# Passivity and Synchronization of Multiple Multi-Delayed Neural Networks via Impulsive Control

Authors: Yong Wang; Zhichun Yang; Tonglai Liu; Hong-An Tang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

This paper is concerned with the passivity and synchronization for multiple multi-delayed neural networks (MMDNNs) under impulsive control. To ensure the passivity, input strict passivity, and output strict passivity in MMDNNs, a suitable impulsive controller is designed. Moreover, an impulsive time-dependent Lyapunov functional is exploited to obtain the synchronization criterion of MMDNNs, where the criterion is formulated by linear matrix inequalities. Numerical examples are given to verify the validity of the theoretical results.


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## Body

## 1. Introduction

Neural networks (NNs) have received extensive attention due to their successful applications in vision system [1], associative memory [2], pattern recognition [3], and image compression [4]. NNs always require stability, which is a prerequisite for many applications. Therefore, the stability of NNs has become a hot issue in recent years [5–11]. Zhang et al. [7] considered some asymptotic stability criteria of NNs with distributed delays based on Lyapunov–Krasovskii functionals. By improving the auxiliary polynomial-based functions, Li et al. [9] solved the stability problem in delayed NNs. Since the main property of passivity is to keep the system internally stable, some researchers have focused on the passivity for NNs [12–17]. Lian et al. [14] proposed a kind of switched NNs with time-varying delays and stochastic disturbances, and the passivity of networks was analyzed by designing a state-dependent switching law and a hysteresis switching law. Cao et al. [17] addressed the robust passivity issue of uncertain NNs with additive time-varying delays and leakage delay, and a general activation function was utilized to ensure that the proposed network model was passive.In addition, multi-weighted network models [18–21] can be used to describe many real-world networks including public transportation road networks, communication networks, social networks, and so forth. Recently, some researchers have investigated the dynamical behaviors of complex networks with multiple weights [20, 21]. Wang et al. [20] concentrated on two types of multi-weighted complex networks with several different weights between two nodes, and sufficient conditions ensuring the synchronization were developed by utilizing the pinning control method. Under the help of pinning adaptive control techniques, the passivity of multi-weighted complex networks with different dimensions of input and output was discussed in [21]. However, a few authors have considered the stability and passivity of NNs with multiple delays.Compared with continuous control, there are many advantages of impulsive control strategies, which include low maintenance costs, high reliability, ease of installation, and high efficiency [22]. So far, a series of investigations in regard to the stability [23–28] and passivity [29–31] for impulsive NNs have been reported. Zhu and Cao [25] dealt with the stability problem of impulsive stochastic BAM NNs with mixed time delays and Markovian jump parameters by exploiting Itô’s formula and stochastic analysis theory. According to the comparison principle and compression mapping theorem, the global exponential stability of periodic solution was considered in an array of Cohen–Grossberg NNs with time-varying delays and periodic coefficients via impulsive control in [26]. Zhou [31] took into account the passivity of recurrent NNs with multiproportional delays and impulse. But very few authors have discussed the stability and passivity problems of multi-delayed NNs under impulsive control.At present, most of the literatures with respect to identifying network structures from the observation and control of dynamical behavior are concentrated in a single neural network [5, 7, 12, 13]. In practical applications, some tasks are difficult to complete by a single neural network; even if the network can accomplish these tasks, it may result in high costs. But multiple NNs can solve some difficult problems through cooperation with each other so that the cost can be reduced. Recently, some cooperative control problems [32–35] involving passivity and synchronization [36–38] have been concerned in multiple NNs. Unfortunately, as far as we know, the passivity and synchronization of multiple multi-delayed NNs (MMDNNs) via impulsive control have never been considered. Inspired by the above discussion, this paper aims to further study the passivity and global exponential synchronization of MMDNNs by using impulsive control techniques. The contributions of this paper are as follows. First, compared with the traditional impulse-time-independent Lyapunov functional, the impulse-time-dependent feature of the Lyapunov functional in this paper can capture more dynamical behaviors of MMDNNs. Second, with the help of some piecewise linear functions and inequality techniques, the passivity problems of MMDNNs are addressed via impulsive control. Third, a newly designed impulsive controller is applied to synchronize the proposed networks.


## 2. Preliminaries

### 2.1. Notations

LetN=1,2,…,N, ℕ=0,1,2,…. The fixed moments tk satisfy 0=t0<t1<t2<⋯<tk<⋯ and limk⟶+∞tk=+∞,k∈ℕ. λmin⋅λmax⋅ denotes the smallest (largest) eigenvalue of a matrix. For any ξt=ξ1t,ξ2t,…,ξNtT∈ℝN, ξt=∑i=1Nξi2t1/2, ξtτ=sup−τ≤ϵ≤0ξt+ϵ.The notationSν1,ν2 represents the set of impulse time sequences tk satisfying ν2≥tk+1−tk≥ν1>0 for all k∈ℕ, in which ν1,ν2∈ℝ. For a given impulse time sequence tk∈Sν1,ν2, some piecewise linear functions can be defined as follows:(1)ϕt=1tk+1−tk,t∈tk,tk+1,k∈ℕ,ϕ1t=t−tkϕt,t∈tk,tk+1,k∈ℕ,1,t0>t.Letϕ2t=1−ϕ1t. It is obvious to see that 0,1∋ϕ1t, 0,1∋ϕ2t, for ℝ∋t, and ϕ1tk=ϕ2tk−=0, ϕ1tk−=ϕ2tk=1,ℕ∋k.Fortk∈Sν1,ν2, ϕt can be written as(2)ϕt=ψ1tν1+ψ2tν2,where(3)ψ1t=1−ψ2t,ψ2t=ϕt−1/ν21/ν1−1/ν2,ifν1≠ν2,1,ifν1=ν2.Definition 1.
(see [21]). A system is said to be passive with output yt∈ℝϵN and input vt∈ℝnN, if there is a storage function S:0,+∞⟶0,+∞ and a matrix A∈ℝϵN×nN satisfying(4)Stϱ−St0≤∫t0tϱyTtAvtdt,for any tϱ,t0∈0,+∞ and t0≤tϱ.Definition 2.
(see [21]). A system is said to be strictly passive with output yt∈ℝϵN and input vt∈ℝnN, if there is a storage function S:0,+∞⟶0,+∞, matrices A∈ℝϵN×nN, 0≤A1∈ℝnN×nN, 0≤A2∈ℝϵN×ϵN, and eigenvalues λmA1+λmA2>0 satisfying(5)Stϱ−St0≤∫t0tϱyTtAvt−vTtA1vt−yTtA2ytdt,for any tϱ,t0∈0,+∞ and t0≤tϱ.
The system is input-strictly passive if0<A1 and output-strictly passive if 0<A2.


## 2.1. Notations

LetN=1,2,…,N, ℕ=0,1,2,…. The fixed moments tk satisfy 0=t0<t1<t2<⋯<tk<⋯ and limk⟶+∞tk=+∞,k∈ℕ. λmin⋅λmax⋅ denotes the smallest (largest) eigenvalue of a matrix. For any ξt=ξ1t,ξ2t,…,ξNtT∈ℝN, ξt=∑i=1Nξi2t1/2, ξtτ=sup−τ≤ϵ≤0ξt+ϵ.The notationSν1,ν2 represents the set of impulse time sequences tk satisfying ν2≥tk+1−tk≥ν1>0 for all k∈ℕ, in which ν1,ν2∈ℝ. For a given impulse time sequence tk∈Sν1,ν2, some piecewise linear functions can be defined as follows:(1)ϕt=1tk+1−tk,t∈tk,tk+1,k∈ℕ,ϕ1t=t−tkϕt,t∈tk,tk+1,k∈ℕ,1,t0>t.Letϕ2t=1−ϕ1t. It is obvious to see that 0,1∋ϕ1t, 0,1∋ϕ2t, for ℝ∋t, and ϕ1tk=ϕ2tk−=0, ϕ1tk−=ϕ2tk=1,ℕ∋k.Fortk∈Sν1,ν2, ϕt can be written as(2)ϕt=ψ1tν1+ψ2tν2,where(3)ψ1t=1−ψ2t,ψ2t=ϕt−1/ν21/ν1−1/ν2,ifν1≠ν2,1,ifν1=ν2.Definition 1.
(see [21]). A system is said to be passive with output yt∈ℝϵN and input vt∈ℝnN, if there is a storage function S:0,+∞⟶0,+∞ and a matrix A∈ℝϵN×nN satisfying(4)Stϱ−St0≤∫t0tϱyTtAvtdt,for any tϱ,t0∈0,+∞ and t0≤tϱ.Definition 2.
(see [21]). A system is said to be strictly passive with output yt∈ℝϵN and input vt∈ℝnN, if there is a storage function S:0,+∞⟶0,+∞, matrices A∈ℝϵN×nN, 0≤A1∈ℝnN×nN, 0≤A2∈ℝϵN×ϵN, and eigenvalues λmA1+λmA2>0 satisfying(5)Stϱ−St0≤∫t0tϱyTtAvt−vTtA1vt−yTtA2ytdt,for any tϱ,t0∈0,+∞ and t0≤tϱ.
The system is input-strictly passive if0<A1 and output-strictly passive if 0<A2.


## 3. Passivity of MMDNNs via Impulsive Control

### 3.1. Network Model

The MMDNNs are considered as follows:(6)z˙it=−Dzit+∑ι=1pWιgιzit−τι+J+vit,in which i∈N; zit=zi1t,zi2t,…,zintT∈ℝn is the state vector of node i; D=diagd1,d2,…,dn>0; Wι∈ℝn×n; gιzit−τι=g1ιzi1t−τι,g2ιzi2t−τι,…,gnιzint−τιT∈ℝn; J=J1,J2,…,JnT∈ℝn; vit∈ℝn means the input vector of node i; and τι is the transmission delay and 0≤τι≤τ.In this paper, there isℝ∋ρς>0ς=1,2,…,n such that(7)gςιθ1−gςιθ2≤ρςιθ1−θ2.for any θ1,θ2∈ℝ. Let Λι=diagρ1ι2,ρ2ι2,…,ρnι2.Suppose thatz∗=z1∗,z2∗,…,zn∗T∈ℝn is an equilibrium solution of an isolated node of the MMDNNs (6). Then, one gets(8)−Dz∗+J+∑ι=1pWιgιz∗=0.For the MMDNNs (6), construct the following impulsive controller:(9)uitk=∑j=1NBijΦzjtk−,k∈ℕ,i∈N,in which 0<Φ∈ℝn×n; zitk−=limt⟶tk−zit; B=BijN×N means the impulsive coupling matrix, where Bij is described as follows: if there is a link from node i to node j, then ; otherwise, Bij=0i≠j; and(10)Bii=−∑j=1j≠iNBij,i∈N.It is derived from (6) and (9) that(11)z˙it=−Dzit+∑ι=1pWιgιzit−τι+J+vit,t≠tk,Ikzit=∑j=1NBijΦzjtk−,k∈ℕ,i∈N,where Ikzit=zitk+−zitk−, zitk+=limt⟶tk+zit. Suppose zitk=zitk+.Letζit=zit−z∗. Then, by (8) and (11), we acquire(12)ζ˙it=−Dζit+∑ι=1pWιgιzit−τι−gιz∗+vit,t≠tk,Ikζit=∑j=1NBijΦζjtk−,k∈ℕ,i∈N.The output vectoryit∈ℝϵ of the MMDNNs (12) is chosen as(13)yit=C1ζit+C2vit,in which C1∈ℝϵ×n and C2∈ℝϵ×n are known matrices.


### 3.2. Passivity Criteria

Theorem 1.
Under the impulsive controller (9), the MMDNNs (12) are passive over Sν1,ν2 if there exist matrices 0<F1∈ℝnN×nN, 0<F2∈ℝnN×nN, and A∈ℝϵN×nN and a scalar γ∈0,1 such that(14)Γ1rh=Ω1rhΩ2rΩ2rTΩ3≤0,(15)−γF1InN+B⊗ΦTF2F2InN+B⊗Φ−F2≤0,in which r=1,2,h=1,2, Ω1rh=1/νhF1−F2−FrIN⊗D−IN⊗DFr+∑ι=1pFrIN⊗WιWιTFr+IN⊗Λι, Ω2r=Fr−IN⊗C1TA, and Ω3=−ATIN⊗C2−IN⊗C2TA.Proof.
LetFt=∑r=12ϕrtFr,Ω1t=ϕtF1−F2−IN⊗DFt−FtIN⊗D+∑ι=1pFtIN⊗WιWιTFt+IN⊗Λι,Ω2t=Ft−IN⊗C1TA.
Then, by (14), we have(16)Γ1t≜∑r,h=12ϕrtψhtΓ1rh=Ω1tΩ2tΩ2TtΩ3≤0.
The impulse-time-dependent Lyapunov functional for the MMDNNs (12) is considered as follows:(17)V1t=ζTtFtζt+∑ι=1p∫t−τιtζTsIN⊗Λιζsds,where ζt=ζ1Tt,ζ2Tt,…,ζNTtT.
Then,(18)V˙1t=ζTtϕtF1−F2−IN⊗DFt−FtIN⊗Dζt+2ζTtFt∑ι=1pIN⊗Wι×g^ιzt−τι−g^ιz^∗+vt−∑ι=1pζTt−τιIN⊗Λιζt−τι+∑ι=1pζTtIN⊗Λιζt,where z^∗=z∗T,z∗T,…,z∗TT,g^ιzt−τι=gιz1t−τιT,gιz2t−τιT,…,gιzNt−τιTT,g^ιz^∗=gιz∗T,gιz∗T,…,gιz∗TT,vt=v1Tt,v2Tt,…,vNTtT.
Obviously,(19)2ζTtFtIN⊗Wιg^ιzt−τι−g^ιz^∗≤ζTtFtIN⊗WιWιTFtζt+ζTt−τιIN⊗Λιζt−τι.
Substituting (19) into (18) yields(20)V˙1t≤ζTtϕtF1−F2−IN⊗DFt−Ft×IN⊗D+∑ι=1pFtIN⊗WιWιTFt+∑ι=1pIN⊗Λιζt+2ζTtFtvt.
From (13), (16), and (20), one obtains(21)V˙1t−2yTtAvt≤ζTtϕtF1−F2−IN⊗DFt−FtIN⊗D+∑ι=1p​FtIN⊗WιWιTFt+IN⊗Λιζt+vTt−ATIN⊗C2−IN⊗C2TAvt+2ζTtFt−IN⊗C1TAvt=ϖ1TtΩ1tΩ2tΩ2TtΩ3ϖ1t≤0,t∈tk,tk+1,ℕ∋k,in which ϖ1t=ζTt,vTtT.
Integrating (21) with respect to t from t0 to tϱtη≤tϱ<tη+1,η=0,1,…, we acquire(22)∫t0tϱV˙1t−2yTtAvtdt=∫tηtϱV˙1tdt+∑σ=1η∫tσ−1tσV˙1tdt−2∫t0tϱyTtAvtdt=V1tϱ−V1tη+∑σ=1ηV1tσ−−V1tσ−1−2∫t0tϱyTtAvtdt=V1tϱ−V1t0+∑σ=1ηV1tσ−−V1tσ−2∫t0tϱyTtAvtdt,in which tϱ,t0∈0,+∞ and t0≤tϱ.
At the impulse timetk,k∈ℕ, according to the definition of Ft, one gets(23)Ftk−=F1,Ftk=F2.
It is found from (12), (15), and (17) that(24)V1tk=ζTtkFtkζtk+∑ι=1p∫tk−τιtkζTsIN⊗Λιζsds=ζTtk−InN+B⊗ΦTF2InN+B⊗Φζtk−+∑ι=1p∫tk−τιtkζTsIN⊗Λιζsds≤γζTtk−F1ζtk−+∑ι=1p∫tk−τιtkζTsIN⊗Λιζsds≤ζTtk−Ftk−ζtk−+∑ι=1p∫tk−τιtkζTsIN⊗Λιζsds=V1tk−,ℕ∋k.
By (21)–(24), we have(25)V1tϱ−V1t0≤2∫t0tϱyTtAvtdt.
Hence,(26)Stϱ−St0≤∫t0tϱyTtAvtdt,for any tϱ,t0∈0,+∞ and t0≤tϱ, in which St=V1t/2. □Theorem 2.
Under the impulsive controller (9), the MMDNNs (12) are input-strictly passive over Sν1,ν2 if there exist matrices 0<F1∈ℝnN×nN, 0<F2∈ℝnN×nN, A∈ℝϵN×nN, and 0<A1∈ℝnN×nN and a scalar γ∈0,1 such that(27)Γ2rh=Ω1rhΩ2rΩ2rTΩ4≤0,(28)−γF1InN+B⊗ΦTF2F2InN+B⊗Φ−F2≤0,in which r=1,2,h=1,2, Ω1rh=1/νhF1−F2−FrIN⊗D−IN⊗DFr+∑ι=1pFrIN⊗WιWιTFr+IN⊗Λι, Ω2r=Fr−IN⊗C1TA, Ω4=2A1−ATIN⊗C2−IN⊗C2TA.Proof.
LetFt=∑r=12ϕrtFr,Ω1t=ϕtF1−F2−IN⊗DFt−FtIN⊗D+∑ι=1pFtIN⊗WιWιTFt+IN⊗Λι,Ω2t=Ft−IN⊗C1TA.
Then, from (27), one obtains(29)Γ2t≜∑r,h=12​ϕrtψhtΓ2rh=Ω1tΩ2tΩ2TtΩ4≤0.
Selecting the sameV1 as (17) for the networks (12) and using (29), one has(30)V˙1t−2yTtAvt+2vTtA1vt≤ζTtϕtF1−F2−IN⊗DFt−FtIN⊗D+∑ι=1p​FtIN⊗WιWιTFt+IN⊗Λιζt+vTt2A1−ATIN⊗C2−IN⊗C2TAvt+2ζTtFt−IN⊗C1TAvt=ϖ1TtΩ1tΩ2tΩ2TtΩ4ϖ1t≤0,t∈tk,tk+1,ℕ∋k,in which ϖ1t=ζTt,vTtT.
Integrating (30) with respect to t from t0 to tϱtη≤tϱ<tη+1,η=0,1,…, we acquire(31)∫t0tϱV˙1t−2yTtAvt+2vTtA1vtdt=V1tϱ−V1t0+∑σ=1ηV1tσ−−V1tσ−2∫t0tϱyTtAvtdt−vTtA1vtdt,in which tϱ,t0∈0,+∞ and t0≤tϱ.
At the impulse timetk,k∈ℕ, according to the definition of Ft, one acquires(32)Ftk−=F1,Ftk=F2.
On the basis of (12), (17), and (28), we get(33)V1tk≤V1tk−,ℕ∋k.
Considering (30)–(33), it is obtained that(34)V1tϱ−V1t0≤2∫t0tϱ​yTtAvt−vTtA1vtdt.
Thus,(35)Stϱ−St0≤∫t0tϱyTtAvtdt−∫t0tϱvTtA1vtdt,for any tϱ,t0∈0,+∞ and t0≤tϱ, in which St=V1t/2.Theorem 3.
Under the impulsive controller (9), the MMDNNs (12) are output-strictly passive over Sν1,ν2 if there exist matrices 0<F1∈ℝnN×nN, 0<F2∈ℝnN×nN, A∈ℝϵN×nN, and 0<A2∈ℝϵN×ϵN and a scalar γ∈0,1 such that(36)Γ3rh=Ω5rhΩ6rΩ6rTΩ7≤0,(37)−γF1InN+B⊗ΦTF2F2InN+B⊗Φ−F2≤0,in which r=1,2,h=1,2, Ω5rh=1/νhF1−F2−FrIN⊗D−IN⊗DFr+∑ι=1pFrIN⊗WιWιTFr+IN⊗Λι+2IN⊗C1TA2IN⊗C1, Ω6r=2IN⊗C1TA2IN⊗C2+Fr−IN⊗C1TA, and Ω7=−ATIN⊗C2−IN⊗C2TA+2IN⊗C2TA2IN⊗C2.Proof.
LetFt=∑r=12ϕrtFr,Ω5t=−IN⊗DFt+ϕtF1−F2+∑ι=1pFtIN⊗WιWιTFt+IN⊗Λι+2IN⊗C1TA2IN⊗C1−FtIN⊗D,Ω6t=2IN⊗C1TA2IN⊗C2+Ft−IN⊗C1TA.
Then, it is derived from (36) that(38)Γ3t≜∑r,h=12​ϕrtψhtΓ3rh=Ω5tΩ6tΩ6TtΩ7≤0.
Selecting the sameV1 as (17) for the networks (12) and utilizing (38), we get(39)V˙1t−2yTtAvt+2yTtA2yt≤ζTtϕtF1−F2−IN⊗DFt−FtIN⊗D+2IN⊗C1TA2IN⊗C1+∑ι=1pIN⊗Λι+Ft×IN⊗WιWιTFtζt+2ζTt2IN⊗C1TA2IN⊗C2+Ft−IN⊗C1TAvt+vTt−ATIN⊗C2−IN⊗C2TA+2IN⊗C2TA2IN⊗C2vt=ϖ1TtΩ5tΩ6tΩ6TtΩ7ϖ1t≤0,t∈tk,tk+1,ℕ∋k,in which ϖ1t=ζTt,vTtT.
Integrating (39) with respect to t from t0 to tϱtη≤tϱ<tη+1,η=0,1,…, one has(40)∫t0tϱV˙1t−2yTtAvt+2yTtA2ytdt=V1tϱ−V1t0+∑σ=1ηV1tσ−−V1tσ−2∫t0tϱyTtAvt−yTtA2ytdt,in which tϱ,t0∈0,+∞ and t0≤tϱ.
At the impulse timetk,k∈ℕ, according to the definition of Ft, we derive(41)Ftk−=F1,Ftk=F2.
It is obtained from (12), (17), and (37) that(42)V1tk≤V1tk−,ℕ∋k.
Using (39)–(42), we can acquire(43)V1tϱ−V1t0≤2∫t0tϱ​yTtAvt−yTtA2ytdt.
Therefore,(44)Stϱ−St0≤∫t0tϱyTtAvtdt−∫t0tϱyTtA2ytdt,for any tϱ,t0∈0,+∞ and t0≤tϱ, in which St=V1t/2.Remark 1.
In recent years, as an effective method to study synchronization, the passivity of NNs has been investigated by some researchers [12–17]. Impulsive control is a popular control method among control methods due to its reliability, low cost, and flexibility [22]. Using impulsive control strategies, some authors have focused on the passivity and synchronization of NNs. However, the passivity and synchronization of MMDNNs have not been considered under impulsive control.


## 3.1. Network Model

The MMDNNs are considered as follows:(6)z˙it=−Dzit+∑ι=1pWιgιzit−τι+J+vit,in which i∈N; zit=zi1t,zi2t,…,zintT∈ℝn is the state vector of node i; D=diagd1,d2,…,dn>0; Wι∈ℝn×n; gιzit−τι=g1ιzi1t−τι,g2ιzi2t−τι,…,gnιzint−τιT∈ℝn; J=J1,J2,…,JnT∈ℝn; vit∈ℝn means the input vector of node i; and τι is the transmission delay and 0≤τι≤τ.In this paper, there isℝ∋ρς>0ς=1,2,…,n such that(7)gςιθ1−gςιθ2≤ρςιθ1−θ2.for any θ1,θ2∈ℝ. Let Λι=diagρ1ι2,ρ2ι2,…,ρnι2.Suppose thatz∗=z1∗,z2∗,…,zn∗T∈ℝn is an equilibrium solution of an isolated node of the MMDNNs (6). Then, one gets(8)−Dz∗+J+∑ι=1pWιgιz∗=0.For the MMDNNs (6), construct the following impulsive controller:(9)uitk=∑j=1NBijΦzjtk−,k∈ℕ,i∈N,in which 0<Φ∈ℝn×n; zitk−=limt⟶tk−zit; B=BijN×N means the impulsive coupling matrix, where Bij is described as follows: if there is a link from node i to node j, then ; otherwise, Bij=0i≠j; and(10)Bii=−∑j=1j≠iNBij,i∈N.It is derived from (6) and (9) that(11)z˙it=−Dzit+∑ι=1pWιgιzit−τι+J+vit,t≠tk,Ikzit=∑j=1NBijΦzjtk−,k∈ℕ,i∈N,where Ikzit=zitk+−zitk−, zitk+=limt⟶tk+zit. Suppose zitk=zitk+.Letζit=zit−z∗. Then, by (8) and (11), we acquire(12)ζ˙it=−Dζit+∑ι=1pWιgιzit−τι−gιz∗+vit,t≠tk,Ikζit=∑j=1NBijΦζjtk−,k∈ℕ,i∈N.The output vectoryit∈ℝϵ of the MMDNNs (12) is chosen as(13)yit=C1ζit+C2vit,in which C1∈ℝϵ×n and C2∈ℝϵ×n are known matrices.


## 3.2. Passivity Criteria

Theorem 1.
Under the impulsive controller (9), the MMDNNs (12) are passive over Sν1,ν2 if there exist matrices 0<F1∈ℝnN×nN, 0<F2∈ℝnN×nN, and A∈ℝϵN×nN and a scalar γ∈0,1 such that(14)Γ1rh=Ω1rhΩ2rΩ2rTΩ3≤0,(15)−γF1InN+B⊗ΦTF2F2InN+B⊗Φ−F2≤0,in which r=1,2,h=1,2, Ω1rh=1/νhF1−F2−FrIN⊗D−IN⊗DFr+∑ι=1pFrIN⊗WιWιTFr+IN⊗Λι, Ω2r=Fr−IN⊗C1TA, and Ω3=−ATIN⊗C2−IN⊗C2TA.Proof.
LetFt=∑r=12ϕrtFr,Ω1t=ϕtF1−F2−IN⊗DFt−FtIN⊗D+∑ι=1pFtIN⊗WιWιTFt+IN⊗Λι,Ω2t=Ft−IN⊗C1TA.
Then, by (14), we have(16)Γ1t≜∑r,h=12ϕrtψhtΓ1rh=Ω1tΩ2tΩ2TtΩ3≤0.
The impulse-time-dependent Lyapunov functional for the MMDNNs (12) is considered as follows:(17)V1t=ζTtFtζt+∑ι=1p∫t−τιtζTsIN⊗Λιζsds,where ζt=ζ1Tt,ζ2Tt,…,ζNTtT.
Then,(18)V˙1t=ζTtϕtF1−F2−IN⊗DFt−FtIN⊗Dζt+2ζTtFt∑ι=1pIN⊗Wι×g^ιzt−τι−g^ιz^∗+vt−∑ι=1pζTt−τιIN⊗Λιζt−τι+∑ι=1pζTtIN⊗Λιζt,where z^∗=z∗T,z∗T,…,z∗TT,g^ιzt−τι=gιz1t−τιT,gιz2t−τιT,…,gιzNt−τιTT,g^ιz^∗=gιz∗T,gιz∗T,…,gιz∗TT,vt=v1Tt,v2Tt,…,vNTtT.
Obviously,(19)2ζTtFtIN⊗Wιg^ιzt−τι−g^ιz^∗≤ζTtFtIN⊗WιWιTFtζt+ζTt−τιIN⊗Λιζt−τι.
Substituting (19) into (18) yields(20)V˙1t≤ζTtϕtF1−F2−IN⊗DFt−Ft×IN⊗D+∑ι=1pFtIN⊗WιWιTFt+∑ι=1pIN⊗Λιζt+2ζTtFtvt.
From (13), (16), and (20), one obtains(21)V˙1t−2yTtAvt≤ζTtϕtF1−F2−IN⊗DFt−FtIN⊗D+∑ι=1p​FtIN⊗WιWιTFt+IN⊗Λιζt+vTt−ATIN⊗C2−IN⊗C2TAvt+2ζTtFt−IN⊗C1TAvt=ϖ1TtΩ1tΩ2tΩ2TtΩ3ϖ1t≤0,t∈tk,tk+1,ℕ∋k,in which ϖ1t=ζTt,vTtT.
Integrating (21) with respect to t from t0 to tϱtη≤tϱ<tη+1,η=0,1,…, we acquire(22)∫t0tϱV˙1t−2yTtAvtdt=∫tηtϱV˙1tdt+∑σ=1η∫tσ−1tσV˙1tdt−2∫t0tϱyTtAvtdt=V1tϱ−V1tη+∑σ=1ηV1tσ−−V1tσ−1−2∫t0tϱyTtAvtdt=V1tϱ−V1t0+∑σ=1ηV1tσ−−V1tσ−2∫t0tϱyTtAvtdt,in which tϱ,t0∈0,+∞ and t0≤tϱ.
At the impulse timetk,k∈ℕ, according to the definition of Ft, one gets(23)Ftk−=F1,Ftk=F2.
It is found from (12), (15), and (17) that(24)V1tk=ζTtkFtkζtk+∑ι=1p∫tk−τιtkζTsIN⊗Λιζsds=ζTtk−InN+B⊗ΦTF2InN+B⊗Φζtk−+∑ι=1p∫tk−τιtkζTsIN⊗Λιζsds≤γζTtk−F1ζtk−+∑ι=1p∫tk−τιtkζTsIN⊗Λιζsds≤ζTtk−Ftk−ζtk−+∑ι=1p∫tk−τιtkζTsIN⊗Λιζsds=V1tk−,ℕ∋k.
By (21)–(24), we have(25)V1tϱ−V1t0≤2∫t0tϱyTtAvtdt.
Hence,(26)Stϱ−St0≤∫t0tϱyTtAvtdt,for any tϱ,t0∈0,+∞ and t0≤tϱ, in which St=V1t/2. □Theorem 2.
Under the impulsive controller (9), the MMDNNs (12) are input-strictly passive over Sν1,ν2 if there exist matrices 0<F1∈ℝnN×nN, 0<F2∈ℝnN×nN, A∈ℝϵN×nN, and 0<A1∈ℝnN×nN and a scalar γ∈0,1 such that(27)Γ2rh=Ω1rhΩ2rΩ2rTΩ4≤0,(28)−γF1InN+B⊗ΦTF2F2InN+B⊗Φ−F2≤0,in which r=1,2,h=1,2, Ω1rh=1/νhF1−F2−FrIN⊗D−IN⊗DFr+∑ι=1pFrIN⊗WιWιTFr+IN⊗Λι, Ω2r=Fr−IN⊗C1TA, Ω4=2A1−ATIN⊗C2−IN⊗C2TA.Proof.
LetFt=∑r=12ϕrtFr,Ω1t=ϕtF1−F2−IN⊗DFt−FtIN⊗D+∑ι=1pFtIN⊗WιWιTFt+IN⊗Λι,Ω2t=Ft−IN⊗C1TA.
Then, from (27), one obtains(29)Γ2t≜∑r,h=12​ϕrtψhtΓ2rh=Ω1tΩ2tΩ2TtΩ4≤0.
Selecting the sameV1 as (17) for the networks (12) and using (29), one has(30)V˙1t−2yTtAvt+2vTtA1vt≤ζTtϕtF1−F2−IN⊗DFt−FtIN⊗D+∑ι=1p​FtIN⊗WιWιTFt+IN⊗Λιζt+vTt2A1−ATIN⊗C2−IN⊗C2TAvt+2ζTtFt−IN⊗C1TAvt=ϖ1TtΩ1tΩ2tΩ2TtΩ4ϖ1t≤0,t∈tk,tk+1,ℕ∋k,in which ϖ1t=ζTt,vTtT.
Integrating (30) with respect to t from t0 to tϱtη≤tϱ<tη+1,η=0,1,…, we acquire(31)∫t0tϱV˙1t−2yTtAvt+2vTtA1vtdt=V1tϱ−V1t0+∑σ=1ηV1tσ−−V1tσ−2∫t0tϱyTtAvtdt−vTtA1vtdt,in which tϱ,t0∈0,+∞ and t0≤tϱ.
At the impulse timetk,k∈ℕ, according to the definition of Ft, one acquires(32)Ftk−=F1,Ftk=F2.
On the basis of (12), (17), and (28), we get(33)V1tk≤V1tk−,ℕ∋k.
Considering (30)–(33), it is obtained that(34)V1tϱ−V1t0≤2∫t0tϱ​yTtAvt−vTtA1vtdt.
Thus,(35)Stϱ−St0≤∫t0tϱyTtAvtdt−∫t0tϱvTtA1vtdt,for any tϱ,t0∈0,+∞ and t0≤tϱ, in which St=V1t/2.Theorem 3.
Under the impulsive controller (9), the MMDNNs (12) are output-strictly passive over Sν1,ν2 if there exist matrices 0<F1∈ℝnN×nN, 0<F2∈ℝnN×nN, A∈ℝϵN×nN, and 0<A2∈ℝϵN×ϵN and a scalar γ∈0,1 such that(36)Γ3rh=Ω5rhΩ6rΩ6rTΩ7≤0,(37)−γF1InN+B⊗ΦTF2F2InN+B⊗Φ−F2≤0,in which r=1,2,h=1,2, Ω5rh=1/νhF1−F2−FrIN⊗D−IN⊗DFr+∑ι=1pFrIN⊗WιWιTFr+IN⊗Λι+2IN⊗C1TA2IN⊗C1, Ω6r=2IN⊗C1TA2IN⊗C2+Fr−IN⊗C1TA, and Ω7=−ATIN⊗C2−IN⊗C2TA+2IN⊗C2TA2IN⊗C2.Proof.
LetFt=∑r=12ϕrtFr,Ω5t=−IN⊗DFt+ϕtF1−F2+∑ι=1pFtIN⊗WιWιTFt+IN⊗Λι+2IN⊗C1TA2IN⊗C1−FtIN⊗D,Ω6t=2IN⊗C1TA2IN⊗C2+Ft−IN⊗C1TA.
Then, it is derived from (36) that(38)Γ3t≜∑r,h=12​ϕrtψhtΓ3rh=Ω5tΩ6tΩ6TtΩ7≤0.
Selecting the sameV1 as (17) for the networks (12) and utilizing (38), we get(39)V˙1t−2yTtAvt+2yTtA2yt≤ζTtϕtF1−F2−IN⊗DFt−FtIN⊗D+2IN⊗C1TA2IN⊗C1+∑ι=1pIN⊗Λι+Ft×IN⊗WιWιTFtζt+2ζTt2IN⊗C1TA2IN⊗C2+Ft−IN⊗C1TAvt+vTt−ATIN⊗C2−IN⊗C2TA+2IN⊗C2TA2IN⊗C2vt=ϖ1TtΩ5tΩ6tΩ6TtΩ7ϖ1t≤0,t∈tk,tk+1,ℕ∋k,in which ϖ1t=ζTt,vTtT.
Integrating (39) with respect to t from t0 to tϱtη≤tϱ<tη+1,η=0,1,…, one has(40)∫t0tϱV˙1t−2yTtAvt+2yTtA2ytdt=V1tϱ−V1t0+∑σ=1ηV1tσ−−V1tσ−2∫t0tϱyTtAvt−yTtA2ytdt,in which tϱ,t0∈0,+∞ and t0≤tϱ.
At the impulse timetk,k∈ℕ, according to the definition of Ft, we derive(41)Ftk−=F1,Ftk=F2.
It is obtained from (12), (17), and (37) that(42)V1tk≤V1tk−,ℕ∋k.
Using (39)–(42), we can acquire(43)V1tϱ−V1t0≤2∫t0tϱ​yTtAvt−yTtA2ytdt.
Therefore,(44)Stϱ−St0≤∫t0tϱyTtAvtdt−∫t0tϱyTtA2ytdt,for any tϱ,t0∈0,+∞ and t0≤tϱ, in which St=V1t/2.Remark 1.
In recent years, as an effective method to study synchronization, the passivity of NNs has been investigated by some researchers [12–17]. Impulsive control is a popular control method among control methods due to its reliability, low cost, and flexibility [22]. Using impulsive control strategies, some authors have focused on the passivity and synchronization of NNs. However, the passivity and synchronization of MMDNNs have not been considered under impulsive control.


## 4. Synchronization of MMDNNs via Impulsive Control

Settingvit=0 in the MMDNNs (6), we obtain(45)z˙it=−Dzit+∑ι=1pWιgιzit−τι+J,i∈N.The initial value of (45) is given by(46)zit=χit,t∈−τ,0,i∈N,where χit∈C−τ,0,ℝn is the set of continuous functions from −τ,0 to ℝn,χt=χ1Tt,χ2Tt,…,χNTtT.Suppose thatz∗=z1∗,z2∗,…,zn∗T∈ℝn is an equilibrium solution of an isolated node of the MMDNNs (45). Then, one has(47)−Dz∗+J+∑ι=1pWιgιz∗=0.For the MMDNNs (45), select the following impulsive controller:(48)uitk=∑j=1NBijΦzjtk−,k∈ℕ,i∈N,in which Bij and Φ have the same meanings as in the third section. Assume zitk−=limt⟶tk−zit.It is obtained from (45) and (48) that(49)z˙it=−Dzit+∑ι=1p​Wιgιzit−τι+J,t≠tk,Ikzit=∑j=1NBijΦzjtk−,k∈ℕ,i∈N,where Ikzit=zitk+−zitk−, zitk+=limt⟶tk+zit. Suppose zitk=zitk+.Letζit=zit−z∗. Then, it is found from (47) and (49) that(50)ζ˙it=−Dζit+∑ι=1p​Wιgιzit−τι−gιz∗,t≠tk,Ikζit=∑j=1NBijΦζjtk−,k∈ℕ,i∈N.Definition 3.
(see [20]). The MMDNNs (45) achieve synchronization if(51)limt⟶+∞zit−z∗=0,i∈N.Theorem 4.
If there exist matrices0<M1∈ℝn×n and 0<M2∈ℝn×n and scalars β∈0,1,ξ>0 such that(52)Γ4rh=Ω8rhΨr1Ψr2⋯ΨrpΨr1T−In0⋯0Ψr2T0−In⋯0⋮⋮⋮⋱⋮ΨrpT00⋯−In≤0,(53)−βIN⊗M1Y1Y1T−IN⊗M2≤0,where r=1,2,h=1,2,Ω8rh=ξ+lnβ/νhMr−MrD−DMr+1/νhM1−M2+∑ι=1peξτΛι, Ψr1=β−1/2MrW1,Ψr2=β−1/2MrW2,…,Ψrp=β−1/2MrWp, and Y1=InN+B⊗ΦTIN⊗M2,i∈N, then under the impulsive controller (48), the MMDNNs (45) achieve global exponential synchronization over Sν1,ν2 with convergence rate ξ/2.Proof.
LetMt=∑r=12ϕrtMr,Ψ1t=β−1/2MtW1,Ψ2t=β−1/2MtW2,…,Ψpt=β−1/2MtWp, Ω8t=−MtD−DMt+ξ+ϕtlnβMt+∑ι=1peξτΛι+ϕtM1−M2.
Then, by (52), we have(54)Γ4t≜∑r,h=12ϕrtψhtΓ4rh=Ω8tΨ1tΨ2t⋯ΨptΨ1Tt−In0⋯0Ψ2Tt0−In⋯0⋮⋮⋮⋱⋮ΨpTt00⋯−In≤0.
The impulse-time-dependent Lyapunov functional for the MMDNNs (50) is considered as follows:(55)V2t=∑ι=1p∑i=1N∫t−τιteξs+τ−t0φsζiTsΛιζisds+∑i=1Neξt−t0φtζiTtMtζit,in which φt=βϕ1t.
Fortk,tk+1∋t, ℕ∋k, differentiating V2t along the solution of the MMDNNs (50) and applying the fact that φt−s/φt≥β, for all 0≤s and t0<t, one gets(56)V˙2t=∑i=1Neξt−t0φtζiTtξ+ϕtlnβMt+ϕtM1−M2−MtD−DMtζit+2∑ι=1pMtWιgιzit−τι−gιz^∗+∑ι=1p∑i=1Neξt+τ−t0φtζiTtΛιζit−∑ι=1p∑i=1Neξt−τι+τ−t0φt−τι×ζiTt−τιΛιζit−τι≤∑i=1Neξt−t0φtζiTtξ+ϕtlnβMt+ϕtM1−M2+∑ι=1p​eξτΛι−MtD−DMtζit−β∑ι=1p​ζiTt−τιΛιζit−τι+2∑ι=1p​MtWιgιzit−τι−gιz^∗.
Moreover,(57)2ζiTtMtWιgιzit−τι−gιz^∗≤β−1ζiTtMtWιWιTMtζit+βζiTt−τιΛιζit−τι.
It is obtained from (54), (56), and (57) that(58)V˙2t≤∑i=1Neξt−t0φtζiTtξ+ϕtlnβMt+ϕtM1−M2−DMt−MtD+∑ι=1peξτΛι+β−1MtWιWιTMtζit≤0,t∈tk,tk+1,ℕ∋k.
Therefore,(59)V2t≤V2tk,∀t∈tk,tk+1,ℕ∋k.
At the impulse timetk,k∈ℕ, on the basis of the definitions of Mt and φt, one has(60)φtk−=β,φtk=1,Mtk−=M1,Mtk=M2.
Combining (50), (53), and (55) together, we derive that(61)V2tk=∑i=1Neξtk−t0φtkζiTtkMtkζitk+∑ι=1p∑i=1N∫tk−τιtkeξs+τ−t0φsζiTsΛιζisds=eξtk−t0φtkζTtkIN⊗Mtkζtk+∑ι=1p∑i=1N∫tk−τιtkeξs+τ−t0φsζiTsΛιζisds=eξtk−t0ζTtk−InN+B⊗ΦTIN⊗M2InN+B⊗Φζtk−+∑ι=1p∑i=1N∫tk−τιtkeξs+τ−t0φsζiTsΛιζisds≤eξtk−t0βζTtk−IN⊗M1ζtk−+∑ι=1p∑i=1N∫tk−τιtkeξs+τ−t0φsζiTsΛιζisds=∑i=1Neξtk−t0φtk−ζiTtk−Mtk−ζitk−+∑ι=1p∑i=1N∫tk−τιtkeξs+τ−t0φsζiTsΛιζisds=V2tk−,ℕ∋k.
By (59) and (61), one obtains V2t0≥V2t,∀t0≤t. Hence, we acquire(62)ζt≤λ¯M+∑ι=1pλmaxΛιeξτ−1/ξβλ¯Mχt0−z^∗τe−ξ/2t−t0,t0≤t,in which λ¯M=maxλmaxM1,λmaxM2,λ¯M=minλminM1,λminM2. Thus, the MMDNNs (45) reach global exponential synchronization over Sν1,\\ν2 with convergence rate ξ/2. □Remark 2.
Compared with [37], the sufficient condition for the synchronization of MMDNNs depends on the length of impulsive intervals, and the time-varying Lyapunov function in Theorem 4 can capture the dynamical characteristics of MMDNNs.


## 5. Numerical Examples

Example 1.
The MMDNNs are given by(63)z˙it=−Dzit+∑ι=13Wιgιzit−τι+J+vit,in which i=1,2,…,5,gς1θ=1/8θ+1−θ−1,gς2θ=1/16θ+1−θ−1,gς3θ=1/32θ+1−θ−1,ς=1,2,3, J=0,0,0T, D=diag0.9,0.8,0.9, τ1=0.3,τ2=0.5,τ=τ3=0.7,(64)W1=0.10.20.10.30.10.30.40.20.4,W2=0.20.30.40.10.40.50.30.50.1,W3=0.20.50.50.40.30.30.10.60.3.
Apparently,gς1⋅,gς2⋅, and gς3,⋅,ς=1,2,3, respectively, satisfy the Lipschitz condition with ρς1=0.25,ρς2=0.125, and ρς3=0.0625. z∗=0,0,0T∈ℝ3 is an equilibrium solution of the MMDNNs (63).
Taketk=k,Φ=diag0.2,0.3,0.1,ν1=1.12,ν2=2.12,γ=0.6,(65)C1=0.40.30.20.30.50.30.20.10.10.60.30.1,C2=0.20.50.30.40.10.40.10.50.40.20.10.3,B=−0.70.200.20.30.1−0.40.100.200.1−0.70.10.50.100.2−0.40.10.10.20.10.2−0.6.Case 1.
Utilizing MATLAB YALMIP Toolbox, one obtains(66)F1=I5⊗0.3631−0.44130.4421−0.44132.1340−1.39110.4421−1.39111.4273>0,F2=I5⊗0.2043−0.23270.2396−0.23271.1282−0.71700.2396−0.71700.7490>0,A=I5⊗−17.798382.4589−36.12295.1034−28.01237.452816.4980−40.502456.82564.0410−26.63111.9003,satisfying (14) and (15). From Theorem 1, the MMDNNs (12) under the impulsive controller (9) are passive.Case 2.
Utilizing MATLAB YALMIP Toolbox, one gets(67)F1=I5⊗0.3501−0.41080.4248−0.41082.0450−1.31800.4248−1.31801.3634>0,F2=I5⊗0.1991−0.22030.2343−0.22031.0979−0.69210.2343−0.69210.7281>0,A=I5⊗−37.232983.8487−129.001610.8454−29.462434.196733.2126−33.9720141.52928.5271−28.854322.1856,A1=I5⊗1.1069−0.35485.0976−0.35484.84491.30805.09761.308025.3992,satisfying (27) and (28). From Theorem 2, the MMDNNs (12) under the impulsive controller (9) are input-strictly passive.Case 3.
Utilizing MATLAB YALMIP Toolbox, one derives(68)F1=I5⊗0.3936−0.47350.4538−0.47352.2493−1.45820.4538−1.45821.4735>0,F2=I5⊗0.2258−0.25310.2484−0.25311.2036−0.76130.2484−0.76130.7839>0,A=I5⊗17.206097.5401−23.84824.7611−33.84672.173317.1505−43.038058.53883.6457−32.6881−3.9551,A2=I5⊗20.6353−7.8392−4.8377−7.9339−7.83923.4784−1.43603.7580−4.8377−1.436025.2331−4.0463−7.93393.7580−4.04634.5656>0,satisfying (36) and (37). According to Theorem 3, the MMDNNs (12) under the impulsive controller (9) are output-strictly passive. Figure 1 shows the results of simulation.Figure 1
ζit,yit,vit,i=1,2,…,5.Example 2.
The MMDNNs are given by(69)z˙it=−Dzit+∑ι=13Wιgιzit−τι+J,in which i=1,2,…,5,gς1θ=1/8θ+1−θ−1,gς2θ=1/16θ+1−θ−1,gς3θ=1/32θ+1−θ−1,ς=1,2,3, D=diag0.7,0.9,0.8, J=0,0,0T, τ1=0.2,τ2=0.3,τ=τ3=0.4,(70)W1=0.30.40.50.50.60.10.20.30.2,W2=0.50.60.20.20.30.30.10.20.5,W3=0.20.10.30.30.40.40.40.50.6.
Apparently,gς1⋅,gς2⋅, and gς3,⋅,ς=1,2,3, respectively, satisfy the Lipschitz condition with ρς1=0.25,ρς2=0.125, and ρς3=0.0625. z∗=0,0,0T∈ℝ3 is an equilibrium solution of the MMDNNs (69).
Lettk=1.8k,ν1=1.3,ν2=3.1,β=0.8,ξ=0.2,Φ=diag0.5,0.4,0.2,(71)B=−0.40.100.20.10.3−0.60.200.100.2−0.80.20.40.200.1−0.50.20.20.30.30.1−0.9.
By using MATLAB YALMIP Toolbox, we derive(72)M1=1.1830−0.8947−0.2582−0.89472.9926−1.5950−0.2582−1.59502.0166>0,M2=0.7928−0.5673−0.1779−0.56731.6871−0.7982−0.1779−0.79821.1134>0,satisfying (52) and (53). According to Theorem 4, the MMDNNs (45) under the impulsive controller (48) are globally exponentially synchronized. Figure 2 shows the results of simulation.Figure 2
ζit,i=1,2,…,5.


## 6. Conclusion

In this paper, a class of multiple multi-delayed neural network models has been introduced, and the model can describe the dynamics of neurons more accurately. On the one hand, several passivity criteria for MMDNNs have been established by means of inequality techniques. On the other hand, the synchronization of the proposed model has also been researched based on stability theory. Finally, the correctness of the synchronization and passivity criteria has been verified by two numerical examples. In future work, we will design a suitable impulsive controller to further study the passivity and synchronization of multiple reaction-diffusion neural networks.



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