H∞ state-feedback control is proposed to explain the stability of neutral-type CVNNs; to our knowledge, there is little research about the stabilization of neural networks in the complex field. Secondly, the stabilization of Markov jumping neutral-type CVNNs with H∞ performance is considered under the framework of a SBETcontroller. And it is the first time to study CVNNs with a sampling-based event-triggered mechanism while avoid splitting the system into two parts, which reduces the computational complexity greatly.
Notations: throughout this work, C denotes the complex field and C n is the n dimensional complex space. A > 0 (or where A H and A T , respectively, mean the conjugate transpose matrix and transpose matrix of A, * in a matrix denotes the selfconjugate part of the Hermitian matrix.

Complexity
where ? expresses the transition rates which are unknown and, for ∀p ∈ ℘, the set P p is expressed as P p � P p k ⋃ P p uk with ℵ p k ≜ q: π pq is known for q ∈ ℘ , ℵ p uk ≜ q: π pq is uknown for q ∈ ℘ .

(4)
With r t � p and the controllers being considered as u(t) � K p ϑ(t), then system (1) would be reexpressed as . . , ι k ∈ Z + , describe the eventtriggered sequence, it is clear that 5 2 ⊆5 1 , and the samplingbased event-triggered scheme is given as where As a matter of fact, the network-induced time delay is inevitable when signals are transmitted between the event generator and the actuator at time t, which can be expressed as ς(ι k h), and ς(ι k h) ∈ [0, ς]. In [t k , t k+1 ), the control signal is hold by a zero-order-hold (ZOH) function. For j � 0, 1, . . . , l m− 1 , set , and the SBET controllers are represented as with t ∈ T k ; system (1) would be transformed to Remark 1. e parameter ϖ can make a significant difference in whether the data would be released or not. As ϖ gets smaller, conditions (6) are more likely to be violated.
that all the sampled data would be transmitted, and the SBET scheme (6) reduces to a periodic time-triggered one. Case 2: if the right-hand side of the inequality is a constant ♭ > 0, the SBET scheme would be changed into (10) and then, the SBET scheme (6) is converted from a relative threshold event-trigger condition to a fixed threshold eventtrigger condition. However, an event-triggered control with the condition in case 2 may cause the system to be unstable and fail to achieve the control intention. e following assumptions are put forward to draw the main results. Assumption 1. Each activation function φ p (·) in (5) satisfies the following condition: where ϑ 1 ≠ ϑ 2 and L � diag l 1 , l 2 , . . . , l n with l i > 0, i � 1, 2, . . . , n.

Complexity 3
Definition 1 (see [28]). System (5) is said to be stochastically stable if the following inequality holds: Definition 2. If the CVNNs satisfy Definition 1 under the initial condition, and ‖G‖ ∞ < c, that is, then we say system (5) possesses H∞ performance with attenuation index c. Several essential lemmas are given before the proof of our theorems.
Lemma 1 (see [12]). For any vector ζ ∈ C m , there is a scalar c ∈ (0, 1), and H ∈ C n×n is a positive defined matrix, ma- If Lemma 2 (see [41,42]). R > 0 is a Hermitian matrix, for any continuous and differentiable function ϑ: [c, d] ⟶ C n ; the following inequality holds where By combining Lemma 1 with Lemma 2, we have the following lemma.

Lemma 3. For Hermitian matrix S and arbitrary matrices
where Proof. By dividing the left-hand side of inequality (18) into two parts, the following formula is obtained: estimating the two parts of the right-hand side of the above inequality, respectively, by Lemma 2, and then, combining the obtained parts via Lemma 1, we have with 4 Complexity and this ends the proof. □ Remark 2. A similar conclusion has been obtained in real domain. Lemma 3 is an extension which can apply to the complex domain with complex-valued matrices and vectors ϑ(t), ϑ(t − σ(t)). It is worth noting that the quadratic form _ ϑ H (s)S _ ϑ(s) is in the real number field because S is a Hermitian matrix.
Lemma 4 (see [35]). Set U > 0 as an arbitrary matrix in R n , and υ(t) is a vector with appropriate dimensional. us, the following inequality is obtained: for any V and vector λ(t) with appropriate dimensional are not dependent on integral variables.
is lemma is currently only used in the real number field; the following corollary will extend it to the domain of complex numbers.

Corollary 1. U is a positive defined Hermitian matrix, and ξ(t) is a complex vector with appropriate dimensional. us, the following inequality is obtained:
for any V and vector λ(t) with appropriate dimensional are not dependent on integral variables.
Proof. U > 0 is a Hermitian matrix; for complex vectors a, b with appropriate dimension, we have that is to say, and, then, the following conclusion is obtained: is ends the proof.
□ Remark 3. In the proof of Corollary 1, we can clearly see that ξ(s)ds because it may be an imaginary number. Also, the condition U > 0 in Lemma 4 is changed into U, which is a positive defined Hermitian matrix, which can ensure that both sides of the inequality are real numbers so that the magnitudes can be compared. erefore, the above inequality can be used in the complex domain in the form of Corollary 1.

Main Result
Two main theorems will be presented in this section. In the first place, we study the stabilization of system (5) with H∞ performance without a SBET scheme; and then, by Complexity employing the SBET scheme (6), we derive a sufficient condition for the stability of system (8). Now we define the following vectors and matrices for clarity: where H , e p � 0 n×(p− 1)n I n 0 n×(11− p)n , 6 Complexity

en, system (5) is asymptotically stable via u(t) � K p ϑ(t) with H∞ disturbance attenuation c and the controller gain matrices are designed as
Proof. Consider the following LKF: where with Adopting the weak infinitesimal operator L in [21], which acts on V m (ϑ(t), t, i) as then, similar to the computation in [26,27], we have and considering the transition rate matrix includes not only the known part but also the unknown part, the following equations would hold for matrices Complexity en, (36) and (37) are equal to the following equations: Via easy calculations, the following equations are given: Dealing with the integral term containing the Markov jump, by pre-and postmultiplying (X H ) − 1 and X − 1 with (30), we have and then, the following inequalities are obtained: 8 Complexity For arbitrary matrices S 11 , S 22 , S 12 ∈ C n×n , because of (42) will be processed as follows: by dividing it into two parts, we have estimating the two parts of the right-hand side of the above inequality, respectively, by Lemma 2, and then, combining the obtained parts via Lemma 1, we have (47) us, On the other hand, for diagonal matrices One can also obtain the following equation from (5) for any nonsingular matrix X 2 with the appropriate dimension, we defined X 1 � mX 2 , and it is clearly that X 2 is also nonsingular, such that rough the analysis in (39)-(50), we have where Defining and then, by pre-and postmultiplying X H Z and X Z (Z � 11) with Ω, (29) is obtained. Combining (29) with (51) and using the Schur complement, we have and integrating the formula above from 0 to ∞, we have and, with the zero initial condition ϕ � 0, we have V(φ(θ)) � 0, V(ϑ(∞)) ≥ 0; thus, is completes the proof. Next, based on eorem 1 and Lemma 3 and Corollary 1, a sampling-based event-triggered controller will be used to study the stabilization and H∞ performance of CVNNs with uncertain Markov jump. □ Theorem 2. For the given positive scalars σ, σ d , ρ, η and 0 ≤ ϖ < 1, if there exist Hermitian matrices P p , Q p , Q 4 , Q 5 , Q 6 , R, G, W p in C n×n , R p , R, M p ∈ C 2n×2n , and E 1 , E 2 , E 1 , E 2 are positive diagonal matrices, matrices K p , and nonsingular matrix X ∈ C n * n are proper dimension matrices with p ∈ 1, 2, 3 { }, such that the following conditions hold: 10 Complexity , Complexity System (8) possessing H∞ performance is asymptotically stable under the SBET scheme (6), and the controller gain matrices in (7) are designed as K p � Y 2p X − 1 .
Proof. Choose the following LKF: with being positive defined Hermitian matrices with proper dimension, and From the definition of V(ϑ(t), p), noting V J (ϑ(t), p), J � 1, 2, 3, is a continuous function; it is easy to see that Also, we can derive that   Im (x 1 (t)) Im (x 2 (t))  Re (x (t)) Re (x 1 (t)) Re (x 2 (t))  Re (x 2 (t)) Re (x 1 (t))     Im (x 2 (t)) Im (x 1 (t)) Im (x (t))     applying the controller, and also, the signal of Markov jump is represented in (7). Compared with Figures 5 and 6 which are the state of Example 1 without controller, it is clear that the controller we designed can stabilize an unstable system effectively.
Example 2. For the same system parameters with Example 1, σ(t) � t − (0.3 − 0.1 sin(t)) expresses the time-varying delay, and σ d � 0.1, set ϖ � 0.2, h � 0.036, ς � 0.01, and ρ � 20, with the external disturbance ω � (1/1 − t 2 ). When the initial value is set as ψ(t) � [− 5 + 6i; 10 Under the conditions which are given in this example, the imaginary part and real part of control inputs u(t) are portrayed in Figures 7 and 8 and Figure 9 is the Markov signals. Figures 10 and 11 are given to show the states of the real part and imaginary part, respectively. It is easy to see system (8) is tending towards stability soon under the controllers which are designed in eorem 2. Figure 12 shows the release instants of the controller clearly. With the sample interval h � 0.036, it is not difficult to find that the number of data transfers is reduced dramatically by comparing to the time-driven one.

Conclusions
In this work, the stability and stabilization of a class of neutral-type CVNNs with time-varying delay and partly unknown Markov jump have been addressed. After designing a H∞ state-feedback control to investigate the stabilization of our system, we have studied the stabilization of CVNNs with time-varying delay and partly unknown Markov jump by a sampling-based event-triggered controller for the first time. Moreover, the feasibility and validity of our results have been proved through two examples. However, there are also some issues that need further consideration, for instance, to design a new eventtriggered to reduce the number of data-sampling transmission and to develop novel LKFs to reduce the conservatism of the stabilization conditions, and the derivation of less conservative integral inequalities can also help us to get better results; they will put into practice in our work in the near future.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.