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

Authors: Zhen Wang; Lianglin Xiong; Haiyang Zhang; Yingying Liu
Journal: Complexity
 (2021)

---

## Abstract

This work is devoted to studying the stochastic stabilization of a class of neutral-type complex-valued neural networks (CVNNs) with partly unknown Markov jump. Firstly, in order to reduce the conservation of our stability conditions, two integral inequalities are generalized to the complex-valued domain. Secondly, a state-feedback controller is designed to investigate the stability of the neutral-type CVNNs withH∞ performance, making the stability problem a further extension, and then, the stabilization of the CVNNs with H∞ performance is investigated through a sampling-based event-triggered (SBET) control for the first time that the transmission event is not triggered except when it violates the event-triggered condition. Finally, two examples are given to illustrate the validity and correctness of our obtained theorems.


---

## Body

## 1. Introduction

In terms of the wide application in electromagnetic processing, light wave and sound wave, the neural networks have attracted much attention in recent decades [1–3]. In the meantime, complex-valued signals occur inevitably in practice, and more and more scholars began to make investigations on CVNNs [4–12]. There 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 [8–10]. The 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, 13–15], 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, 16–20]. 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. The 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, 21–27]. 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.In the past few years, the sampled-data control and event-triggered control as a discrete control have attracted much attention from scholars [15, 28–40]. The so-called event-triggered control refers to a control of the tasks; whether to be executed is determined by the given event-triggered conditions in advance rather than according to the time. The 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. There 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, aH∞ 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 SBET controller. 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 Cn is the n dimensional complex space. A>0 (or A<0) means that A is a positive Hermitian matrix (or negative Hermitian matrix), symA=AH+A, where AH and AT, respectively, mean the conjugate transpose matrix and transpose matrix of A, ∗ in a matrix denotes the self-conjugate part of the Hermitian matrix.


## 2. Problem Description and Preliminaries

### 2.1. Problem Description

Consider a class of neutral-type CVNNs with partly unknown Markov jump as follows:(1)ϑ˙t−B1rtϑ˙t−σt=−A1rtϑt+C1rtφϑt+D1rtφϑt−σt+E1rtωt+E2rtut,vt=−A2rtϑt+C2rtφϑt+D2rtφϑt−σt+H1rtωt,ϑt0+θ=ψθ,∀θ∈−σ,0,where ϑt∈Cn is the state vector, which is described as ϑ1t,ϑ2t,…,ϑjt and ϑjt expresses the state of the jth neuron. ωt∈Cn is the external disturbance, ut∈Cn is a control input, vt∈Cm is the control output, 0≤σt≤σ,σ˙t≤σd is the time-varying delay, and φθ∈Cn is the initial value condition. φϑt=colφ1ϑt,φ2ϑt,…,φnϑt∈Cn, rt denotes a Markov process with on the probability space t≥0, which is right-continuous, and it takes values in set ℘=1,2,…,n with the transition rate matrix Π=πpq,p,q∈℘ given by(2)Prrt+Δ=q|rt=p=πpqΔ+oΔ,q≠p,1+πpqΔ+oΔ,q=p.However, not all the information of our transition rates can be obtained; then, the transition rate matrixΠ with ℕ modes can be expressed as(3)Π=π11?π13⋯????⋯π2n⋮⋮⋮⋱⋮?πn2πn3⋯πnn,where ? expresses the transition rates which are unknown and, for ∀p∈℘, the set Pp is expressed as Pp=Pkp∪Pukp with(4)ℵkp≜q:πpqis known forq∈℘,ℵukp≜q:πpqis uknown forq∈℘.Withrt=p and the controllers being considered as ut=Kpϑt, then system (1) would be reexpressed as(5)ϑ˙t=−A1p+E2pKpϑt+B1pϑ˙t−σt+C1pφϑt+D1pφϑt−σt+E1pωt,vt=−A2pϑt+C2pφϑt+D2pφϑt−σt+H1pωt.


### 2.2. The SBET Scheme

To economize resource, we study the stabilization of the CVNNs (1) under the SBET control in this section. Assuming that the sampling device performs periodically sampling on the system state, Ϝ1=0,h,2h,…,kh,… is the sampling sequence with k∈Z+, and h is the constant sampling period, set Ϝ2=0,ι1h,ι2h,…,ιkh,…, ιk∈Z+, describe the event-triggered sequence, it is clear that Ϝ2⊆Ϝ1, and the sampling-based event-triggered scheme is given as(6)ιk+1h=ιkh+inflh∣eHιkh+lhGeιkh+lh≥ϖϑHιkhGϑιkh=ιk+lmh,where lm=1/hinflh|eHιkh+lhGeιkh+lh≥ϖϑHιkhGϑιkh and G is a positive Hermitian matrix, eιkh+lh=ϑιkh+lh−ϑιkh, with l∈Z+, 0≤ϖ<1. 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 ςιkh, and ςιkh∈0,ς. In tk,tk+1, the control signal is hold by a zero-order-hold (ZOH) function. For j=0,1,…,lm−1, set Tk=ιk+jh+ςιk+jh,ιk+j+1h+ςιk+j+1h, and define ηt=t−ιk+jh−ςιk+jh, η∗t=ηt+ςιk+jh, 0<ηt≤η∗t≤η=h+ς, with tk=ιkh+ςιkh, and the SBET controllers are represented as(7)ut=K˜pϑιkh=K˜pϑt−η∗t−K˜pet−η∗t,with t∈Tk; system (1) would be transformed to(8)ϑ˙t=−A1pϑt+B1pϑ˙t−σt+C1pφϑt+D1pφϑt−σt+E1pωt+E2pK˜pϑt−η∗t−E2pK˜pet−η∗t,vt=−A2pϑt+C2pφϑt+D2pφϑt−σt+H1pωt.Remark 1.
The parameterϖ 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.Case1: if ϖ=0, the SBET scheme would be degenerated into(9)ιk+1h=ιkh+inflh∣eHιk+lhGeιk+lh≥0,that all the sampled data would be transmitted, and the SBET scheme (6) reduces to a periodic time-triggered one.Case2: if the right-hand side of the inequality is a constant ♭>0, the SBET scheme would be changed into(10)ιk+1h=ιkh+inflh∣eHιk+lhGeιk+lh≥♭,and then, the SBET scheme (6) is converted from a relative threshold event-trigger condition to a fixed threshold event-trigger 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.
The following assumptions are put forward to draw the main results.Assumption 1.
Each activation functionφp⋅ in (5) satisfies the following condition:(11)φpϑ1−φpϑ2≤lpϑ1−ϑ2,where ϑ1≠ϑ2 and L=diagl1,l2,…,ln with li>0,i=1,2,…,n.Definition 1.
(see [28]). System (5) is said to be stochastically stable if the following inequality holds:(12)E∫0∞ϑs2ds<∞.Definition 2.
If the CVNNs satisfy Definition1 under the initial condition, and G∞<γ, that is,(13)∫0∞vHsvsds≤γ2∫0∞ωHsωsds,then we say system (5) possesses H∞ performance with attenuation index γ.
Several essential lemmas are given before the proof of our theorems.Lemma 1 (see [12]).
For any vectorζ∈Cm, there is a scalar γ∈0,1, and ℋ∈Cn×n is a positive defined matrix, matrices F1,F2∈Cn×m, and Φγ,ℋ is considered as(14)Φγ,ℋ=1γζHF1HℋF1ζ+11−γζHF2HSF2ζ.
If the matrixU∈Cn×n satisfies ℋU∗ℋ>0, then(15)minβ∈0,1Φγ,ℋ≥F1ζF2ζHℋU∗ℋF1ζF2ζ.Lemma 2 (see [41, 42]).
ℜ>0 is a Hermitian matrix, for any continuous and differentiable function ϑ:c,d⟶Cn; the following inequality holds(16)∫cdϑHsℜϑsds≥Θ1HℜΘ1+3Θ2HℜΘ2d−c,where(17)Θ1=ϑd−ϑc,Θ2=ϑd+ϑc−2d−c∫cdϑsds.
By combining Lemma1 with Lemma 2, we have the following lemma.Lemma 3.
For Hermitian matrixS and arbitrary matrices S11,S22,S12, which belong to Cn×n, setting 0<σt≤σ, we have(18)−σ∫t−σtϑ˙HsSϑ˙sds≤−ΓHΩΓ,where(19)Ω=S0S11S12∗3SS12HS22∗∗S0∗∗∗3S≥0,Γ=ϑt−ϑt−σtϑt+ϑt−σt−2σt∫t−σttϑsdsϑt−σt−ϑt−σϑt−σt+ϑt−σ−2σ−σt∫t−σt−σtϑsds.Proof.
By dividing the left-hand side of inequality (18) into two parts, the following formula is obtained:(20)−σ∫t−σtϑ˙HsSϑ˙sds=−σ∫t−σttϑ˙HsSϑ˙sds−σ∫t−σt−σtϑ˙HsSϑ˙sds,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(21)−σ∫t−σtϑ˙HsSϑ˙sds≤−ση1HSη1+3η2HSη2σt−ση3HSη3+3η4HSη4σ−σt≤−η1η2η3η4HS0S11S12∗3SS12HS22∗∗S0∗∗∗3Sη1η2η3η4,with(22)η1H=ϑHt−ϑHt−σt,η2H=ϑHt+ϑHt−σt−2σt∫t−σttϑHsds,η3H=ϑHt−σt−ϑHt−σ,η4H=ϑHt−σt+ϑHt−σ−2σ−σt∫t−σt−σtϑHsds,and this ends the proof.Remark 2.
A similar conclusion has been obtained in real domain. Lemma3 is an extension which can apply to the complex domain with complex-valued matrices and vectors ϑt,ϑt−σt. It is worth noting that the quadratic form ϑ˙HsSϑ˙s is in the real number field because S is a Hermitian matrix.Lemma 4 (see [35]).
SetU>0 as an arbitrary matrix in Rn, and υt is a vector with appropriate dimensional. Thus, the following inequality is obtained:(23)−∫t1t2υTsUυsds≤t2−t1λTtVTU−1Vλt+2λTtVT∫t1t2υsds,for any V and vector λt with appropriate dimensional are not dependent on integral variables.This 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 ξt is a complex vector with appropriate dimensional. Thus, the following inequality is obtained:(24)−∫t1t2ξHsUξsds≤t2−t1λHtVHU−1Vλt+λHtVH∫t1t2ξsds+∫t1t2ξsdsVλt,for any V and vector λt 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(25)a+YbHY−1a+Yb≥0,that is to say,(26)aHY−1a+aHb+bHa+bHYb≥0,and, then, the following conclusion is obtained:(27)−λHtVH∫t1t2ξsds−∫t1t2ξHsdsVλt=∫t1t2−λHtVHξs−ξHsVλtds≤∫t1t2λHtVHU−1Vλtds+∫t1t2ξHsUξsds=t2−t1λHtVHU−1Vλt+∫t1t2ξHsUξsds.
This ends the proof.Remark 3.
In the proof of Corollary1, we can clearly see that −λHtVH∫t1t2ξsds and −∫t1t2ξsdsVλt cannot be merged into −2λHtVH∫t1t2ξsds 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. Therefore, the above inequality can be used in the complex domain in the form of Corollary 1.


## 2.1. Problem Description

Consider a class of neutral-type CVNNs with partly unknown Markov jump as follows:(1)ϑ˙t−B1rtϑ˙t−σt=−A1rtϑt+C1rtφϑt+D1rtφϑt−σt+E1rtωt+E2rtut,vt=−A2rtϑt+C2rtφϑt+D2rtφϑt−σt+H1rtωt,ϑt0+θ=ψθ,∀θ∈−σ,0,where ϑt∈Cn is the state vector, which is described as ϑ1t,ϑ2t,…,ϑjt and ϑjt expresses the state of the jth neuron. ωt∈Cn is the external disturbance, ut∈Cn is a control input, vt∈Cm is the control output, 0≤σt≤σ,σ˙t≤σd is the time-varying delay, and φθ∈Cn is the initial value condition. φϑt=colφ1ϑt,φ2ϑt,…,φnϑt∈Cn, rt denotes a Markov process with on the probability space t≥0, which is right-continuous, and it takes values in set ℘=1,2,…,n with the transition rate matrix Π=πpq,p,q∈℘ given by(2)Prrt+Δ=q|rt=p=πpqΔ+oΔ,q≠p,1+πpqΔ+oΔ,q=p.However, not all the information of our transition rates can be obtained; then, the transition rate matrixΠ with ℕ modes can be expressed as(3)Π=π11?π13⋯????⋯π2n⋮⋮⋮⋱⋮?πn2πn3⋯πnn,where ? expresses the transition rates which are unknown and, for ∀p∈℘, the set Pp is expressed as Pp=Pkp∪Pukp with(4)ℵkp≜q:πpqis known forq∈℘,ℵukp≜q:πpqis uknown forq∈℘.Withrt=p and the controllers being considered as ut=Kpϑt, then system (1) would be reexpressed as(5)ϑ˙t=−A1p+E2pKpϑt+B1pϑ˙t−σt+C1pφϑt+D1pφϑt−σt+E1pωt,vt=−A2pϑt+C2pφϑt+D2pφϑt−σt+H1pωt.


## 2.2. The SBET Scheme

To economize resource, we study the stabilization of the CVNNs (1) under the SBET control in this section. Assuming that the sampling device performs periodically sampling on the system state, Ϝ1=0,h,2h,…,kh,… is the sampling sequence with k∈Z+, and h is the constant sampling period, set Ϝ2=0,ι1h,ι2h,…,ιkh,…, ιk∈Z+, describe the event-triggered sequence, it is clear that Ϝ2⊆Ϝ1, and the sampling-based event-triggered scheme is given as(6)ιk+1h=ιkh+inflh∣eHιkh+lhGeιkh+lh≥ϖϑHιkhGϑιkh=ιk+lmh,where lm=1/hinflh|eHιkh+lhGeιkh+lh≥ϖϑHιkhGϑιkh and G is a positive Hermitian matrix, eιkh+lh=ϑιkh+lh−ϑιkh, with l∈Z+, 0≤ϖ<1. 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 ςιkh, and ςιkh∈0,ς. In tk,tk+1, the control signal is hold by a zero-order-hold (ZOH) function. For j=0,1,…,lm−1, set Tk=ιk+jh+ςιk+jh,ιk+j+1h+ςιk+j+1h, and define ηt=t−ιk+jh−ςιk+jh, η∗t=ηt+ςιk+jh, 0<ηt≤η∗t≤η=h+ς, with tk=ιkh+ςιkh, and the SBET controllers are represented as(7)ut=K˜pϑιkh=K˜pϑt−η∗t−K˜pet−η∗t,with t∈Tk; system (1) would be transformed to(8)ϑ˙t=−A1pϑt+B1pϑ˙t−σt+C1pφϑt+D1pφϑt−σt+E1pωt+E2pK˜pϑt−η∗t−E2pK˜pet−η∗t,vt=−A2pϑt+C2pφϑt+D2pφϑt−σt+H1pωt.Remark 1.
The parameterϖ 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.Case1: if ϖ=0, the SBET scheme would be degenerated into(9)ιk+1h=ιkh+inflh∣eHιk+lhGeιk+lh≥0,that all the sampled data would be transmitted, and the SBET scheme (6) reduces to a periodic time-triggered one.Case2: if the right-hand side of the inequality is a constant ♭>0, the SBET scheme would be changed into(10)ιk+1h=ιkh+inflh∣eHιk+lhGeιk+lh≥♭,and then, the SBET scheme (6) is converted from a relative threshold event-trigger condition to a fixed threshold event-trigger 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.
The following assumptions are put forward to draw the main results.Assumption 1.
Each activation functionφp⋅ in (5) satisfies the following condition:(11)φpϑ1−φpϑ2≤lpϑ1−ϑ2,where ϑ1≠ϑ2 and L=diagl1,l2,…,ln with li>0,i=1,2,…,n.Definition 1.
(see [28]). System (5) is said to be stochastically stable if the following inequality holds:(12)E∫0∞ϑs2ds<∞.Definition 2.
If the CVNNs satisfy Definition1 under the initial condition, and G∞<γ, that is,(13)∫0∞vHsvsds≤γ2∫0∞ωHsωsds,then we say system (5) possesses H∞ performance with attenuation index γ.
Several essential lemmas are given before the proof of our theorems.Lemma 1 (see [12]).
For any vectorζ∈Cm, there is a scalar γ∈0,1, and ℋ∈Cn×n is a positive defined matrix, matrices F1,F2∈Cn×m, and Φγ,ℋ is considered as(14)Φγ,ℋ=1γζHF1HℋF1ζ+11−γζHF2HSF2ζ.
If the matrixU∈Cn×n satisfies ℋU∗ℋ>0, then(15)minβ∈0,1Φγ,ℋ≥F1ζF2ζHℋU∗ℋF1ζF2ζ.Lemma 2 (see [41, 42]).
ℜ>0 is a Hermitian matrix, for any continuous and differentiable function ϑ:c,d⟶Cn; the following inequality holds(16)∫cdϑHsℜϑsds≥Θ1HℜΘ1+3Θ2HℜΘ2d−c,where(17)Θ1=ϑd−ϑc,Θ2=ϑd+ϑc−2d−c∫cdϑsds.
By combining Lemma1 with Lemma 2, we have the following lemma.Lemma 3.
For Hermitian matrixS and arbitrary matrices S11,S22,S12, which belong to Cn×n, setting 0<σt≤σ, we have(18)−σ∫t−σtϑ˙HsSϑ˙sds≤−ΓHΩΓ,where(19)Ω=S0S11S12∗3SS12HS22∗∗S0∗∗∗3S≥0,Γ=ϑt−ϑt−σtϑt+ϑt−σt−2σt∫t−σttϑsdsϑt−σt−ϑt−σϑt−σt+ϑt−σ−2σ−σt∫t−σt−σtϑsds.Proof.
By dividing the left-hand side of inequality (18) into two parts, the following formula is obtained:(20)−σ∫t−σtϑ˙HsSϑ˙sds=−σ∫t−σttϑ˙HsSϑ˙sds−σ∫t−σt−σtϑ˙HsSϑ˙sds,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(21)−σ∫t−σtϑ˙HsSϑ˙sds≤−ση1HSη1+3η2HSη2σt−ση3HSη3+3η4HSη4σ−σt≤−η1η2η3η4HS0S11S12∗3SS12HS22∗∗S0∗∗∗3Sη1η2η3η4,with(22)η1H=ϑHt−ϑHt−σt,η2H=ϑHt+ϑHt−σt−2σt∫t−σttϑHsds,η3H=ϑHt−σt−ϑHt−σ,η4H=ϑHt−σt+ϑHt−σ−2σ−σt∫t−σt−σtϑHsds,and this ends the proof.Remark 2.
A similar conclusion has been obtained in real domain. Lemma3 is an extension which can apply to the complex domain with complex-valued matrices and vectors ϑt,ϑt−σt. It is worth noting that the quadratic form ϑ˙HsSϑ˙s is in the real number field because S is a Hermitian matrix.Lemma 4 (see [35]).
SetU>0 as an arbitrary matrix in Rn, and υt is a vector with appropriate dimensional. Thus, the following inequality is obtained:(23)−∫t1t2υTsUυsds≤t2−t1λTtVTU−1Vλt+2λTtVT∫t1t2υsds,for any V and vector λt with appropriate dimensional are not dependent on integral variables.This 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 ξt is a complex vector with appropriate dimensional. Thus, the following inequality is obtained:(24)−∫t1t2ξHsUξsds≤t2−t1λHtVHU−1Vλt+λHtVH∫t1t2ξsds+∫t1t2ξsdsVλt,for any V and vector λt 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(25)a+YbHY−1a+Yb≥0,that is to say,(26)aHY−1a+aHb+bHa+bHYb≥0,and, then, the following conclusion is obtained:(27)−λHtVH∫t1t2ξsds−∫t1t2ξHsdsVλt=∫t1t2−λHtVHξs−ξHsVλtds≤∫t1t2λHtVHU−1Vλtds+∫t1t2ξHsUξsds=t2−t1λHtVHU−1Vλt+∫t1t2ξHsUξsds.
This ends the proof.Remark 3.
In the proof of Corollary1, we can clearly see that −λHtVH∫t1t2ξsds and −∫t1t2ξsdsVλt cannot be merged into −2λHtVH∫t1t2ξsds 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. Therefore, the above inequality can be used in the complex domain in the form of Corollary 1.


## 3. 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 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:(28)v1=1σt∫t−σttϑsds,v2=1σ−σt∫t−σt−σtϑsds,v3=1ηt∫t−ηttϑsds,v4=1η−ηt∫t−ηt−ηtϑsds,v5=1η∗t∫t−η∗ttϑsds,v6=1η−η∗t∫t−ηt−η∗tϑsds,Q1¯=Q^30S^11S^12∗3Q^3S^12HS^22∗∗Q^30∗∗∗3Q^3≥0,Q2¯=Q^40S^33S^34∗3Q^4S^34HS^44∗∗Q^40∗∗∗3Q^4≥0,Q3¯=Q^60S^55S^56∗3Q^6S^56HS^66∗∗Q^60∗∗∗3Q^6≥0,Q1=Q30S11S12∗3Q3S12HS22∗∗Q30∗∗∗3Q3≥0,Q2=Q40S33S34∗3Q4S34HS44∗∗Q40∗∗∗3Q4≥0,Q3=Q60S55S56∗3Q6S56HS66∗∗Q60∗∗∗3Q6≥0.Theorem 1.
Under the condition of Assumption1, for given scalar σ,σd and m, if there exist positive defined Hermitian matrices P^p,Q^p,Q^3,W^p in Cn×n, R^p,R^,ℳ^p∈C2n×2n, E^1,E^2 are positive diagonal matrices, and any matrices Y1p,S^11,S^12,S^22,Kp, nonsingular matrix X with proper dimension such that the following LMIs, are established with p∈1,2,3:(29)Ω1=Ξ1+Ξ2Λ∗−I<0,(30)∑q∈ℵkpnπpqR^q−ℳ^p≤R^,P^q−W^p≤0,q∈ℵukp,p≠q,R^q−ℳ^p≤0,q∈ℵukp,p≠q,P^q−W^p≥0,q∈ℵukp,p=q,R^q−ℳ^p≥0,q∈ℵukp,p=q,where(31)Ξ1=e10HQ^1e10−1−σde8HQ^1e8+e1HQ^2e1−1−σde3HQ^2e3+σ2e10HQ^3e10−Γ1HQ^1Γ1+σΓ2HR^Γ2+syme1HP^pe10+Γ2HR^pΓ2+Γ3HR^pΓ3+e1H∑q∈ℵkpnπpqP^q−W^pe1+e1HLHE^1Le1−e4HE^1e4+e3HLHE^2Le3−e5HE^2e5−γ2e11HI^e11−e12HI^e12,Ξ2=symme1H+e10H−Xe10−A1pX−Ε2pYpe1+B1pXe8+C1pXe4+D1pXe5+E1pXe11,Λ=−XA2p,0,0,XC2p,XD2p,0,0,0,XH1p,ξ1Ht=ϑHt,ϑHt−σ,ϑHt−σt,φHϑt,φHϑt−σt,v1H,v2H,ϑ˙Ht−σt,φHϑt−σ,ϑ˙t,ωtH,ξ2Ht=ϑHt,φHϑtH,ep=0n×p−1nIn0n×11−pn,Γ1=e1−e3e1+e3−2e6e3−e2e3+e2−2e7,Γ2=e1,e4,Γ3=e2,e9.
Then, system (5) is asymptotically stable via ut=Kpϑt with H∞ disturbance attenuation γ and the controller gain matrices are designed as Kp=Y1pX−1.Proof.
Consider the following LKF:(32)Vϑt,t,p=∑m=15Vmϑt,t,p,where(33)V1ϑt,p=ϑHtPpϑt,V2ϑt,p=∫t−σtξ2HsRpξ2sds,V3ϑt,p=∫t−σttϑ˙HsQ1ϑ˙sds+∫t−σttϑHsQ2ϑsds,V4ϑt,p=σ∫σ0∫t−θtϑ˙HsQ3ϑ˙sdsdθ,V5ϑt,p=∫t−σt∫θ0ξ2HsRξ2sdsdθ,with(34)Pp=XH−1P^pX−1,Qp=XH−1Q^pX−1,Rp=XH−1R^pX−1,R=XH−1R^X−1,p∈1,2,3.
Adopting the weak infinitesimal operatorL in [21], which acts on Vmϑt,t,i as(35)LVϑt,p=limΔ⟶∞1ΔEVϑt+Δ,t,rt+Δ|ϑt,rt=p−Vϑt,t,p,then, similar to the computation in [26, 27], we have(36)LV1ϑt,p=ϑHtPpϑ˙t+ϑ˙HtPpϑt+ϑHt∑q=1nπpqPqϑt,(37)LV2ϑt,p=ξ2HtRpξ2t−ξ2Ht−σRpξ2t−σ+∫t−σtξ2Hs∑q=1nπpqRqξ2sds,and considering the transition rate matrix includes not only the known part but also the unknown part, the following equations would hold for matrices Wp=XH−1W^pX−1,ℳp=XH−1ℳ^pX−1:(38)0=−ϑHt∑q=1nπpqWpϑt,∀p∈℘,0=−∫t−σtϑHs∑q=1nπpqℳpϑsds,∀p∈℘.
Then, (36) and (37) are equal to the following equations:(39)LV1ϑt,p=ϑHtPpϑ˙t+ϑ˙HtPpϑt+ϑHt∑q∈ℵkpπpqPq−Wpϑt+ϑHt∑q∈ℵukpπpqPq−Wpϑt,(40)LV2ϑt,p=ξ2HtRpξ2t−ξ2Ht−σRpξ2t−σ+∫t−σtξ2Hs∑q∈ℵkpπpqRq−ℳpξs2ds+∫t−σtξ2Hs∑q∈ℵukpπpqRq−ℳpξ2sds.
Via easy calculations, the following equations are given:(41)LV3ϑt,p≤ϑ˙HtQ1ϑ˙t−1−σdϑ˙Ht−σtQ1ϑ˙t−σt+ϑHtQ2ϑt−1−σdϑHt−σtQ2ϑt−σt,(42)LV4ϑt,p=σ2ϑ˙HtQ3ϑ˙t−σ∫t−σtϑ˙HsQ3ϑ˙sds,(43)LV5ϑt,p=σξ2HtRξ2t−∫t−σtξ2HsRξ2sds.
Dealing with the integral term containing the Markov jump, by pre- and postmultiplyingXH−1 and X−1 with (30), we have(44)∑q∈ℵkpnπpqRq−ℳp≤R,Pq−Wp≤0,q∈ℵukp,p≠q,Rq−ℳp≤0,q∈ℵukp,p≠q,Pq−Wp≥0,q∈ℵukp,p=q,Rq−ℳp≥0,q∈ℵukp,p=q,and then, the following inequalities are obtained:(45)∫t−σtξ2Hs∑q∈ℵkpπpqRq−ℳpξ2sds−∫t−σtξ2HsRξ2sds≤0,ϑHt∑q∈ℵukpπpqPq−Wpϑt+∫t−σtξ2Hs∑q∈ℵukpπpqRq−ℳpξ2sds≤0.
For arbitrary matricesS11,S22,S12∈Cn×n, because of Lemma 3, −σ∫t−σtϑ˙HsQ3ϑ˙sds in (42) will be processed as follows: by dividing it into two parts, we have(46)−σ∫t−σtϑ˙HsQ3ϑ˙sds=−σ∫t−σttϑ˙HsQ3ϑ˙sds−σ∫t−σt−σtϑ˙HsQ3ϑ˙sds,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)−σ∫t−σtϑ˙HsQ3ϑ˙sds≤−ση1HQ3η1+3η2HQ3η2σt−ση3HQ3η3+3η4HQ3η4σ−σt≤−η1η2η3η4HQ30S11S12∗3Q3S12HS22∗∗Q30∗∗∗3Q3η1η2η3η4=−ξ1HtΓ1HQ1Γ1ξ1t.
Thus,(48)LV4ϑt,p≤σ2ϑ˙HtQ3ϑ˙t−ξ1HtΓ1HQ1Γ1ξ1t.
On the other hand, for diagonal matricesE1=XH−1E^1X−1,E2=XH−1E^2X−1, Assumption 1 implies that(49)ϑHtLE1Lϑt−φHϑtE1φϑt≥0,ϑHt−σtLE2Lϑt−σt−φHϑt−σtE2φϑt−σt≥0.
One can also obtain the following equation from (5) for any nonsingular matrix X2 with the appropriate dimension, we defined X1=mX2, and it is clearly that X2 is also nonsingular, such that(50)0=ϑHtX1H+ϑ˙HtX2H−ϑ˙t+−A1p+Ε2pKpϑt+B1pϑ˙t−σt+C1pφϑt+D1pφϑt−σt+E1pωt+−ϑ˙t+−A1p+E2pKpϑt+B1pϑ˙t−σt+C1pφϑt+D1pφϑt−σt+E1pωtHX1ϑt+X2ϑ˙t.
Through the analysis in (39)–(50), we have(51)LVϑt,p+vHtvt−γ2ωHtωt≤ϑHtPpϑ˙t+ϑ˙HtPpϑt+ϑHt∑q∈ℵkpπpqPq−Wpϑt+ξ2HtRpξ2t−ξ2Ht−σRpξ2t−σ+ϑ˙HtQ1+σ2Q3ϑ˙t−1−σdϑ˙Ht−σtQ1ϑ˙t−σt+ϑHtQ2ϑt−1−σdϑHt−σtQ2ϑt−σt−ξ1HtΓ1HQ1Γ1ξ1t+σξ2HtRξ2t+ϑHtLE1Lϑt−φHϑtE1φϑt+ϑHt−σtLE2Lϑt−σt−φHϑt−σtE2φϑt−σt+symϑHtX1H+ϑ˙HtX2H−ϑ˙t+−A1p+E2pKpϑt+B1pϑ˙t−σt+C1pφϑt+D1pφϑt−σt+E1pωt+vHtvt−γ2ωHtωt=ξ1HΩξ1,where(52)Ω=syme1HPpe10+e1H∑q∈ℵkpnπpqPq−Wpe1+Γ2HRpΓ2+Γ3HRpΓ3+e10HQ1e10+σ2e10HQ3e10−1−σde8HQ1e8+e1HQ2e1−1−σde3HQ2e3−Γ1HQ1Γ1+σΓ2HRΓ2+e1HLHE1Le1−e4HE1e4+syme1HX1H+e10HX2H−e10−A1p−E2pKpe1+B1pe8+C1pe4+D1pe5+E1pe11+e3HLHE2Le3−e5HE2e5−γ2e11HIe11+χHχ,χ=−A2pe1+C2pe4+D2pe5+H1pe11.
Defining(53)X=X2−1,Y1p=KpX,Xℏ=diagX,X,…,X︸ℏ,P^p=XHPpX,W^p=XHWpX,ℳ^p=XHℳpX,P^q=XHPqX,R^q=XHRqX,R^p=XHℛpX,Q^p=XHQpX,Q^4=XHQ4X,R^=XHRX,E^1=XHE1X,E^2=XHE2X,I^=XHIX,S^11=XHS11X,S^12=XHS12X,S^22=XHS22X,and then, by pre- and postmultiplying XℏH and Xℏℏ=11 with Ω, (29) is obtained. Combining (29) with (51) and using the Schur complement, we have(54)LVϑt,p+vHtvt−γ2ωHtωt<0,and integrating the formula above from 0 to ∞, we have(55)Vϑ∞−Vφθ≤−∫0∞vHtvtdt+γ2∫0∞ωHtωtdt,and, with the zero initial condition ϕ=0, we have Vφθ=0, Vϑ∞≥0; thus,(56)−∫0∞vHtvtdt+γ2∫0∞ωHtωtdt>Vϑ∞≥0.
This completes the proof.
Next, based on Theorem1 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 scalarsσ,σd,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 Cn×n, R^p,R^,ℳ^p∈C2n×2n, and E1,E2,E^1,E^2 are positive diagonal matrices, matrices K˜p, and nonsingular matrix X^∈Cn∗n are proper dimension matrices with p∈1,2,3, such that the following conditions hold:(57)Ω˜20=Ξ^Λ^∗−I<0,(58)Ω˜2η=Ξ^+ηΞ^1Λ^∗−I<0,(59)Ω˜3η=Ξ^−ηe^17HU^22e^17Λ^e^1He^10He^17HV˜∗−I0∗∗−1ηU11<0,(60)∑q∈ℵkpnπpqR^q−ℳ^p≤R^,P^q−W^p≤0,q∈ℵukp,p≠q,R^q−ℳ^p≤0,q∈ℵukp,p≠q,P^q−W^p≥0,q∈ℵukp,p=q,R^q−ℳ^p≥0,q∈ℵukp,p=q,where(61)Ξ^=symρe^1H+e^10H−X^e^10−A1pX^e^1+B1pX^e^8+C1pX^e^4+D1pX^e^5+E1pX^e^9+E2pY2pe^12−e^14+e^10HQ^1e^10−1−σde^8HQ^1e^8+e^1HQ^2e^1−1−σde^3HQ^2e^3+σ2e^10HQ^3e^10−Γ1HQ¯1Γ1+σΓ2HR^Γ2+syme^1HP^pe^10+Γ2HR^pΓ2+Γ3HR^pΓ3+e^1H∑q∈ℵkpnπpqP^q−W^pe^1+e^1HLHE^1Le^1−e^4HE^1e^4+e^3HLHE^2Le^3−e^5HE^2e^5+η2e^10HQ^4+Q^6e^10−Γ4HQ¯2Γ4−Γ5HQ¯3Γ5−e^1−e^17HQ^5e^1−e^17−Γ7HV^Γ7−syme^1−e^17HU^12e^17+syme^1He^10He^17HV^e^1−e^17−γ2e^11HIe^11−e^20HIe^20−e^14HG^e^14+ϖe^12−e^14HG^e^12−e^14Ξ^1=syme^1−e^17HQ^5e^10+e^10symV^1e^1+symV^2−V^1e^17+Γ8HU^Γ8,(62)ξ4Ht=ϑ˙Ht,ϑHt−ηt,ξ5Ht=ϑHt,ϑHt−σ,ϑHt−σt,φHϑt,φHϑt−σt,v1H,v2H,ϑ˙Ht−σt,φHϑt−σ,ϑ˙Ht,ωHt,ϑHt−η∗t,ϑHt−η,eHt−η∗tv3H,v4H,ϑHt−ηt,v5H,v6H,ξ6Ht=ϑtH,ϑ˙Ht,ϑt−ηtH,ξ7Ht=ϑHt,ϑHt−ηt,Λ^=−X^A2p,0,0,X^C2p,X^D2p,0,0,0,0,0,X^H1p,0,0,0,0,0,0,0,0,(63)Γ4=e^1−e^17e^1+e^17−2e^15e^17−e^13e^17+e^13−2e^16,Γ5=e^1−e^12e^1+e^12−2e^18e^12−e^13e^12+e^13−2e^19,Γ6=e^1e^10e^17,Γ7=e^1e^17,Γ8=e^10e^17,e^p=0n×p−1nIn0n×19−pn.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=Y2pX^−1.Proof.
Choose the following LKF:(64)Vϑt,t,p=∑m=15Vmϑt,t,p,V1ϑt,t,p=ϑHtPpϑt+∫t−σtξ2HsRpξ2sds+∫t−σttϑ˙HsQ1ϑ˙sds+∫t−σttϑHsQ2ϑsds+σ∫σ0∫t−θtϑ˙HsQ3ϑ˙sdsdθ+∫t−σt∫θ0ξ2HsRξ2sdsdθV2ϑt,t,p=η∫η0∫t−θtϑ˙HsQ4ϑ˙sdsdθV3ϑt,t,p=η∫η0∫t−θtϑ˙HsQ6ϑ˙sdsdθ,V4ϑt,t,p=hk−ηt∫t−ηttξ4HsUξ4sds,V5ϑt,t,p=hk−ηtϑt−ϑt−ηtHQ5ϑt−ϑt−ηt+hk−ηtξ7HtVξ7t,with(65)hk=h+ςιk+j+1h−ςιk+jh,Pp=X^H−1P^pX^−1,Qp=X^H−1Q^pX^−1,Q4=X^H−1Q^4X^−1,Q5=X^H−1Q^5X^−1,Rp=X^H−1R^pX^−1,R=X^H−1R^X^−1,p∈1,2,3,being positive defined Hermitian matrices with proper dimension, and(66)U=U11U12∗U22,V=symV12−V1+V2∗symV12−V2.
From the definition ofVϑt,p, noting VJϑt,p,J=1,2,3, is a continuous function; it is easy to see that(67)limt⟶ιk+jh+ςιk+jh−VJϑt,p=VJϑιk+jh+ςιk+jh,p.
Also, we can derive that(68)limt⟶ιk+jh+ςιk+jh−V4ϑt,p=limt⟶ιk+jh+ςιk+jh+V4ϑt,p=V4ϑιk+jh+ςιk+jh,p=0,limt⟶ιk+jh+ςιk+jh−V5ϑt,p=limt⟶ιk+jh+ςιk+jh+V5ϑt,p=V5ϑιk+jh+ςιk+jh,p=0.
Therefore,Vϑt,p is continuous at t=ιk+jh+ςιk+jh. Moreover, VJϑt,p≥0, that Vϑt,p=VJϑt,p+V4ϑt,p+V5ϑt,p≥0 for any k. By pre- and postmultiplying X^H−1 and X^−1 with (60), we have(69)∑q∈ℵkpnπpqRq−ℳp≤R,Pq−Wp≤0,q∈ℵukp,p≠q,Rq−ℳp≤0,q∈ℵukp,p≠q,Pq−Wp≥0,q∈ℵukp,p=q,Rq−ℳp≥0,q∈ℵukp,p=q,and following the same procedure to Theorem 1, we have(70)LV1ϑt,p≤ϑHtPpϑ˙t+ϑ˙HtPpϑt+ϑHt∑q∈ℵkpπpqPq−Wpϑt+ξ2HtRpξ2t−ξ2Ht−σRpξ2t−σ+ϑ˙HtQ1+σ2Q3ϑ˙t−1−σdϑ˙Ht−σtQ1ϑ˙t−σt+ϑHtQ2ϑt−1−σdϑHt−σtQ2ϑt−σt−ξ5HtΓ1HQ¯1Γ1ξ5t+σξ2HtRξ2t+ϑHtLE1Lϑt−φHϑtE1φϑt+ϑHt−σtLE2Lϑt−σt−φHϑt−σtE2φϑt−σt.
The following inequalities can be obtained because of Lemma3 with arbitrary matrices S33,S44,S34,S55,S66, S56∈Cn×n:(71)LV2ϑt,p=η2ϑ˙HtQ4ϑ˙t−η∫t−ηtϑ˙HsQ4ϑ˙sds=η2ϑ˙HtQ4ϑ˙t−η∫t−ηttϑ˙HsQ4ϑ˙sds−η∫t−ηt−ηtϑ˙HsQ4ϑ˙sds≤η2ϑ˙HtQ4ϑ˙t−ξ5HtΓ4HQ2Γ4ξ5t,LV3ϑt,p=η2ϑ˙HtQ6ϑ˙t−η∫t−ηtϑ˙HsQ6ϑ˙sds=η2ϑ˙HtQ6ϑ˙t−η∫t−η∗ttϑ˙HsQ6ϑ˙sds−η∫t−ηt−η∗tϑ˙HsQ6ϑ˙sds≤η2ϑ˙HtQ6ϑ˙t−ξ5HtΓ5HQ3Γ5ξ5t,LV4ϑt,p=hk−ηtξ4HUξ4−∫t−ηttξ4HUξ4ds=hk−ηtξ4HUξ4−∫t−ηttϑ˙HsU11ϑ˙sds−∫t−ηttϑ˙HsdsU12ϑt−ηt−ϑHt−ηtU12∫t−ηttϑ˙sds−ηtϑHt−ηtU22ϑt−ηt.
From the corollary of Lemma4, the following formula is established with V of appropriate dimensions:(72)−∫t−ηttϑ˙HsU11ϑ˙sds≤ηtξ6HtVHU11−1Vξ6t+ξ6HtVH∫t−ηttϑ˙sds+∫t−ηttϑ˙sdsVξ6t.
Then, we have(73)LV4ϑt,p=hk−ηtξ4HUξ4−∫t−ηttξ4HUξ4ds≤hk−ηtξ4HUξ4+ηtξ6HtVHU11−1Vξ6t+ξ6HtVH∫t−ηttϑ˙sds+∫t−ηttϑ˙sdsVξ6t−ηtϑHt−ηtU22ϑt−ηt−∫t−ηttϑ˙HsdsU12ϑt−ηt−ϑHt−ηtU12∫t−ηttϑ˙sds.
Noticing thatηt=t−ιk+jh−ςιk+jh, the following inequality is obtained:(74)LV5ϑt,p≤−ϑt−ϑt−ηtHQ5ϑt−ϑt−ηt−ξ7HsVξ7s+hk−ηtϑt−ϑt−ηtHQ5ϑ˙t+hk−ηtϑ˙HtQ5ϑt−ϑt−ηt+hk−ηtϑ˙HtV1+V1Hϑt+hk−ηtϑ˙HtV2−V1ϑt−ηt+hk−ηtϑHt−ηtV2−V1ϑ˙t.
From the event-triggered mechanism (6), when t∈Tk, it is clear that(75)eHιkh+lhGeιkh+lh<ϖϑιkh+lh−eιkh+lhHGϑιkh+lh−eιkh+lh.
It means that the data of the current sampling moment is not transmitted. Moreover, one can obtain the following zero equality based on system (8) with any nonsingular matrix X4 with appropriate dimension, defining X3=ρX4, that X4 is nonsingular:(76)0=ϑHtX3H+ϑ˙HtX4H−ϑ˙t−A1pϑt+B1pϑ˙t−σt+C1pφϑt+D1pφϑt−σt+Ε1pωt+E2pK˜pϑt−η∗t−E2pK˜pet−η∗t+−ϑ˙t−A1pϑt+B1pϑ˙t−σt+C1pφϑt+D1pφϑt−σt+E1pωt+E2pK˜pϑt−η∗t−E2pK˜pet−η∗tHX3ϑt+X4ϑ˙t.
Combining with (69)–(76), we have the following equation:(77)LVϑt,p+vHtvt−γ2ωHtωt≤LVϑt,p+vHtvt−γ2ωHtωt−eHtkh+lhGetkh+lh+ϖϑιkh+lh−eιkh+lhHGϑιkh+lh−eιkh+lh+symϑHtX3H+ϑ˙HtX4H−ϑ˙t−A1pϑt+B1pϑ˙t−σt+C1pφϑt+D1pφϑt−σt+E1pωt+E2pK˜pϑt−ηt−E2pK˜pet−ηt.
By dividing the right-hand side of the above inequality into three parts, we have(78)LVϑt,p+vHtvt−γ2ωHtωt≤ξ5HtΞ˜+hk−ηtΞ˜1+ηtΞ˜2+ϱHϱξ5t=ξ5Hthk−ηthkΩ2hk+ηthkΩ3hkξ5t,with(79)Ω2hk=Ξ˜+hkΞ˜1+ϱHϱ,Ω3hk=Ξ˜+hkΞ˜2+ϱHϱ,Ξ˜=syme^1HX3H+e^10HX4H−e^10−A1pe^1+B1pe^8+C1pe^4+D1pe^5+E1pe^9+E2pK˜pe^12−e^14+syme^1HPpe^10+e^1H∑q∈ℵkpnπpqPq−Wpe^1+Γ2HRpΓ2+Γ3HRpΓ3+e^10HQ1e^10−1−σde^8HQ1e^8+e^1HQ2e^1−1−σde^3HQ2e^3+σ2e^10HQ3e^10−Γ1HQ1Γ1+σΓ2HRΓ2+e^1HLHE1Le^1−e^4HE1e^4+e^3HLHE2Le^3−e^5HE2e^5+η2e^10HQ4+Q6e^10−Γ4HQ2Γ4−Γ5HQ3Γ5−e^1−e^17HQ5e^1−e^17−Γ7HVΓ7−syme^1−e^17HU12e^17+syme^1He^10He^17HVe^1−e^17−γ2e^11HIe^11−e^14HGe^14+ϖe^12−e^14HGe^12−e^14,Ξ˜1=syme^1−e^17HQ5e^10+e^10symV1e^1+symV2−V1e^17+Γ8HUΓ8,Ξ˜2=−e^17HU22e^17+Γ6HVU11−1VHΓ6,ϱ=−A2pe^1+C2pe^4+D2pe^5+H1pe^11.
Define(80)X^=X4−1,Y2p=K˜pX^,V˜=X^HVH,X^ℏ=diagX^,X^,…,X^︸ℏ,P^p=X^HPpX^,W^p=X^HWpX^,ℳ^p=X^HℳpX^,P^q=X^HPqX^,R^q=X^HRqX^,R^p=X^HRpX^,Q^p=X^HQpX^,Q^4=X^HQ4X^,Q^5=X^HQ5X^,Q^6=X^HQ6X^,R^p=X^HRpX^,E^1=X^HE1X^,E^2=X^HE2X^,S^11=X^HS11X^,S^12=X^HS12X^,S^22=X^HS22X^,S^33=X^HS33X^,S^34=X^HS34X^,S^44=X^HS44X^,S^55=X^HS55X^,S^56=X^HS56X^,S^66=X^HS66X^,U^11=X^HU11X^,U^12=X^HU12X^,U^22=X^HU22X^,I^=X^HIX^,G^=X^HGX^,V^=X^HX^,V^1=X^HV1X^,V^2=X^HV2X^,U^=X^HUX^,V^=X^HVX^,and then, by pre- and postmultiplying X^ℏH and X^ℏℏ=19 with Ω2hk and Ω3hk, combining (57)–(61) and (78), then applying Schur complement, we have Ξ˜+hkΞ˜1+ϱHϱ<0, Ξ˜+hkΞ˜2+ϱHϱ<0, that LVϑt,p+vHtvt−γ2ωHtωt<0 for t∈Tk. Therefore, if the LMIs in Theorem 2 hold, according to the similar proof line as Theorem 1, the CVNNs (8) with Markovian jump are asymptotically stable and possess H∞ performance based on a SBET controller; this completes the proof.Remark 4.
As far as we know, a discrete sampling-based event-triggered controller which is designed for the neutral-type CVNNs with partly unknown Markov jump in this work has not received much attention, which reduces the data transmission rate greatly for it only needs to monitor the state of our system at the discrete moment. Also, the partial unknown Markov phenomenon is taken into account which makes the results more general and practical.Remark 5.
The segmented sawtooth structure caused by sampling hold is fully considered byV4ϑt,p and V5ϑt,p that the conservatism of our results would be reduced. Furthermore, we analyze the CVNNs under the sampling-based event-driven framework while not dividing it into the real part and the imaginary part; it is able to reduce computational complexity greatly while the desired effect is achieved.Remark 6.
V2ϑt,p and V3ϑt,p in Theorem 2 have a very similar presentation, but in fact, they are different, and different interval segmentations are adopted for the processing of the two LKFs, which can also be merged together as V6ϑt,t,p=η∫η0∫t−θtϑ˙HsQ7ϑ˙sdsdθ:(81)LV6ϑt,t,p=η2ϑ˙HtQ7ϑ˙t−η∫t−ηtϑ˙HsQ7ϑ˙sds=η2ϑ˙HtQ7ϑ˙t−η∫t−ηttϑ˙HsQ7ϑ˙sds−η∫t−η∗tt−ηtϑ˙HsQ7ϑ˙sds−η∫t−ηt−η∗tϑ˙HsQ7ϑ˙sds.
According to Lemma1,(82)LV6ϑt,t,p≤η2ϑ˙HtQ7ϑ˙t−ηηtα1HQ7α1−ηη∗t−ηtα2HQ7α2−ηη−η∗tα3HQ7α3≤η2ϑ˙HtQ7ϑ˙t−α1α2α3HQ7S1S2∗Q7S3∗∗Q7α1α2α3,with(83)α1=ϑt−ϑt−ηt,α2=ϑt−ηt−ϑt−η∗t,α3=ϑt−η∗t−ϑt,S1,S2,S3∈Cn×n.
However, the vectors that are contained inLV6ϑt,t,p are not as comprehensive as those in LV2ϑt,t,p and LV3ϑt,t,p.


## 4. Numerical Examples

Through numerical simulations, two examples are given to demonstrate the superiority and effectiveness of our conclusions derived from previous chapters.Example 1.
Consider the CVNNs (5) with the following parameters:(84)A11=0.8000.7,A12=0.6000.5,A13=0.5000.4,A21=0.6000.5,A22=0.7000.6,A23=0.7000.6,B11=−0.03+0.08i−0.05+0.02i−0.03−0.02i0.08+0.02i,B12=−0.04−0.02i−0.04+0.02i−0.04−0.03i0.06+0.04i,B13=−0.05+0.07i−0.02+0.05i−0.05−0.08i−0.05+0.02i,C11=0.5−0.4i−0.3+0.3i0.4+0.3i−0.7−0.9i,C12=0.6−0.5i−0.4+0.4i0.5+0.4i−0.8−0.8i,C13=0.7−0.6i−0.5+0.5i0.6+0.5i−0.9−0.7i,D11=0.4−0.4i0.2−0.2i−0.4+0.3i0.3+0.2i,D12=0.5−0.5i−0.4−0.5i−0.5+0.4i0.4+0.3i,D13=−0.5−0.6i0.4−0.4i−0.6+0.5i0.5+0.4i,E11=−0.4−0.6i0.3−0.5i−0.6+0.5i0.5−0.6i,E12=−0.3−0.6i0.6−0.3i−0.7+0.5i0.3−0.5i,E13=−0.5−0.4i0.5−0.5i−0.4+0.5i0.4−0.5i,E21=−0.5−0.6i0.4−0.2i−0.6+0.5i0.5−0.7i,E22=0.7−0.8i−0.4−0.3i−0.6+0.5i0.1−0.4i,E23=−0.2−0.5i0.4−0.1i−0.5+0.5i0.6+0.2i,C21=1.2−1.5i−1.6+0.3i1.5+0.4i1.6−0.4i,C22=1.3−0.4i−1.4+0.6i1.8+0.5i1.3−0.7i,C23=1.4−1.3i−1.3+0.2i1.4+0.6i1.4−0.6i,D21=0.5−0.4i−0.3+0.3i−0.4−0.3i0.3−0.3i,D22=0.6−0.5i−0.4+0.4i−0.5−0.4i0.4−0.4i,D23=0.7−0.6i−0.5+0.5i−0.6−0.5i0.5−0.5i,H11=−0.4−0.6i0.4−0.4i−0.6+0.5i0.8−0.9i,H12=0.5−0.2i0.3+0.4i−0.4+0.5i0.3−0.4i,H13=−0.5−0.6i0.4−0.2i−0.1+0.5i0.3−0.2i.
Π represents the partly unknown transition rate matrix, which is considered as(85)Π=−6??5????−4.
In the following, aH∞ controller will be designed based on Theorem 1, where the activation function is set as φϑt=tanhϑt+itanhϑt, σ=0.4,σd=0.1, L=10;01, and σt=t−0.3+0.1∗sint expresses the time-varying delay. If the initial value is set as ψt=−0.3+0.5i;0.5−0.3i, the solution could be obtained by the LMI toolbox, and γ=7.6623 is obtained, the corresponding controller gain matrices are computed as(86)K1=9.4442−38.5585i−10.3759+26.9077i21.1045−23.0361i−31.4861−1.6866i,K2=−7.6922−13.4025i16.2192+6.3072i4.0202−11.3110i5.6539−22.0753i,K3=−4.3350−31.0012i4.9612+25.3198i−10.1567−23.3295i−13.8700+12.1124i,and then system (5) is stochastically stable under the controllers ut=Kpϑt,p∈1,2,3, and the states are shown as in Figures 1 and 2 .
Figures3 and 4 depict the imaginary part and real part of the states of ut, respectively, Figures 1 and 2 depict the states of the imaginary and the real part, respectively, via 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.Figure 1
Responses the state of the imaginary part.Figure 2
Responses the state of the real part.Figure 3
Responses inputut of the imaginary part.Figure 4
Responses control inputut of the real part.Figure 5
The imaginary part of the state withoutut.Figure 6
The real part of the state withoutut.Example 2.
For the same system parameters with Example1, σt=t−0.3−0.1sint expresses the time-varying delay, and σd=0.1, set ϖ=0.2,h=0.036,ς=0.01, and ρ=20, with the external disturbance ω=1/1−t2. When the initial value is set as ψt=−5+6i;10−4i that the SBET weighted matrix and control gain matrices are obtained,(87)G^=103×4.1883+0.0000i−0.6747−0.2273i−0.6747+0.2273i3.1574+0.0000i,K˜1=0.9748−6.5659i−1.3351+2.9435i3.3333−4.7188i−3.5523+0.1313i,K˜2=−1.2784−2.0467i1.6042+0.7371i1.5378−2.2989i0.1075−2.3443i,K˜3=−1.2773−5.5737i0.3576+3.1295i−2.5166−4.9823i−1.6298+1.9885i.
Under the conditions which are given in this example, the imaginary part and real part of control inputsut 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 Theorem 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.Figure 7
Responses inputut of the imaginary part.Figure 8
Responses control inputut of the real part.Figure 9
Markov signal.Figure 10
Responses the state of the real part.Figure 11
Responses the state of the imaginary part.Figure 12
The interval of the event-triggered.


## 5. 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 aH∞ 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 event-triggered 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.



---

*Source: 5563888-2021-05-24.xml*