# Stochastic Exponential Stabilization for Markov Jump Neural Networks with Time-varying Delays via Adaptive Event-Triggered Impulsive Control

Authors: Xiaoman Liu; Haiyang Zhang; Tao Wu; Jinlong Shu
Journal: Complexity
 (2020)

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## Abstract

This paper focuses on the exponential stabilization problem for Markov jump neural networks with Time-varying Delays (TDs). Firstly, we provide a new Free-matrix-based Exponential-type Integral Inequality (FMEII) containing the information of attenuation exponent, which is helpful to reduce the conservativeness of stability criteria. To further save control cost, we introduce a sample-based Adaptive Event-triggered Impulsive Control (AEIC) scheme, in which the trigger threshold is adaptively varied with the sampled state. By fully considering the information about sampled state, TDs, and Markov jump parameters, a suitable Lyapunov–Krasovskii functional is constructed. With the virtue of FMEII and AEIC scheme, some novel stabilization criteria are presented in the form of linear matrix inequalities. At last, two numerical examples are given to show the validity of the obtained results.


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## Body

## 1. Introduction

Recent decades have witnessed the fast development of neural networks since its wide applications in many practical fields, such as pattern recognition [1], smart antenna arrays [2], and circuit design [3]. In the implementation of such applications, both Time-varying Delays (TDs) caused by the inherent communication time among neurons or the finite switching speed of amplifier [4–8] and random abrupt arisen from the external environment sudden change or the information latching [9–11] are inevitably encountered, which often lead to some undesirable dynamic behaviors, such as chaotic, oscillation, and even unstable [12–14]. Markov Jump Neural Networks (MJNNs) with Time-varying Delays (TDs), as a special kind of hybrid system, has a powerful ability in describing those complicated behaviors. Stability is a precondition for the normal operation of systems, and sometimes fast convergence of the networks is essential for real-time computation. As it is well known, the exponential convergence rate is generally used to determine the speed of neural computations [15]. Thus, it is of great theoretical and practical importance to study the exponential stability for MJNNs with TDs, and many fruit results have been reported in the literature [16–18].In particular, under an assumption that the proportional delay is unbounded time-varying, some global stochastic exponential stability conditions for MJNNs with proportional TDs are derived in [16]. By considering a more general uncertain transition rates, Liu et al. [17] investigated the stochastic exponential stability for neutral-type impulsive MJNNs with mixed TDs. With the aid of random analysis method and Lyapunov functional techniques, Zhang et al. [18] presented some exponential stability results for MJNNs with TDs and general disturbance. It is worth mentioning that the Exponential-type Integral Quadratic Terms (EIQTs), i.e., −∫δ1δ2eαs−δ2φTsΞφsds, which play crucial role in deriving less conservative stability criteria [16–21]. In most of the existing literature, the upper bound of EIQTs are usually estimated by using some nonexponential-type integral inequalities, such as Free-matrix-based Integral Inequality (FMII) [19], Wirtinger’s integral inequality [16–18, 20], and other kind of integral inequality [21]. However, there exists a common shortcoming in such kinds of inequalities, that is, the information of attenuation exponent is not considered fully, which brings some conservativeness to some extent. Recently, some Exponential-type Integral Inequalities (EIIs) have been proposed and successfully applied to the exponential stability analysis for time-delay systems [22, 23]. Nerveless, it cannot find more tighter upper bound for the EIQTs, since the parameter matrices in EIIs are not adjustable. Consequently, it should be possible to reduce the conservativeness of results based on the ideas of FMII [19] and EIIs [23], which is the first motivation of our paper.On the contrary, Event-triggered Impulsive Control (EIC), as a powerful tool, plays an important role in many different science and engineering fields. Especially, in the field of artificial neural networks, the study for stabilization is more complicated due to the state-dependent nonlinear switching behaviors of neural networks. Compared with the common impulsive control, EIC scheme may avoid some undesirable network behaviors, such as some “unnecessary” data are sent frequently, and the network resources are excessively used to some extent, especially in the case that the networks resources are limited [24]. Therefore, EIC scheme has attracted much more attention in many different fields, multiagent systems [25], nonlinear systems [26, 27], neural networks [28, 29], and so on [30, 31]. It should be pointed out that there exist some disadvantages in the above EIC schemes in [25–31]. For example, the supervising time of sensors is in the mean of continuous time, which implies that the sensors are required to monitor the system state all the time [25, 30, 31], but it is not necessary because the worst scenario rarely happens. Besides, the triggering threshold is assumed to be a constant [27–29], which reduces the flexibility of EIC schemes and limits the applications ability of them. Furthermore, to overcome the above disadvantages, a sample-based AEIC scheme is introduced in this paper, where the trigger threshold is adaptively varied with the sampled state. So far as it is known to the authors, few literature studies have devoted to the exponential stabilization problem for MJNNs with TDs via a sample-based AEIC scheme, which is another motivation of our paper.Motivated by the aforementioned discussions, the main contributions of our paper are summarized as follows. (1) Inspired by the ideas of FMII [19] and EIIs [23], a new FMEII is proposed to derive more tighter bound of the EIQTs in this paper, which is more general and flexible than the existing EIIs. (2) In order to overcome the shortcomings in existing EIC schemes, a sample-based AEIC scheme is introduced, where the triggering threshold is governed by a well designed adaptive rule corresponded to the sampled state. (3) The FMEII and AEIC scheme are successfully applied to investigate the exponential stabilization issue for MJNNs with TDs, and some novel stabilization results are derived in terms of linear matrices inequalities.Notations: letℕ denote the set of positive integers, ℝ the set of real numbers, ℝn the n-dimensional real space equipped with the Euclidean norm ⋅, ℝm×n the set of all m×n real matrices, and R+n and Rn the set of symmetric positive definite and symmetric matrices of ℝn×n, respectively. Denote f,gw=∫abfsgswsds for any integrable functions f⋅,g⋅,w⋅ in a,b. The symbol “∗” in a block matrix signifies the symmetric terms, col⋯ and diag⋯ express a column vector and a diagonal matrix, respectively. For any matrix X∈ℝn×n, ℍX means that X+XT and λmaxX and λminX stand for the maximum and minimum eigenvalue of X, respectively. The zero and identity matrices with appropriate dimensions are described by 0 and I, respectively. The symbol e¯i=0,…,0,I︸i,0,…,0,,i=1,…,14.


## 2. Description of Problem and Preliminaries

Letrt,t≥0 be a continuous-time Markov process taking values in a finite state space N=1,2,…,N. The evolution of rt,t≥0 is governed by the following transition probability:(1)Prrt+Δ=jrt=i=πijΔ+oΔ,i≠j,1+πiiΔ+oΔ,i=j,where Δ≥0, limΔ⟶0oΔ/Δ=0; πij≥0 for i≠j∈N is the transition rate from mode i at time t to mode j at time t+Δ, and πii=−∑j=1,j≠iNπij.Consider the following impulsive MJNNs with TDs:(2)x˙t=−Brtxt+Artfxt+Crtxtk+Drtfxt−δt,t∈tk,tk+1,xt=1+qkxt−,t=tk,k∈ℕ,ϕθ=xt0+θ,rt0=r0,θ∈−maxδ,η,0,where xt=colx1t,…,xnt is the state; fxt=colf1x1t,…,fnxnt is the neuron activation function and satisfies(3)λl−≤flx1−flx2x1−x2≤λl+,fl0=0,l∈ℕ,∀x1≠x2∈ℝ,where λl−andλl+ are scalars which can be positive, negative, and zero; ϕθ is the initial condition; and δt is a time-varying delay and satisfies(4)0≤δ1≤δt≤δ2,δ˙t≤μ,where δ1,δ2, and μ are known constants; Brt is a positive diagonal matrix and Art and Drt are connection weighted matrices; Crt is control gain matrix to be determine, qk is impulsive intensity, tk is impulsive instant, xtk=xtk+=limε⟶0+xtk+ε,xtk−=limε⟶0−xtk+ε, and r0∈N is the initial mode as t=t0.To further mitigate unnecessary waste of network resources, a sample-based AEIC scheme is introduced in this paper. Assume that the system’s state is periodically sampled and the sampling sequence is depicted by the setΠs=0,h,2h,…,kh with k∈ℕ, where h is constant sampling period, and the event-triggered sequence is described by the set Πe=0,b1h,b2h,…,bkh⊆Πs with bk∈ℕ. Suppose the event-triggered instants to be the impulsive instants, i.e., tk=bkh, then the next impulsive instant tk+1=tk+lmh, where(5)lm=minleTtk+lhΩetk+lh>σtk+lhxTtk+lhΩxtk+lh,l∈ℕ,and Ω∈R+n is an unknown weighted matrix, etk+lh=xtk+lh−xtk expresses the error between the two states at the latest trigger instant and the current sampling one, and trigger threshold σtk+lh∈0,1 is governed by the following adaptive rule:(6)σtk+l+1h=maxσtk+lhΔtk+lh,σm,where Δtk+lh=1−2Δ1/πarctanΔ2xtk+lh−xtk, arctan⋅ is the invert tangent function, Δ1,Δ2>0 are given constants to adjust the output of arctan⋅, σm is the given lower bound of σtk+lh, and σ0=σm.Remark 1.
Note that the functionarctan⋅ in adaptive rule (6) has the lower and upper bounds, i.e., actan⋅∈−π/2,π/2, and this property can be used in this paper combining with adjustable parameters Δ1 and Δ2 to adaptively adjust the trigger threshold σtk+lh. Specifically, when the norm of the current sampling state is larger than the one of the last triggered state, that is, xtk+lh>xtk, then we can obtain that Δtk+lh∈0,1 and σtk+l+1h<σtk+lh, which means that, in this case, the adaptive rule (6) uses smaller σtk+l+1h to reduce the error between xtk+l+1h and xtk at next sampling instant, and if the trigger condition is satisfied, the trigger threshold will be updated as σtk+1 at the same time. The inverse case is similar with the above discussions, and it is omitted here. Besides, the existence of parameters Δ1andΔ2 can ensure that the threshold σtk+lh are always belonging to 0,1 as the parameters Δ1andΔ2 are adjustable.Remark 2.
It is familiar to use the following EIC scheme:(7)tk+1=inftt>tk,xt−xtk≥σxtk,which has been widely used in recent years [28, 30, 31]. However, there are some undesirable traits in scheme (7). For example, (i) the EIC scheme (7) does not need to be carried out if the system state decreases very fast in a very short time, even though it satisfies the trigger condition. (ii) It requires the sensors to monitor the system state all the time. Clearly, methods that require continuous monitoring of the systems could waste network resources much more because the worst errors that can happen, such as external disturbance, network congestion, or packet dropout are rarely evident. Hence, the sample-based AEIC scheme (5) saves more communication cost than the scheme (7), because it only needs supervision at sampling instants. Furthermore, the Zeno behavior can be naturally avoided by using the sample-based AEIC scheme (5), since the impulse interval satisfies that inftk+1−tk≥h>0 for all t∈tk,tk+1,k∈ℤ+. (iii) Moreover, under the pregiven constant threshold σ, the EIC scheme (7) cannot deal with the case that the trajectory of the system oscillates along with a certain line and does not converge in a relatively large interval. In these senses, the trigger threshold σtk+lh in scheme (5) is more reasonable, flexible, and can be adaptively adjusted by according to the adaptive rule (6).
For the sake of introducing the sample-based AEIC scheme to determine whether the current sampled-data should be transmitted, an effective way is to consider the sampled-data error at every sampling instant. Decompose the impulsive intervaltk,tk+1 into the following subinterval tk,tk+1=∪l=0lm−1Ikl, where Ikl=tk+lh,tk+l+1h. Define a function:(8)ηt=t−tk+lh,t∈Ikl.
Note thatηt is a linear piecewise function and satisfies 0≤ηt≤η,η˙t=1,and∀t∈Ikl. For simplifying some notations, denote Brt=Bi, Art=Ai, Drt=Di, and Crt=Ci when rt=i. Then, combining with (5) and (8), system (2) can be rewritten as(9)x˙t=−Bixt+Aifxt+Difxt−δt+Cixt−ηt−Ciet−ηt,t∈tk,tk+1,xt=1+qkxt−,t=tk,k∈ℕ,ϕθ=xt0+θ,r0=r0,θ∈−maxδ,η,0.
The following definition and lemmas need to be recalled, which play a key role to demonstrate our main results.Definition 1.
(see [9]). System (9) is said to be stochastically exponentially stable in the mean square sense with convergence rate α>0, if there exist scalar M>0 for ∀t≥t0 such that(10)Ext2≤Me−αt−t0Esupθ∈−maxδ,η,0ϕθ2,ϕ˙θ2.Lemma 1 (see [22]).
For a given matrixΞ∈R+n, scalars a<b, α>0, and a differentiable vector function ξs:a,b⟶ℝn, the following inequality holds:(11)−∫abeαs−bξTsΞξsds≤−αeαb−a−1∫abξsdsTΞ∫abξsds.Remark 3.
Clearly, whenφs=x˙s, one can obtain from (11) that(12)−∫abeαs−bx˙TsΞx˙sds≤−αeαb−a−1xb−xaTΞxb−xa.
Furthermore, whenα⟶0, EII (11) will reduce to the well-known Jensen’s integral inequality in [32]. Besides, it has been shown in [22] that a more tighter bound for EIQT can be derived by using EII (11). Very recently, some more accurate EIIs have been proposed in [23]; however, the FMEII that can enhance the flexibility of EII to some extent has not been found in the existing literature so far. Therefore, inspired by the ideas in [19, 23], we attempt to establish a simple kind of FMEII to further estimate the bound of EIQTs in this paper.Lemma 2.
For given matricesΞ∈R+n, N∈ℝmn×n, any vector ζ∈ℝmn,m∈ℕ, scalars a<b and a differentiable vector function φs:a,b⟶ℝn, and there exist scalar auxiliary function ws≥0 with s∈a,b such that the following inequality holds:(13)−∫abφTsΞφswsds≤1,1wζTNΞ−1NTζ+2ζTN1,φw.Proof.
Note the fact that, for given matricesΞ∈R+n, N∈ℝmn×n, any vector ζ∈ℝmn,m∈ℕ, scalars a<b, and scalar auxiliary function ws≥0 with s∈a,b, one has(14)−2φTsNTζws≤ζTNΞ−1NTηws+φTsΞφsws.
Integrating both sides of (14) from a to b, we have(15)−∫abφTsΞφswsds≤∫abζTNΞ−1NTζwsds+2∫abζTNφswsds.
Then, inequality (13) can be obtained after expanding the right side of inequality (15).Remark 4.
Clearly, whenφs=w−1sξs, ws=e−αs−b, N=Ξ, ζ=−γ0/α∫abξsds, and γ0=eαb−a−1, inequality (13) will reduce to inequality (11), which implies that Lemma 1 can be regarded as a special case of Lemma 2. On the contrary, when φs=w−1sx˙sandws=e−αs−b, inequality (13) can be rewritten as follows:(16)−∫abeαs−bx˙TsΞx˙sds≤γ0αζTNΞ−1NTζ+2ζTNxb−xa.
Due to the existence of vectorζ and free matrix N, the flexibility of (13) can be enhanced to some degree. For example, if ζ=colxb,x˙b,xa,x˙a and N=colN1,N2,N3,N4, one can obtain that(17)−∫abeαs−bx˙TsΞx˙sds≤γ0αζTNΞ−1NTζ+ζTN^ζ,where(18)N^=ℍN1N2T−N1+N3TN4T∗0−N20∗∗ℍN3−N4T∗∗∗0.
It is easy to see that some connections between vectorsxbandxa and other augment vectors are established, which may facilitate the estimation for the bounds of EIQTs.


## 3. Main Results

In this section, our purpose is to establish new stochastic exponential stabilization conditions for systems (9) via the AEIC scheme (5). Before presenting the main results, the following vectors are defined for convenience:(19)ξt=colxt,fxt,xt−δt,fxt−δt,xt−δ1,xt−δ2,xt−ηt,xt−η,et−ηt,∫t−δtt−δ1xsδt−δ1ds∫t−δ2t−δtxsδ2−δtds,∫t−ηttxsηtds,∫t−ηt−ηtxsη−ηtds,x˙t,ζ21=cole¯5−e¯3,e¯5+e¯3−2e¯10,ζ22=cole¯3−e¯6,e¯3+e¯6−2e¯11,ζ31=cole¯1−e¯7,e¯1+e¯7−2e¯12,ζ32=cole¯7−e¯8,e¯7+e¯8−2e¯13.Theorem 1.
For given positive scalarsδ1,δ2,μ,η,α, and σm, system (9) is said to be stochastically exponentially stable in the mean square sense, if there exist matrices Pi, Si, Q1, Q2, Q3, Q4, Q5, Q6, R1, R2, R3∈R+n, X1∈ℝ6n×n, X2, X3∈ℝ2n×2n, Ni, Ki∈ℝn×n and diagonal matrix M1, M2∈R+n such that(20)1+qk2λmaxPi+Si−λminPi>0,k∈ℕ,(21)inftk+1−tk=β>lnqα,k∈ℕ,(22)ℛ2X2∗ℛ2>0,ℛ3X3∗ℛ3>0,(23)Φ1i+Φ2i+Φ3i+Φ4i+Φ5iζ1TX1∗−αγ0R1<0,where q=1+qk2λmaxPi+Si/λminPi,, γ0=eαδ1−1, and(24)Φ1i=ℍe¯1TPie¯14+e¯1TαPi+ΠPje¯1+e¯7−e¯9TαSi+ΠSje¯7−e¯9,Φ2i=e¯1TQ1+Q3+Q4+Q5+Q6e¯1+e¯2TQ2e¯2−e−αδ21−μe¯3TQ1e¯3−e−αδ21−μe¯4TQ2e¯4−e−αδ1e¯5TQ3e¯5−e−αδ2e¯6TQ4e¯6−e−αηe¯8TQ6e¯8,Φ3i=e¯14Tδ1R1+δ2−δ12R2+η2R3e¯14+ℍζ1TX1e¯1−e¯5−e−αδ2ζ21ζ22Tℛ2X2∗ℛ2ζ21ζ22−e−αηζ31ζ32Tℛ3X3∗ℛ3ζ31ζ32,Φ4i=σme¯7TΩe¯7−e¯9TΩe¯9+ℍe¯14+e¯1TKie¯7−e¯9+ℍe¯14+e¯1TNi−e¯14−Bie¯1+Aie¯2+Die¯4,Φ5i=−ℍe¯2−Λ1e¯1TM1e¯2−Λ2e¯1+e¯4−Λ1e¯3TM2e¯4−Λ2e¯3,ΠPj=∑j=1NπijPj,ΠSj=∑j=1NπijSj,ℛl=diagRl,3Rl,l=2,3.In addition, the control gain is designed byCi=Ni−1Ki,i∈N.Proof.
Consider the following stochastic Lyapunov–Krasovskii Functional (LKF):(25)Vxt,rt=V1xt,rt+V2xt,rt+V3xt,rt,where(26)V1xt,rt=xTtPrtxt+xTtkSrtxtk,V2xt,rt=∫t−δtteαs−txTsQ1xs+fTxsQ2fxsds+∫t−δ1teαs−txTsQ3xsds+∫t−δ2teαs−txTsQ4xsds+∫t−ηtteαs−txTsQ5xsds+∫t−ηteαs−txTsQ6xsds,V3xt,rt=∫−δ10∫t+uteαs−tx˙TsR1x˙sdsdu+δ2−δ1∫−δ2−δ1∫t+uteαs−tx˙TsR2x˙sdsdu+η∫−η0∫t+uteαs−tx˙TsR3x˙sdsdu.
Letℒ be the week infinitesimal operator acting on LKF (25), i.e.,(27)ℒVxt,rt=limΔ⟶0+EVxt+Δ,rt+Δ=jxt,rt=i−Vxt,iΔ,then along with the solution of systems (9), we have(28)ℒV1xt,i=2x˙TtPixt+xTtΠPjxt+xTtkΠSjxtk=ξTtΦ1iξt−αV1xt,i,(29)ℒV2xt,i≤xTtQ1+Q3+Q4+Q5+Q6xt+fTxtQ2fxt−1−μe−αδxTt−δtQ1xt−δt−1−μe−αδfTxt−δtQ2fxt−δt−e−αδ1xTt−δ1Q3xt−δ1−e−αδ2xTt−δ2Q4xt−δ2−e−αηxTt−ηQ6xt−η−αV2xt,i=ξTtΦ2iξt−αV2xt,i,(30)ℒV3xt,i≤x˙Ttδ1R1+δ2−δ12R2+η2R3x˙t−∫t−δ1teαs−tx˙TsR1x˙sds−ηe−αη∫t−ηtx˙TsR3x˙sds−δ2−δ1e−αδ2∫t−δ2t−δ1x˙TsR2x˙sds−αV3xt,i.
For vectorζ=colxt,xt−δt,xt−δ1,xt−δ2,xt−ηt,xt−η and matrix X1∈ℝ6n×n, it follows from Lemma 2 that(31)−∫t−δ1teαs−tx˙TsR1x˙sds≤ξTtγ0αζ1TX1R1−1X1Tζ1+2ζ1TX1e¯1−e¯5ξt,where ζ1=cole¯1,e¯3,e¯5,e¯6,e¯7,e¯8. Meanwhile, by utilizing Lemmas 1 and 3 in [33], one can obtain that(32)−δ2−δ1e−αδ2∫t−δ2t−δ1x˙TsR2x˙sds≤ξTt−e−αδ2ζ21ζ22Tℛ2X2∗ℛ2ζ21ζ22ξt,(33)−ηe−αη∫t−ηtx˙TsR3x˙sds≤ξTt−e−αηζ31ζ32Tℛ3X3∗ℛ3ζ31ζ32ξt.
Combining (30)–(33), we have(34)ℒV3xt,i≤ξTtγ0αζ1TX1R1−1X1Tζ1+Φ3iξt−αV3xt,i.
In addition, when the current data does not need to be sent out, it is easy to know from the AEIC scheme (5) that(35)0<σmxTt−ηtΩxt−ηt−eTt−ηtΩet−ηt=ξTtσme¯7TΩe¯7−e¯9TΩe¯9ξt.
Furthermore, from (9), for any matrices Ni,Ki∈ℝn×n, we have(36)0=2x˙t+xtTNi−Bixt+Aifxt+Difxt−dt+2x˙t+xtTKixt−ηt−et−ηt=ξTtℍe¯14+e¯1TKie¯7−e¯9+ℍe¯14+e¯1TNi−e¯14−Bie¯1+Aie¯2+Die¯4ξt.
According to (3), there exist diagonal matrices M1,M2∈R+n such that(37)0≤−2fxt−Λ1xtTM1fxt−Λ2xt=ξTt−ℍe¯2−Λ1e¯1TM1e¯2−Λ2e¯1ξt,(38)0≤−2fxt−dt−Λ1xt−dtTM2fxt−dt−Λ2xt−dt=ξTt−ℍe¯4−Λ1e¯3TM2e¯4−Λ2e¯3ξt,where Λ1=diagλ1−,…,λn−andΛ2=diagλ1+,…,λn+.
Therefore, combining with (28)–(38) and (22), one can obtain(39)ℒVt≤−αVt.
By utilizing Dynkin formula [10] and Grownwall inequality, one has(40)EVt≤e−αt−tkEVtk,∀t∈tk,tk+1.
From (9), (19), and (23), it follows that(41)EVtk−Vtk−≤Extk−1+qk2Pi+Sixtk−−Extk−Pixtk−≤1+qk2λmaxPi+Si−λminPiExtk−2≤q−1Extk−Pixtk−≤q−1Vtk−,which implies that Vtk≤qVtk−.
Hence, combining with (40) and (21), we have(42)EVt≤qke−αt−t0EVt0≤e−α−lnq/βt−t0EVt0,∀t≥t0.
It is easy to know from (25) that(43)EVt≥λminPiExt2,(44)EVt0≤γ1+γ2+γ3supϕθ2,ϕ˙θ2,where(45)γ1=λmaxPi+Si,Λ=diagλ1,…,λn,λν˜=maxλν˜−,λν˜+,ν˜=1,2,…,n,γ2=1−e−αδ2αλmaxQ1+λmaxQ2Λ2+λmaxQ4+1−e−αδ1αλmaxQ3+1−e−αηαλmaxQ5+λmaxQ6,γ3=e−αδ1+αδ1−1α2λmaxR1+ηe−αη+αη−1α2λmaxR3+δ2−δ1e−αδ2−e−αδ1+αδ2−δ1α2λmaxR2.
From (42)–(44), one has(46)Ext2≤Me−γt−t0Esupϕθ2,ϕ˙θ2,where M=γ1+γ2+γ3/λminPiandγ=α−lnq/β.
Therefore, according to Definition 2, system (9) is said to be stochastically exponentially stable in the mean square sense with the convergence rate γ>0. In addition, the control gain is designed by Ci=Ni−1Ki,i∈N. This completes the proof.Remark 5.
It should be pointed out that the constructed LKFVxt,i plays a key role in deriving the stochastic exponential stabilization result. Specifically, the impulsive signal related to Markov jump parameters is considered in V1xt,i, and the information about the lower bound and upper bound of time-varying delay is taken into account in V2xt,i,V3xt,i, which is more general than others given in [16–18] and is helpful to obtain less conservative stability criterion. Besides, in the proof of Theorem 1, we utilize a new FMEII to estimate the bound of EIQT, which was shown more tighter than the ones based on Jensen’s integral inequality [32] and EII in [22]. Meanwhile, the reciprocally convex approach is employed to consider the relationship between the time-varying delay and its interval, which is effective to reduce the conservatism.
In what follows, as a special case, when Markov jump parameters are not considered, system (9) will be reduced to the following equation:(47)x˙t=−Bxt+Afxt+Cxtk+Dfxt−δt,t∈tk,tk+1,xt=1+qkxt−,t=tk,k∈ℕ,ϕθ=xt0+θ,θ∈−maxδ,η,0.
Based on Theorem1, the following criterion can be readily derived.Theorem 2.
For given positive scalarsδ1,δ2,μ,η,α, and σm, system (47) is said to be exponentially stable, if there exist matrices P, S, Q1, Q2, Q3, Q4, Q5, Q6, R1, R2, R3∈R+n, X1∈ℝ6n×n, X2, X3∈ℝ2n×2n, N, K∈ℝn×n and diagonal matrix M1, M2∈R+n such that(48)1+qk2λmaxP+S−λminP>0,k∈ℕ,inftk+1−tk=β>lnqα,k∈ℕ,ℛ2X2∗ℛ2>0,ℛ3X3∗ℛ3>0,Φ^1+Φ^2+Φ^3+Φ^4+Φ^5ζ1TX1∗−αγ0R1<0,where q=1+qk2λmaxP+S/λminP, γ0=eαδ1−1, and(49)Φ^1=ℍe¯1TPe¯14+αe¯1TPe¯1+αe¯7−e¯9TSe¯7−e¯9,Φ^2=e¯1TQ1+Q3+Q4+Q5+Q6e¯1+e¯2TQ2e¯2−e−αδ21−μe¯3TQ1e¯3−e−αδ21−μe¯4TQ2e¯4−e−αδ1e¯5TQ3e¯5−e−αδ2e¯6TQ4e¯6−e−αηe¯8TQ6e¯8,Φ^3=e¯14Tδ1R1+δ2−δ12R2+η2R3e¯14+ℍζ1TX1e¯1−e¯5−e−αδ2ζ21ζ22Tℛ2X2∗ℛ2ζ21ζ22−e−αηζ31ζ32Tℛ3X3∗ℛ3ζ31ζ32,Φ^4=σme¯7TΩe¯7−e¯9TΩe¯9+ℍe¯14+e¯1TKe¯7−e¯9+ℍe¯14+e¯1TN−e¯14−Be¯1+Ae¯2+De¯4,Φ^5=−ℍe¯2−Λ1e¯1TM1e¯2−Λ2e¯1+e¯4−Λ1e¯3TM2e¯4−Λ2e¯3.In addition, the control gain is designed byC=N−1K.


## 4. Numerical Examples

In this section, we aim to demonstrate the feasibility and validity of the obtained results in this paper by two numerical examples.Example 1.
Consider system (9) with the following parameters [34–36]:(50)B1=1001,A1=2−0.1−53,D1=−1.5−0.1−0.2−2.5,(51)B2=0.8001,A2=2−0.11−53.2,D2=−1.6−0.1−0.18−2.4,(52)πij=−335−5,Λ1=diag0,0,Λ2=diag0.5,0.5,(53)δ1=0.1,δ2=0.3,μ=0.2,α=0.1,σm=0.05,η=0.1.
Under the above parameters, applying Theorem1 and using the MATLAB LMI toolbox, the impulsive intensity and the control gains matrices can be derived, respectively, as qk=−0.7440 and(54)C1=−0.66230.24420.9344−2.2286,C2=−0.59640.23820.9783−2.2877.
Meanwhile, the weight matrixΩ in AEIC scheme and other unknown matrices in Theorem 1 can be obtained:(55)Ω=430.7870−9.5302−9.5302141.3639,P1=138.90719.23099.230917.8354,P2=147.83329.58999.589918.2709,S1=118.804922.692222.69229.4962,S2=118.534423.214923.21499.2227,R1=60.94474.82614.82615.2283,R2=81.1484−0.7781−0.778122.8792,R3=126.4894−49.3995−49.3995119.2005.
Let the neuron activation functionfix=0.5x+1−x−1, the time-varying delay δt=0.2+0.2sint, the initial value x0=−0.2;0.3, and the parameters Δ1=1,Δ2=50 in adaptive rule (6); Figures 1–4 are presented to show the feasibility and validity of the obtained result in this paper. Specifically, Figures 1 and 2 stand for the state response of system (9) with parameters (50) and (51) under no control and the well-designed AEIC scheme (5), respectively. Clearly, AEIC scheme (5) is effective. Furthermore, it is not hard to see from Figure 3 that the frequency of impulsive control is reduced to large extent, which means that more network resources are saved by using the AEIC scheme. Moreover, compared with Figure 4, one can find that the triggering threshold adjusted adaptively by adaptive rule (6) is varying with the evolution of the system state, i.e., the threshold gradually decrease when the norm of the system state goes up, which enhances the flexibility of the AEIC scheme to some extent.
On the contrary, if the adaptive thresholdσtk+lh is a constant, that is, Δ2=0, then the AEIC scheme (5) will reduce to a EIC scheme (7). Without loss of generality, take the threshold in EIC scheme as σm, and under the same parameters with above mentioned, Figures 5 and 6 are given to show the results of ECI control scheme. It is clear from Figures 2 and 5 that both AEIC and EIC scheme can stabilise the underlying system (9), while comparing Figures 3 and 6, and the control frequency of AEIC scheme is 27/50 and the control frequency of EIC scheme is 46/50, where 50 is the number of sampling data. Clearly, AEIC scheme is better to decrease the frequency of impulsive control than the EIC scheme. Thus, it is concluded that the control cost can be saved more effectively by the AEIC scheme than the EIC scheme.Figure 1
Curve ofxt for Example 1 without any control.Figure 2
Curve ofxt for Example 1 under AEIC.Figure 3
Release instants and intervals for Example1 under AEIC.Figure 4
Adaptive parameters of AEIC scheme for Example1.Figure 5
Curve ofxt for Example 1 under EIC.Figure 6
Release instants and intervals for Example1 under EIC.Example 2.
Consider system (47) with the following parameters [34–36]:(56)A=0.9+π4190.110.9+π4,D=−1.22π40.30.2−1.22π4,(57)B=diag1,0.9,Λ1=diag0,0,Λ2=diag0.4,0.8,(58)δ1=0.1,δ2=0.4,μ=0.2,α=0.2,σm=0.05,η=0.1.
Under the above parameters, applying Theorem2 and using the MATLAB LMI toolbox, one can derive that the impulsive intensity qk=−0.8884 and the control gains matrix C=−2.5040−6.1983−0.1509−1.6798. Meanwhile, the weight matrix Ω in the AEIC scheme and other unknown matrices in Theorem 2 can be obtained:(59)Ω=104×0.0424−0.0567−0.05671.4377,P=103×0.0514−0.1625−0.16252.5712,W=24.3347−119.3639−119.3639691.4778,R1=103×0.0154−0.1028−0.10281.0734,R2=103×0.0269−0.1600−0.16002.1631,R3=103×0.28300.40310.40317.7438.
Let the neuron activation functionfix=tanhx, the time-varying delay δt=0.2+0.2cost, the initial value x0=−0.4;0.7, and the parameters Δ1=1,Δ2=20 in adaptive rule (6); Figures 7–10 are presented to reflect intuitively the feasibility and validity of the obtained result in this paper. Figures 7 and 8 show the state response of systems (47) with parameters (56) and (57) under no any control and the well designed AEIC scheme (5), respectively. Clearly, AEIC scheme (5) is effective. Furthermore, it is not hard to see from Figure 9 that the frequency of impulsive control is reduced to a large extent, which means that more networks resources are saved by using the AEIC scheme. Moreover, compared with Figures 8 and 10, one can find that the triggering threshold adjusted adaptively by adaptive rule (6) is varying with the evolution of system state, i.e., the threshold gradually decreases when the norm of the system state goes up, which enhances the flexibility of the AEIC scheme to some extent.Figure 7
Curve ofxt without control for Example 2.Figure 8
Curve ofxt with AEIC for Example 2.Figure 9
Release instants and intervals for Example2.Figure 10
Adaptive parameters for Example2.


## 5. Conclusion

In this paper, we have studied the stochastic exponential stabilization problem for MJNNs with TDs. A general LKF involving more information about sample data, Markov jump parameters, and TDs are constructed, and some exponential stabilization criteria for two kinds of NNs with TDs are established by employing a novel FMEII and a new AEIC scheme. The feasibility and validity of the obtained results are illustrated by two numerical examples. More importantly, the methods proposed in this paper can be employed to deal with other control issues for time delayed systems in our future work.



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