# H∞ Synchronization of Semi-Markovian Jump Neural Networks with Randomly Occurring Time-Varying Delays

Authors: Mengping Xing; Hao Shen; Zhen Wang
Journal: Complexity
 (2018)

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## Abstract

Based on the Lyapunov stability theory, this paper mainly investigates theH∞ synchronization problem for semi-Markovian jump neural networks (semi-MJNNs) with randomly occurring time-varying delays (TVDs). The continuous-time semi-MJNNs, where the transition rates are dependent on sojourn time, are introduced to make the issue under our consideration more general. One of the main characteristics of our work is the handling of TVDs. In addition to using the improved Jensen inequality and the reciprocal convexity lemma to deal with the integral inequality, we also employ Schur complement and the projection lemma to achieve the decoupling between the square term of TVDs. Finally, we verify the validity and feasibility of our method by a couple of simulation examples.


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## Body

## 1. Introduction

In recent years, with the unceasingly thorough research on large data and artificial intelligence, the theory and application of neural networks have been greatly developed. It has tremendous application prospect, especially in robotics [1], pattern recognition [2, 3], associative memory [4–6], identification [7, 8], and combinatorial optimization [9–12]. Neural networks can be simply divided into the deterministic neural networks and stochastic neural networks based on whether they are disturbed by outside noise [13]. When the system is undisturbed, the deterministic neural network can describe the actual system accurately [13]. Nevertheless, as far as we know, the actual system is generally uncertain and most of the physical system will be affected by random parameter variation and structure change [14–16]. These changes may be caused by some sudden phenomena, such as components or connection failure and the deviation of parameter. In this circumstance, the stochastic neural networks can be described by a hybrid model, where a discrete stochastic variable called mode or pattern is attached to continuous state variables to describe the random jump of system parameters as well as the appearance of discontinuous points. It allows policymakers to respond to discrete events, which significantly perturb or alter the normal working condition of the system, by combining the empirical knowledge of events and the statistical information of their rates, adequately [17, 18].Markovian jump neural networks (MJNNs), as we all know, as a kind of typical hybrid dynamic systems are widely used in the field of aerospace, industrial production, and biological, medical, and social construction in the past few decades due to its strong modeling ability and therefore draw great attention from researchers. For instance, the stability analysis, state estimation, filter design, passivity analysis, and stochastic synchronization for MJNNs were discussed in [19, 20], respectively. But one obvious drawback of MJNNs is that its jump time obeys the exponential distribution, which is a memoryless distribution and makes the transition probability of jump system an invariant function matrix; that is, the transition probability of the system obeys a stochastic process which is not relevant with the mode of the past [21]. Because of this limitation, it brings great restriction to the application of MJNNs. Therefore, semi-MJNNs were put forward later, where a transition probability matrix and a fixed dwell time probability density function matrix are used to represent the stochastic neural networks [22]. It has a wider range of application background due to the relaxation to the constraint condition where the probability density distribution function obeys exponential distribution. Compared with the abundant research achievements on MJNNs, the research efforts devoted to semi-MJNNs are relatively scarce. The robust stochastic stability condition for semi-MJNNs was derived in [23] and the relevant controller was also designed there. The synchronization controller for the semi-MJNNs was designed in [24] where the semi-MJNNs were transformed into associated MJNNs. An exponential passive filter was designed, and a cone complementarity linearization method was applied to manage the nonconvex feasibility issue in [25]. As mentioned above, the semi-MJNNs have more extensive application, such as in complex medical procedures [26].Due to the finite signal transition speed as well as the limited switching speed of hardware facilities, the time-delay phenomenon exists in various practical industrial control systems widely, such as chemical system, process control system, and network control system [27–36]. It is known that delay argument existing in the system is often unknown or time-varying and the occurrence of delay tends to be random, which makes the analysis and control of the system more difficult. Also, the existence of time-delay tends to result in the degradation of the performance index and can even make the system unstable [37]. Therefore, it has important theoretical significance and practical applying value to study the system with time-varying delays (TVDs). As the system that considers that time-delay is more in line with the actual situation, an increasing number of researches have been made on the time-delay systems in recent years, and considerable results have been presented. To mention a few, Gun and Niculescu discussed the problem of stability analysis for the systems with time-delay and gave a summary on literature about the stability analysis and controller design of systems with time-delay in [38]. Park and Wan Ko studied the stability and robust stability criteria for TVD systems in [39], and then the reciprocally convex approach and the second-order reciprocally convex approach were proposed for stability analysis of TVD systems in [40].Synchronization refers to two or more dynamic systems whose properties are identical or close to each other. Through the interaction between the systems, the state of the dynamic system that evolves under different initial conditions is gradually close to each other and finally reaches the same. Synchronization analysis is particularly important in many dynamic behaviors of neural networks and therefore has been widely studied. Exponential synchronization, adaptive synchronization, finite-timeH∞ synchronization, mixed H∞/passive synchronization, and new delay-dependent exponential H∞ synchronization were considered in [41, 42], respectively.This paper mainly studies theH∞ synchronization of semi-MJNNs with randomly occurring TVDs. First of all, by Lyapunov stability theory, we can get that the key point to establish a Lyapunov functional is to contain more useful information about the delays, which is useful to obtain the results with less conservatism [43]. As a result of the existence of TVDs, some novel inequality techniques derived from the Park inequality and the improved Jensen inequality [44] are employed to handle the time-varying items. At the same time, considering the existence of the square term of TVDs in the formulas, we use the projection lemma to achieve the decoupling between time-varying items. By using convex optimization techniques, the synchronization control of semi-MJNNs is investigated in this paper. The corresponding main results are presented by three theorems: Theorem 1 provides sufficient conditions for the stochastic stability and H∞ synchronization of the closed-loop dynamic error system; Theorem 2 conducts the decoupling arithmetic; and Theorem 3 is then presented to get strict LMI-based conditions, and a numerical method to calculate controller gains is presented, which is simple and easily conducted.Compared with the existing literature, this article has the following characteristics: (1) Different from the previous literature, a more general system model is introduced in this paper, in which both the semi-MJNNs and the random TVDs are taken into account simultaneously; (2) with the introduction of some advanced inequalities, combining with Schur complement lemma and projection lemma,H∞ synchronization conditions with less conservatism are derived; (3) we use the LMI control toolbox to carry out the relevant simulation, and the corresponding controller can be obtained which can verify the correctness and feasibility of the proposed method. Throughout this work, the notations used are standard.


## 2. Problem Formulation

Firstly, given the following semi-MJNNs with randomly occurring TVDsΣ,
(1)ẋt=−Aξtxt+Bξtfxt+βtBθξtfxt−θt+It,(2)zt=Cξtxt,where xt=x1Tt,x2Tt,…,xnTtT∈ℝn is the system state vector which is associated with the n neurons; fxt=f1Tx1t,f2Tx2t,…,fnTxntT∈ℝn denotes the neuron activation functions of the system, which is assumed to be bounded and satisfies
(3)lq−≤fqa−fqba−b≤lq+,q=1,2,…,n,where fq0=0, a,b∈ℝ, a≠b.lq−, and lq+ are real known scalars, and they could be zero, positive, or negative. For the purpose of simplifying the symbols, we set
(4)L1≜diagl1+l1−,l2+l2−,…,ln+ln−,L2≜diagl1++l1−2,l2++l2−2,…,ln++ln−2.It stands for external input; θt denotes the TVDs satisfying 0≤θ1≤θt≤θ2<∞ and θ̇t≤μ<∞, where the nonnegative scalars θ1 and θ2 refer to the minimum and maximum time-delay, respectively. ξt,σt≥0=ξm,σmm∈M≥1 (M is a positive integer) represents a continuous-time and discrete-state homogeneous semi-Markovian process whose trajectories are right continuous. Assuming ξt takes value in a finite state space ϒ=1,2,…,N, the transition rate matrix Λ¯σ≜ρijσN×Ncan be given by
(5)Prξm+1=j,σ+ε≥σm+1∣ξm=i,σ<σm+1=ρijσε+oε,i≠j,Prξm+1=j,σ+ε<σm+1∣ξm=i,σ<σm+1=1+ρiiσε+oε,i=j,where ε>0,limε→0oε/ε=0 and ρijσ≥0ξt=i,ξt+ε=j; i≠j stands for the transition rates from i to j, and ρiiσ=−∑j=1,j≠iNρijσ. βk is a Bernoulli-distributed white sequence that takes values of 0 and 1 and obeys the following probability distribution laws
(6)Prβk=1=β¯,ℰβk=β¯,Prβk=0=1−β¯,where β¯∈0,1 is a known constant.Remark 1.
Different from the previous literature, a more general system model is introduced in this paper, in which both the semi-MJNNs and the random TVDs are taken into account. The time-delay phenomenon, which occurs randomly and tends to be time-varying, exists in various practical neural networks. Therefore, the stochastic variableβt is introduced to express the randomly occurring TVDs in this paper to make the issue under consideration more practical and more reasonable.In this paper, the slave systemΣ^ could be represented as the following forms
(7)x^̇t=−Aξtx^t+Bξtfx^t+βtBθξtfx^t−θt+It+Dξtωt+ut,(8)z^k=Cξtx^k,where x^t and z^k are the response state vector and the response output, respectively; ut is the advisable control input.For presenting a better explanation to the addressed problem, we introduceϱt=xt−x^t as the synchronization error vector, ϱ~t=zt−z^t=Cξtϱt as the output error, and fϱt=fxt−fx^t as the nonlinear error. Then, the following error system is obtained:
(9)ϱ̇t=−Aξtϱt+Bξtfϱt+βtBθξtfϱt−θt+Dξtωt+ut.The controller inputut for the error system is established as
(10)ut=Kξtϱt,where the controller gain matrix Kξt∈ℝn×n will be designed in the sequel. In order to simplify the notation, Aξt,Bξt,Bθξt,Cξt,Dξt and Kξt are denoted by Ai, Bi, Bθi, Ci, Di, and Ki, for each i∈ϒ, respectively. Then, the closed-loop dynamic error system Σ¯ can be obtained:
(11)ϱ̇t=−Aiϱt+Bifϱt+βtBθifϱt−θt+Diωt+Kiϱt=−Ai−Kiϱt+Bifϱt+βtBθifϱt−θt+Diωt.Before making further derivation, we need the definition and lemmas as shown in the following.Lemma 1.
[24] The following inequalities hold for any diagonal matrices Vli>0,l=1,2:
(12)ϱtfϱtT−L1V1iL2V1i∗−V1iϱtfϱt≥0,ϱt−θtfϱt−θtT−L2V2iL2V2i∗−V2iϱt−θtfϱt−θt≥0.Lemma 2.
[45] (projection lemma) For a symmetric matrix F, two matrices U,V with the same column dimension of F, the problem
(13)F+UTXTV+VTXU<0,can be solved with respect to matrix X if and only if
(14)NUTFNU<0,NVTFNV<0,where NUandNV are any basis of the nullspace of U,V.Lemma 3.
[30] (Schur complement lemma). Given constant matrices M11, M12, and M22, where M11=M11T<0 and M22=M22T<0, then, M11−M12M22−1M12<0 if and only if
(15)M11M12∗M22<0orM22M12T∗M11<0.Lemma 4.
[40] Given a matrix ℛ>0, a differentiable function xu∣u∈a1,a2, it is easy to obtain the following inequalities
(16)∫a1a2xTϖℛxϖdϖ≥1a2−a1Ω1TℛΩ1+3a2−a1Ω2TℛΩ2,∫a1a2ẋTϖℛẋϖdϖ≥1a2−a1Ω3TℛΩ3+3a2−a1Ω4TℛΩ4+5a2−a1Ω5TℛΩ5,∫a1a2∫a1βẋTϖℛẋϖdϖdβ≥2Ω8TℛΩ8+4Ω9TℛΩ9,where
(17)Ω1=∫a1a2xϖdϖ,Ω2=∫a1a2xϖdϖ−2a2−a1∫a1a2∫βa2xϖdϖdβ,Ω3=xa2−xa1,Ω4=xa2+xa1−2a2−a1∫a1a2xϖdϖ,Ω5=xa2−xa1+6a2−a1∫a1a2xϖdϖ−12a2−a12∫a1a2∫βa2xϖdϖdβ,Ω6=xa2−1a2−a1∫a1a2xϖdϖ,Ω7=xa2+2a2−a1∫a1a2xϖdϖ−6a2−a12∫a1a2∫βa2xϖdϖdβ,Ω8=xa1−1a2−a1∫a1a2xϖdϖ,Ω9=xa1−4a2−a1∫a1a2xϖdϖ+6a2−a12∫a1a2∫βa2xϖdϖdβ.Lemma 5.
[40] (reciprocal convexity lemma) For any vector ξ∈ℝm; matrices R1,R2∈Sn+, S∈ℝn×n, W1,W2∈ℝn×m; and nonnegative real scalars α and β meeting α+β=1, the following inequality holds
(18)1αξTW1TR1W1ξ+1βξTW2TR2W2ξ≥ξTW1W2TR1SSTR2W1W2ξ,subject to R1SSTR2≥0.Definition 1.
(H∞ synchronization) The system Σ¯ is said to be H∞ synchronization with the disturbance attenuation γ under the condition that the following requirements are met:
(1)
The systemΣ¯ is stochastically stable when the disturbance input ωt is always equal to 0.
(2)
For a positive scalarγ, the following inequality is satisfied under zero initial conditions
(19)Θ=∫0∞ϱ~Ttϱ~t−γ2ωTtωtdt≤0i.e.,supω≠0,ω∈L20,∞ϱ~t2ωt2≤γ.


## 3. Main Results

Theorem 1.
Given scalarsμ, γ>0, 0≤θ1≤θ2<∞, θ=θ2−θ1, δ1>0, δ2>0, and β¯∈0,1, the considered system Σ¯ is stochastically stable and H∞ synchronized; if there exist symmetric matrices Pi∈ℝ3n>0, positive definite matrices Q1i,Q2i,Q3i,R1,R2,G1i,G2i,Z1,Z2, positive definite diagonal matrices V1i,V2i, matrices Yi, and Sij∈ℝn,i,j∈ϒ, such that for each i∈ϒ, the following matrix inequalities hold
(20)Φ^i=Φ¯i+ψiθt+Πθt∑k=1Nρ¯ikPkΠTθt<0,(21)Ψi=G2i+W2i00S11S12S1303G2i+W2i0S21S22S23005G2i+W2iS31S32S33S11TS21TS31TG2i00S12TS22TS32T03G2i0S13TS23TS33T005G2i>0,(22)M1i=R1+R2−∑k=1Nρ¯ikQ1k−∑k=1Nρ¯ikQ2k−∑k=1Nρ¯ikQ3k>0,(23)M2i=R2−∑k=1Nρ¯ikQ2k−∑k=1Nρ¯ikQ3k>0,(24)M3i=R2−∑k=1Nρ¯ikQ2k>0,(25)W1i=Z1−θ1∑k=1Nρ¯ikG1k>0,(26)W2i=Z2−θ∑k=1Nρ¯ikG2k>0,where
(27)Φ¯i=SymE1P11iE2T+θ1E1P12iE8T+E2P12iE2T−E3P12iE2T+θ1E2P22iE8T−θ1E3P22iE8T+E3P13iE2T−E5P13iE2T+θ1E3P23iE8T−θ1E5P23iE8T+E2QiE2T−1−μE4Q3iE4T−E5Q2iE5T+E1GiE1T−E2−E3G1iE2−E3T−3E2+E3−2E8G1iE2+E3−2E8T−5E2−E3+6E8−6E11G1iE2−E3+6E8−6E11T−2E2−E8W1iE2−E8T−4E2+2E8−3E11W1iE2+2E8−3E11T−θθ1E2−E3W2iE2−E3T−3θθ1E2+E3−2E8W2iE2+E3−2E8T−5θθ1E2−E3+6E8−6E11W2iE2−E3+6E8−6E11T−2E3−E9W2iE3−E9T−4E3+2E9−3E12W2iE3+2E9−3E12T−2E4−E10W2iE4−E10T−4E4+2E10−3E13W2iE4+2E10−3E13T−Γ1Ψ^iΓ1T−Symδ1E1YiE1T+δ1E1YiAiE2T−δ1E1YiKiE2T−δ1E1YiBiE6T−δ1β¯E1YiBθi×E7T−δ1E1YiDiE14T+δ2E2YiE1T+δ2E2YiAiE2T−δ2E1YiKiE2T−δ2E2YiBiE6T−δ2β¯E2×YiBθiE7T−δ2E2YiDiE14T−E2L1V1iE2T+E4L1V2iE4T+SymE2L2V1iE6T+E4L2V2iE7T−E6V1iE6T−E7V2iE7T+E2CiTCiE2T−γ2E14E14T,ψiθt=Symθt−θ1E1P13iE9T+θ2−θtE1P13iE10T+θt−θ1E2P23iE9T+θ2−θt×E2P23iE10T+θt−θ1E3P33i−P23iE9T+θ2−θtE3P33i−P23iE10T−θt−θ1E5P33iE9T−θ2−θtE5P33iE10T−θt−θ1E9M2iE9T−3θt−θ1E9−E12×M2iE9−E12T−θ2−θtE10M3iE10T−3θ2−θtE10−E13M3iE10−E13T,Πθt=E2θ1E8θt−θ1E9+θ2−θtE10,with
(28)Ei=0…0⏟i−1nIn0…0⏟14−inT∈ℝ14n×n,Qi=θ1R1+θ2R2+Q1i+Q2i+Q3i,Gi=θ12G1i+θ2G2i+θ122Z1+θ22−θ122Z2,Γ1=E3−E4E3+E4−2E9E3−E4+6E9−6E12E4−E5E4+E5−2E10E4−E5+6E10−6E13,Ψ^i=G2i00S11S12S1303G2i0S21S22S23005G2iS31S32S33S11TS21TS31TG2i00S12TS22TS32T03G2i0S13TS23TS33T005G2i.Proof 1.
We firstly consider a stochastic semi-Markovian Lyapunov-Krasovskii functional as follows:(29)Vϱst,ξt=∑m=15Vmϱst,ξt,where
(30)V1ϱst,ξt=ϱ¯TtPiϱ¯t,V2ϱst,ξt=∫t−θ1tϱϖQ1iϱϖdϖ+∫t−θ2tϱTϖQ2iϱϖdϖ+∫t−θttϱTϖQ3iϱϖdϖ,V3ϱst,ξt=∫−θ10∫t+βtϱTϖR1ϱϖdϖdβ+∫−θ20∫t+βtϱTϖR2ϱϖdϖdβ,V4ϱst,ξt=θ1∫−θ10∫t+βtϱ̇TϖG1iϱ̇ϖdϖdβ+∫−θ10∫y0∫t+βtϱ̇TϖZ1ϱ̇ϖdϖdβdy,V5ϱst,ξt=θ∫−θ2−θ1∫t+βtϱ̇TϖG2iϱ̇ϖdϖdβ+∫−θ2−θ1∫y0∫t+βtϱ̇TϖZ2ϱ̇ϖdϖdβdy,with
(31)Pi=P11iP12iP13i∗P22iP23i∗∗P33i,ϱ¯t=ϱTt∫t−θ1tϱTϖdϖ∫t−θ2t−θ1ϱTϖdϖT.First of all, we define(32)ς1t=ϱ̇tϱtϱt−θ1ϱt−θtϱt−θ2fϱtfϱt−θt,ς2t=1θ1∫t−θ1tϱϖdϖ1θt−θ1∫t−θtt−θ1ϱϖdϖ1θ2−θt∫t−θ2t−θtϱϖdϖ2θ12∫−θ10∫t+βtϱϖdϖdβ2θt−θ12∫−θt−θ1∫t+βt−θ1ϱϖdϖdβ2θ2−θt2∫−θ2−θt∫t+βt−θtϱϖdϖdβωt,ςt=ς1tς2t,Ei=0…0⏟i−1nIn0…0⏟14−inT∈ℝ14n×n,ρ¯ij=ℰρijσ=∫0∞ρijσgiσdσ,where giσ is the probability density function of sojourn time σ resting on mode i.Before further analysis, we consider the weak infinitesimal operatorℒ as the following forms
(33)ℒVϱst,ξt=limε→0+1εℰVϱst+ε,ξt+ε∣ϱst,ξt=i−Vϱst,ξt,and we have
(34)ℒV1ϱst,ξt=SymςTtE1E2−E3E3−E5PiTΠθtςt+ςTtΠθt∑k=1Nρ¯ikPkΠTθtςt,(35)ℒV2ϱst,ξt=ϱTtQ1iϱt−ϱTt−θ1Q1iϱt−θ1+∫t−θ1tϱTϖ∑k=1Nρ¯ikQ1kϱϖdϖ+ϱTtQ2iϱt−ϱTt−θ2Q2iϱt−θ2+∫t−θ2tϱTϖ∑k=1Nρ¯ikQ2kϱϖdϖ+ϱTtQ3iϱt−1−μϱTt−θtQ3iϱt−θt+∫t−θttϱTϖ∑k=1Nρ¯ikQ3kϱϖdϖ,(36)ℒV3ϱst,ξt=ϱTtθ1R1+θ2R2ϱt−∫t−θ1tϱTϖR1ϱϖdϖ−∫t−θ2tϱTϖR2ϱϖdϖ.Considering the integral items in (35) and (36), we can get that
(37)∫t−θ1tϱTϖ∑k=1Nρ¯ikQ1kϱϖdϖ+∫t−θ2tϱTϖ∑k=1Nρ¯ikQ2kϱϖdϖ+∫t−θttϱTϖ∑k=1Nρ¯ikQ3kϱϖdϖ−∫t−θ1tϱTϖR1ϱϖdϖ−∫t−θ2tϱTϖR2ϱϖdϖ=−∫t−θ1tϱTϖR1+R2−∑k=1Nρ¯ikQ1k−∑k=1Nρ¯ikQ2k−∑k=1Nρ¯ikQ3kϱϖdϖ−∫t−θtt−θ1ϱTϖR2−∑k=1Nρ¯ikQ2k−∑k=1Nρ¯ikQ3kϱϖdϖ−∫t−θ2t−θtϱTϖR2−∑k=1Nρ¯ikQ2kϱϖdϖ=−∫t−θ1tϱTϖM1iϱϖdϖ−∫t−θtt−θ1ϱTϖM2iϱϖdϖ−∫t−θ2t−θtϱTϖM3iϱϖdϖ.For the integral items−∫t−θ1tϱTϖM1iϱϖdϖ, −∫t−θtt−θ1ϱTϖM2iϱϖdϖ, and −∫t−θ2t−θtϱTϖM3iϱϖdϖ in (37), by using Lemma 4, we can obtain that
(38)−∫t−θ1tϱTϖM1iϱϖdϖ≤−1θ1∫t−θ1tϱϖdϖTM1i∫t−θ1tϱϖdϖ−3θ1∫t−θ1tϱϖdϖ−2θ1∫t−θ1t∫βtϱϖdϖdβT×M1i∫t−θ1tϱϖdϖ−2θ1∫t−θ1t∫βtϱϖdϖdβ=ςTt−4θ1E8M1iE8T+Sym3θ1E8M1iE11T−3θ1E11M1iE11Tςt.Similarly, it is easy for us to get the following conditions:(39)−∫t−θtt−θ1ϱTϖM2iϱϖdϖ≤−θt−θ1ςTtE9M2iE9T+3E9−E12M2iE9−E12Tςt,−∫t−θ2t−θtϱTϖM3iϱϖdϖ≤−θ2−θtςTtE10M3iE10T+3E10−E13M3iE10−E13Tςt.Synthesizing the above results, we can get that(40)ℒV2ϱst,ξt+ℒV3ϱst,ξt≤ςTtE2QiE2T−E3Q1iE3T−E5Q2iE5T−1−μE4Q3iE4T−4θ1E8M1iE8T+Sym3θ1E8M1iE11T−θt−θ1E9M2iE9T+3E9−E12M2iE9−E12T−3θ1E11M1iE11T−θ2−θtE10M3iE10T+3E10−E13M3iE10−E13Tςt.(41)ℒV4ϱst,ξt=θ12ϱ̇TtG1iϱ̇t+θ122ϱ̇TtZ1ϱ̇t−θ1∫t−θ1tϱ̇TϖG1iϱ̇ϖdϖ+θ1∫−θ10∫t+βtϱ̇Tϖ∑k=1Nρ¯ikG1kϱ̇ϖdϖdβ−∫−θ10∫t+βtϱ̇TϖZ1ϱ̇ϖdϖdβ,(42)ℒV5ϱst,ξt=θ2ϱ̇TtG2iϱ̇t+θ22−θ122ϱ̇TtZ2ϱ̇t−θ∫t−θ2t−θ1ϱ̇TϖG2iϱ̇ϖdϖ+θ∫−θ2−θ1∫t+βtϱ̇Tϖ∑k=1Nρ¯ikG2kϱ̇ϖdϖdβ−∫−θ2−θ1∫t+βtϱ̇TϖZ2ϱ̇ϖdϖdβ.Considering the single integral items in (41) and (42), we can get that
(43)−θ1∫t−θ1tϱ̇TϖG1iϱ̇ϖdϖ−θ∫t−θ2t−θ1ϱ̇TϖG2iϱ̇ϖdϖ=−θ1∫t−θ1tϱ̇TϖG1iϱ̇ϖdϖ−θ∫t−θ2t−θtϱ̇TϖG2iϱ̇ϖdϖ−θ∫t−θtt−θ1ϱ̇TϖG2iϱ̇ϖdϖ.For the integral items−θ1∫t−θ1tϱ̇TϖG1iϱ̇ϖdϖ, −θ∫t−θ2t−θtϱ̇TϖG2iϱ̇ϖdϖ, and −θ∫t−θtt−θ1ϱ̇TϖG2iϱ̇ϖdϖ in (43), by using Lemma 4, we can obtain that
(44)−θ1∫t−θ1tϱ̇TϖG1iϱ̇ϖdϖ≤−ϱt−ϱt−θ1TG1iϱt−ϱt−θ1−3ϱt+ϱt−θ1−2θ1∫t−θ1tϱϖdϖTG1iϱt+ϱt−θ1−2θ1∫t−θ1tϱϖdϖ−5ϱt−ϱt−θ1+6θ1∫t−θ1tϱϖdϖ−12θ12∫t−θ1t∫βtϱϖdϖdβTG1i×ϱt−ϱt−θ1+6θ1∫t−θ1tϱϖdϖ−12θ12∫t−θ1t∫βtϱϖdϖdβ=−ςTtE2−E3G1iE2−E3T+3E2+E3−2E8G1iE2+E3−2E8T+5E2−E3+6E8−6E11G1iE2−E3+6E8−6E11Tςt.By using a similar method, it is not difficult to get that(45)−θ∫t−θ2t−θtϱ̇TϖG2iϱ̇ϖdϖ≤−θθt−θ1ςTtE3−E4G2iE3−E4T+3E3+E4−2E9G2iE3+E4−2E9T+5E3−E4+6E9−6E12G2iE3−E4+6E9−6E12Tςt,(46)−θ∫t−θtt−θ1ϱ̇TϖG2iϱ̇ϖdϖ≤−θθ2−θtςTtE4−E5G2iE4−E5T+3E4+E5−2E10G2iE4+E5−2E10T+5E4−E5+6E10−6E13G2iE4−E5+6E10−6E13Tςt.Considering the double integral items in (41) and (42), we can get that
(47)θ1∫−θ10∫t+βtϱ̇Tϖ∑k=1Nρ¯ikG1kϱ̇ϖdϖdβ−∫−θ10∫t+βtϱ̇TϖZ1ϱ̇ϖdϖdβ+θ∫−θ2−θ1∫t+βtϱ̇Tϖ∑k=1Nρ¯ikG2kϱ̇ϖdϖdβ−∫−θ2−θ1∫t+βtϱ̇TϖZ2ϱ̇ϖdϖdβ=−∫−θ10∫t+βtϱ̇TϖZ1−θ1∑k=1Nρ¯ikG1kϱ̇ϖdϖdβ−∫−θ2−θ1∫t+βtϱ̇TϖZ2−θ∑k=1Nρ¯ikG2kϱ̇ϖdϖdβ=−∫−θ10∫t+βtϱ̇TϖW1iϱ̇ϖdϖdβ−∫−θ2−θ1∫t+βtϱ̇TϖW2iϱ̇ϖdϖdβ.For−∫−θ10∫t+βtϱ̇TϖW1iϱ̇ϖdϖdβ in (47), by using Lemma 4, we can obtain that
(48)−∫−θ10∫t+βtϱ̇TϖW1iϱ̇ϖdϖdβ≤−2ϱt−1θ1∫t−θ1tϱϖdϖTW1iϱt−1θ1∫t−θ1tϱϖdϖ−4ϱt+2θ1∫t−θ1tϱϖdϖ−6θ12∫t−θ1t∫βtϱϖdϖdβTW1iϱt+2θ1∫t−θ1tϱϖdϖ−6θ12∫t−θ1t∫βtϱϖdϖdβ=ςTt−2E2−E8W1iE2−E8T−4E2+2E8−3E11W1iE2+2E8−3E11Tςt.For−∫−θ2−θ1∫t+βtϱ̇TϖW2iϱ̇ϖdϖdβ in (47), we firstly divide it into two parts
(49)−∫−θ2−θ1∫t+βtϱ̇TϖW2iϱ̇ϖdϖdβ=−∫−θ2−θ1∫t−θ1tϱ̇TϖW2iϱ̇ϖdϖdβ−∫−θ2−θ1∫t+βt−θ1ϱ̇TϖW2iϱ̇ϖdϖdβ.It is easy to find that the first integral term in (49) can be reduced to a single integral. So we can use Lemma 4 to handle it:
(50)−∫−θ2−θ1∫t−θ1tϱ̇TϖW2iϱ̇ϖdϖdβ=−θ∫t−θ1tϱ̇TϖW2iϱ̇ϖdϖ≤−θθ1ςTtE2−E3W2iE2−E3T+3E2+E3−2E8W2iE2+E3−2E8T+5E2−E3+6E8−6E11W2iE2−E3+6E8−6E11Tςt.For the second integral term in (49), we firstly divide it into three parts:
(51)−∫−θ2−θ1∫t+βt−θ1ϱ̇TϖW2iϱ̇ϖdϖdβ=−∫−θt−θ1∫t+βt−θ1ϱ̇TϖW2iϱ̇ϖdϖdβ−∫−θ2−θt∫t+βt−θtϱ̇TϖW2iϱ̇ϖdϖdβ−θ2−θt∫t−θtt−θ1ϱ̇TϖW2iϱ̇ϖdϖ.Then, we consider using Lemma4 to deal with (51) and the following inequalities can be obtained:
(52)−∫−θt−θ1∫t+βt−θ1ϱ̇TϖW2iϱ̇ϖdϖdβ−∫−θ2−θt∫t+βt−θtϱ̇TϖW2iϱ̇ϖdϖdβ≤ςTt−2E3−E9W2iE3−E9T−4E3+2E9−3E12W2iE3+2E9−3E12T−2E4−E10W2iE4−E10T−4E4+2E10−3E13W2iE4+2E10−3E13Tςt,(53)−θ2−θt∫t−θtt−θ1ϱ̇TϖW2iϱ̇ϖdϖ≤−θθt−θ1ςTtE3−E4W2iE3−E4T+3E3+E4−2E9W2iE3+E4−2E9T+5E3−E4+6E9−6E12W2iE3−E4+6E9−6E12Tςt+ςTtE3−E4W2iE3−E4T+3E3+E4−2E9W2iE3+E4−2E9T+5E3−E4+6E9−6E12W2iE3−E4+6E9−6E12Tςt.Let us defineα=θt−θ1/θ and β=θ2−θt/θ. Using Lemma 5, we can obtain the following relation from inequalities (45), (46), and (53):
(54)−1αςTtE3−E4G2i+W2iE3−E4T+3E3+E4−2E9G2i+W2iE3+E4−2E9T+5E3−E4+6E9−6E12G2i+W2iE3−E4+6E9−6E12Tςt−1βςTtE4−E5G2iE4−E5T+3E4+E5−2E10G2iE4+E5−2E10T+5E4−E5+6E10−6E13G2iE4−E5+6E10−6E13Tςt+ςTtE3−E4W2iE3−E4T+3E3+E4−2E9W2iE3+E4−2E9T+5E3−E4+6E9−6E12W2iE3−E4+6E9−6E12Tςt≤−Γ1Ψ^iΓ1T.Synthesizing inequalities (41), (42), (44), (48), (50), (52), and (54), we can get that
(55)ℒV4ϱst,ξt+ℒV5ϱst,ξt≤ςTt−E2−E3G1iE2−E3T−3E2+E3−2E8G1iE2+E3−2E8T−5E2−E3+6E8−6E11G1iE2−E3+6E8−6E11T−2E2−E8W1iE2−E8T−4E2+2E8−3E11W1iE2+2E8−3E11T−θθ1E2−E3W2iE2−E3T+3E2+E3−2E8W2iE2+E3−2E8T+5E2−E3+6E8−6E11W2iE2−E3+6E8−6E11T−2E3−E9W2iE3−E9T−4E3+2E9−3E12W2iE3+2E9−3E12T−2E4−E10W2iE4−E10T−4E4+2E10−3E13W2iE4+2E10−3E13T−Γ1Ψ^iΓ1T+E1GiE1Tςt.After that, some free-weight matrices are utilized. For any scalarsδ1 and δ2 and the matrix Yi of appropriate dimensions, we have
(56)2δ1ϱ̇TtYi+δ2ϱTtYi−ϱ̇t−Ai−Kiϱt+Bifϱt+β¯Bθifϱt−θt+Diωt=ςTtSymδ1E1YiE1T+δ1E1YiAiE2T−δ1E1YiKiE2T−δ1E1YiBiE6T−δ1β¯E1YiBθiE7T−δ1E1YiDiE14T+δ2E2YiE1T+δ2E2YiAiE2T−δ2E1YiKiE2T−δ2E2YiBiE6T−δ2β¯E2YiBθiE7T−δ2E2YiDiE14Tςt.On the other hand, considering the nonlinear part and according to Lemma1, one has that for any matrices Vki>0, k=1,2:
(57)ϱtfϱtT−L1V1iL2V1i∗−V1iϱtfϱt=ςTt−E2L1V1iE2T+SymE2L2V1iE6T−E6V1iE6Tςt≥0,ϱt−θtfϱt−θtT−L2V2iL2V2i∗−V2iϱt−θtfϱt−θt=ςTt−E4L1V2iE4T+SymE4L2V2iE7T−E7V2iE7Tςt≥0.Furthermore, considering theH∞ synchronization, in view of Definition 1, we can get
(58)ϱ~Ttϱ~t−γ2ωTtωt=ςTtE2CiTCiE2T−γ2E14E14Tςt.Above all, we can get that(59)ℰℒVϱst,ξt+ϱ~Ttϱ~t−γ2ωTtωt≤ςTtΦ¯i+∏θt∑k=1Nρ¯ikPk∏Tθtςt.According to (20), one can obtain that
(60)ℰℒVϱst,ξt+ϱ~Ttϱ~t−γ2ωTtωt<0.Withωt=0, we can get from (59) that ℒVϱst,ξt<0, which implies that the system Σ¯ is stochastically stable.Furthermore, considering the systemΣ¯, from (60) one could obtain the following inequality under zero initial conditions for any γ>0:
(61)ℰ∫0∞ϱ~Ttϱ~t−γ2ωTtωtdt≤ℰ∫0∞ℒVϱst,ξt+ϱ~Ttϱ~t−γ2ωTtωtdt≤0,which means that
(62)Θ=∫0∞ϱ~Ttϱ~t−γ2ωTtωtdt≤0.Therefore, the condition in Definition1 holds.In view of the above, we can conclude that the considered system is stochastically stable and satisfiesH∞ synchronization with the disturbance attenuation γ.From the stochastic stability andH∞ synchronization analysis criterion for the system in Theorem 1, it is not difficult to find that the inequalities may not be easily verified as the existence of TVD term θt and nonlinear delay term θ2t, especially the quadratic term ∏θt∑k=1Nρ¯ikPk∏Tθt. So, we utilize projection lemma and Schur complement to achieve matrix decoupling.Remark 2.
For the quadratic term∏θt∑k=1Nρ¯ikPk∏Tθt, it is not difficult to find that ρ¯ii,i∈ϒ are negative terms while ρ¯ij,i,j∈ϒ,i≠j are positive terms. Therefore, we use different methods to deal with this problem. When i≠j,ρ¯ij>0, we can use Schur complement to handle it. When i=j,ρ¯ii<0, it is quite clear that Schur complement cannot be applied to it directly. So we use projection lemma to cope with it. By using this way, we can achieve the decoupling between nonlinear terms properly.Remark 3.
From stochastic Lyapunov functional (29) as well as inequality (16), sufficient conditions which guarantee the stochastically stable and H∞ synchronization of system Σ¯ are obtained. The term ϱ¯t=ϱTt∫t−θ1tϱTϖdϖ∫t−θ2t−θ1ϱTϖdϖT in V1ϱst,ξt is used to include more information about state delays. The semi-Markovian parameters are introduced to V2ϱst,ξt, V4ϱst,ξt, and V5ϱst,ξt for the consideration of mode-dependent Lyapunov functional. Inequality (16) is used to deal with the integral terms. By using this way, we can obtain sufficient conditions with less conservatism.Theorem 2.
Given scalarsμ, γ>0, 0≤θ1≤θ2<∞, θ=θ2−θ1, δ1>0, δ2>0, and β¯∈0,1, the considered system Σ¯ is stochastically stable and H∞ synchronized; if there exist symmetric matrices Pi∈ℝ3n>0, positive definite matrices Q1i,Q2i,Q3i,R1,R2,G1i,G2i,Z1,Z2, positive definite diagonal matrices V1i,V2i, matrices Yi and Sij∈ℝn,i,j∈ϒ such that for each i∈ϒ,k=1,2, conditions (21), (22), (23), (24), (25), and (26) and the following inequality hold
(63)Φ~ik=Φ¯i+ψik+SymΠkX−XTΠkNiPi∗ρ¯iiPi0∗∗−Pi<0,where
(64)ψi1=symθE1P13iE10T+θE2P23iE10T+θE3P33i−P23iE10T−θE5P33iE10T−θE10M3iE10T−3θE10−E13M3iE10−E13T,ψi2=SymθE1P13iE9T+θE2P23iE9T+θE3P33i−P23iE9T−θE5P33iE9T−θE9M2iE9T−3θE9−E12M2iE9−E12T,Π1=E2θ1E8θE10,Π2=E2θ1E8θE9,Ni=ρ¯i1I,ρ¯i2I,…,ρ¯i,i−1I,ρ¯i,i+1I,…,ρ¯iNI,Pi=diagP1,P2,…,Pi−1,Pi+1,…,PN,
with the other notations are the same as in Theorem1.Proof 2.
From Theorem1, we can know that when conditions (20), (21), (22), (23), (24), (25), and (26) are satisfied, the considered system Σ¯ is stochastically stable. In order to achieve decoupling between nonlinear time-delay terms in (20), further processing will be made for inequality (20).
First of all, the inequality (20) can be rewritten as
(65)NPTΞiNP<0,where
(66)NP=I14nΠθtT,Ξi=Φ¯i+ψiθt+Πθt∑k=1,k≠iNρ¯ikPkΠTθt00ρ¯iiPi,From inequality (65) we can also get from quadratic form knowledge that Ξi<0, then we can obtain the following inequality
(67)NQTΞiNQ<0,where NQ=0I3nT.
We defineP=ΠTθt−I14n,Q=I14n0. It is not difficult to know that PNP=0, QNQ=0. Therefore, NP and NQ are the elements of the nullspace of P and Q.
Then, by using projection lemma to inequalities (65) and (67), we can know that
(68)Ξi+symPTXQ<0,which means
(69)Φ¯i+ψiθt+Πθt∑k=1,k≠iNρ¯ikPkΠTθt+symΠθtX−XT∗ρ¯iiPi<0.
Then, we use Schur complement to deal with the coupling termΠθt∑k=1,k≠iNρ¯ikPkΠTθt in (69) and we can obtain that
(70)Φ¯i+ψiθt+symΠθtX−XTΠθtNiPi∗ρ¯iiPi0∗∗−Pi<0,where
(71)Ni=ρ¯i1I,ρ¯i2I,…,ρ¯i,i−1I,ρ¯i,i+1I,…,ρ¯iNI,Pi=diagP1,P2,…,Pi−1,Pi+1,…,PN.By using the above decoupling method, we achieve that there is only linear TVD termθt in inequality (70). As 0≤θ1≤θt≤θ2<∞, therefore inequality (70) is satisfied under the condition that θt=θ1,θt=θ2,, respectively. It can guarantee that once inequality (70) is satisfied under the condition that θ1≤θt≤θ2, then the correctness of inequality (70) can be readily deduced as (63) holds. So, (20) holds if inequality (63) is satisfied. This completes the proof.Since a stochastic stability andH∞ synchronization analysis criterion for the system in Theorem 2 is developed, the procedure of H∞ synchronization can be developed in Theorem 3 in the further analysis. Therefore, in this section we derive H∞ synchronization conditions for the drive-response dynamic systems. The main result about the design of a desired controller will be presented, and the controller gain will be presented.Theorem 3.
Fixed scalarsμ, γ>0, 0≤θ1≤θ2<∞, θ=θ2−θ1, δ1>0, δ2>0, and β¯∈0,1, the considered system Σ¯ is stochastically stable and H∞ synchronized, if there exist symmetric matrices Pi∈ℝ3n>0, positive definite matrices Q1i,Q2i,Q3i,R1,R2,G1i,G2i,Z1,Z2, positive definite diagonal matrices V1i,V2i, matrices Yi and K~i, and Sij∈ℝn,i,j∈ϒ, such that for each i∈ϒ, and k=1,2,, the conditions (21), (22), (23), (24), (25), and (26) and the following inequality hold:
(72)Φˇi+ψik+symΠkX−XTΠkNiPi∗ρ¯iiPi0∗∗−Pi<0,where
(73)Φˇi=SymE1P11iE2T+θ1E1P12iE8T+E2P12iE2T−E3P12iE2T+θ1E2P22iE8T−θ1E3P22iE8T+E3P13iE2T−E5P13iE2T+θ1E3P23iE8T−θ1E5P23iE8T+E2QiE2T−1−μE4Q3iE4T−E5Q2iE5T+E1GiE1T−E2−E3G1iE2−E3T−3E2+E3−2E8G1iE2+E3−2E8T−5E2−E3+6E8−6E11G1iE2−E3+6E8−6E11T−2E2−E8W1iE2−E8T−4E2+2E8−3E11W1iE2+2E8−3E11T−θθ1E2−E3W2iE2−E3T−3θθ1E2+E3−2E8W2iE2+E3−2E8T−5θθ1E2−E3+6E8−6E11W2iE2−E3+6E8−6E11T−2E3−E9W2iE3−E9T−4E3+2E9−3E12W2iE3+2E9−3E12T−2E4−E10W2iE4−E10T−4E4+2E10−3E13W2iE4+2E10−3E13T−Γ1Ψ^iΓ1T−Symδ1E1YiE1T+δ1E1YiAiE2T−δ1E1K~iE2T−δ1E1YiBiE6T−δ1β¯E1YiBθi×E7T−δ1E1YiDiE14T+δ2E2YiE1T+δ2E2YiAiE2T−δ2E1K~iE2T−δ2E2YiBiE6T−δ2β¯E2Yi×BθiE7T−δ2E2YiDiE14T−E2L1V1iE2T+E4L1V2iE4T+SymE2L2V1iE6T+E4L2V2iE7T−E6V1iE6T−E7V2iE7T−E2CiTCiE2T−γ2E14E14T,then the system Σ¯ is stochastically stable. In this regard, the available gain of the controller can be calculated by
(74)Ki=Yi−1K~i,i∈ϒ.Proof 3.
We define(75)K~i=YiKi.
By using this way, the freedom variableYi and the controller gain Ki can be merged, which makes sure inequality (72) can be solved by dealing with a convex optimization problem. This completes the proof.


## 4. Numerical Examples

In this section, two numerical examples are given for demonstrating the feasibility and validity of the proposed method. In the first example, the existence of the desiredH∞ synchronization controller is verified. The second example gives a set of figures to reveal the correctness of our method.Example 1.
For the proposed systemΣ and Σ^ with three modes i=1,2,3, the parameters are chosen as follows:
(76)A1=2.20001.80,A2=0.20003.40,A3=1.00000.80,B1=0.800.40−0.200.10,B2=0.701.100.20−0.05,B3=1.001.00−1.00−1.00,Bθ1=1.201.00−0.200.30,Bθ2=−2.40−4.80−0.322.00,Bθ3=0.881.001.001.00,C1=0.10000.10,C2=0.20−0.30−0.300.20,C3=−0.801.001.000.60,D1=0.20−0.10−0.100.20,D2=0.20−0.32−0.320.20,D3=0.100.100.100.10.Considering the activation functionsfis,i=1,2,3 with the parameters lq−=0 and lq+=0.4, then it is easy to get that
(77)L1=0000,L2=0.2000.2.The expectation ofβt is taken as β¯=0.6. Assuming that It=0, and the TVDs θt=1.1+sin0.5t, then we can obtain that 0.1≤θt≤2.1 and θ̇t≤0.5, which means θ1=0.1, θ2=2.1, and μ=0.5. Besides, the transition rates of ξt are selected as the same as in [24], where the probability distribution function is also chosen as Weibull distribution. Therefore, the corresponding mathematical expectation of Λ¯σ is
(78)ℰΛ¯σ=−1.77240.88620.88621.7725−3.54501.77252.65872.6587−5.3174.We setδ1=1, δ2=2, and γ2=0.4. Then, solving LMIs (21), (22), (23), (24), (25), (26) and (72) by using the LMI control toolbox, the H∞ synchronization controller gain matrixes can be obtained as follows:
(79)K1=−8.63983.10223.1631−7.3662,K2=−28.0246−13.5891−13.6663−17.7993,K3=−10.4938−2.2998−2.7691−9.9076.Remark 4.
When we setβ¯=1 and D1=D2=D3=0 and do not consider the H∞ performance index and the other parameters remain the same, then our system is the same as Example 13 in [24]. In this condition, we can get the controller gain matrixes as follows:
(80)K1=−1.5988−0.07530.0621−0.3183,K2=−6.93601.13151.41770.8950,K3=−1.6030−0.34020.0094−1.3144.
Then, we can find that whenμ=0.5, θ1=0.1, the maximum allowable upper delay bound θ2 in [24] is 0.984, while we set θ2=2.1. The desired controller gains can also be obtained in this paper. This indicates that our method is superior to [24] and can lead to less conservatism for time-delay systems.Example 2.
The parameters of the systemsΣ and Σ^ are given with N=3 as follows:
(81)A1=1.00001.00,A2=1.10000.90,A3=1.20000.80,B1=1.40−0.301.051.50,B2=1.75−0.400.951.65,B3=1.25−0.251.101.30,Bθ1=−2.201.00−0.80−2.10,Bθ2=−2.701.10−0.70−2.30,Bθ3=−2.801.20−0.50−2.10,C1=0.10000.10,C2=0.20−0.30−0.300.20,C3=−0.801.001.000.60,D1=0.20−0.10−0.100.20,D2=−0.20−0.32−0.32−0.20,D3=0.100.100.100.10.The activation functions are assumed asfis=tanhs,i∈ϒ with the parameters lq−=0 and lq+=1; then, it is easy to get that
(82)L1=0000,L2=0.5000.5.The expectation ofβt is chosen as β¯=1. Assuming that It=0,ωt=1/1+t21/1+t2T and the TVDs θt=1+0.3sin2t, then we can obtain that θ1=0.7, θ2=1.3, and μ=0.6. The mathematical expectation of the transition rate matrix is taken as the same as Example 1. Then, we set δ1=1, δ2=5, γ2=0.4, and we can get the desired controller gain matrixes as follows:
(83)K1=−14.8258−0.4952−0.4985−14.6980,K2=−22.0351−1.4217−0.7621−20.2739,K3=−24.5083−0.68440.2891−21.1654.Under the obtained gain matrixes andx0=0.20.25T and x^0=0.05−0.15T, we simulate the system and obtain a set of figures. Figure 1 shows the mode transitions ξt of the system. Figures 2 and 3 show the state vector’s behaviors of systems Σ and Σ^, respectively. Figure 4 shows the state responses of the system Σ¯. In view of Figure 4, we can observe that although under different initial conditions, the error between systems Σ and Σ^ gradually tends to zero which indicates that the gain matrixes derived from this paper can guarantee the synchronization between the systems Σ and Σ^ effectively. Therefore, it confirms the validity of the developed method.Figure 1
Modes of transitions of semi-MJNNs.Figure 2
The state vector’s behaviors of the drive system (1).Figure 3
The state vector’s behaviors of the response system (7).Figure 4
The synchronization error between the drive system and the response system.


## 5. Conclusion

In this paper, theH∞ synchronization problem for semi-MJNNs, where the randomly occurring TVDs have been considered to make the neural networks under consideration more practical, has been investigated. By constructing an appropriate semi-Markovian Lyapunov-Krasovskii functional, combining with the derivation of an infinitesimal generator for the Lyapunov functional and the sufficient H∞ synchronization condition for semi-MJNNs has been established. To deal with the TVD items, some improved inequalities, together with Schur complement lemma and projection lemma, have been introduced. By using a linearization technique, the desired controller has been designed and the existence of the desired controller can be verified by the feasibility of a set of LMIs. Finally, two meaningful examples have been given to validate the feasibility and validity of the developed approach. Considering that the conditions established in this paper are sufficient, how to reduce the conservatism of current results will therefore be our future work. In addition, how to extend our results to cope with the controller design problem for more complex networks, such as genetic regulatory networks, will also be one of our future research directions.



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