# Stability and Stabilization of Delayed Neural Networks with Hybrid Impulses

Authors: Kefa Zou; Xuechen Li; Nan Wang; Jungang Lou; Jianquan Lu
Journal: Complexity
 (2020)

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## Abstract

In this paper, the stability and stabilization issues for a class of delayed neural networks with time-varying hybrid impulses are investigated. The hybrid effect of two types of impulses including both stabilizing and destabilizing impulses is considered simultaneously in the analysis of systems. To characterize the occurrence features of impulses, the concepts of average impulse interval and average impulse strength are employed. Based on the analysis of stability, a pinning impulsive controller which can ensure the global exponential stability of the studied neural networks is designed by pinning a small fraction of neurons. Finally, two numerical examples are given to illustrate the effectiveness of the proposed control schemes for delayed neural networks with hybrid impulses.


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## Body

## 1. Introduction

During the past few decades, dynamic networks have been systematically studied due to their broad application background in different areas [1–9]. In recent years, in order to cater to the specific needs of modeling various practical systems, many kinds of dynamic networks with special structures have been developed, for example, [10–13] and the references therein. When networks encounter transient disturbance or abrupt dynamic variation in various instants, systems may display switching or impulsive behaviors [14, 15]. Consequently, impulsive neural networks, which can model various electronic or biological networks encountering instantaneous and abrupt changes frequently, have been extensively investigated in various fields of science and engineering [16–19]. On the contrary, time delays frequently appear in various dynamical systems [20, 21]. The existence of time delays in neural networks may induce more complex dynamical behaviors such as instability, oscillations, and chaos [22–27]. Therefore, it is necessary to investigate effects of time delays and impulses on the stability of neural networks.Generally, impulses can be divided into two categories according to their impact on systems. It is supposed that the impulses are destabilizing if the impulses can potentially destroy the stability of dynamical systems, while the impulses are considered to be stabilizing if they are potentially beneficial for the stabilization of dynamical systems. Stabilizing impulses can be considered as impulsive controllers, which can enhance the stabilization of dynamical systems. In the last several decades, stabilizing impulses and destabilizing impulses have been studied by a great many scholars [28–31]. At the same time, the impulsive control method has received many researchers’ attention (see [32–36] and references therein). Particularly, stability or stabilization issue for dynamical systems with delays and impulses was investigated in [37–41] and references therein. In [42], a unified synchronization criterion for impulsive dynamical networks subject to desynchronizing or synchronizing impulses was derived by using the average impulsive interval approach. In [43], some adequate conditions that can ensure the exponential synchronization of inertial memristor-based neural networks with time delay were given by utilizing the average impulsive interval approach. In [44], the pinning impulsive control strategy was proposed. By utilizing the Lyapunov method combined with the comparison principle, pinning stabilization of probabilistic Boolean networks subject to time delays was investigated in [45]. Synchronization problem for stochastic neural networks was studied by impulsively controlling partial states in [46]. Recently, networks with hybrid impulses were explored in [47, 48]. To reduce conservativeness, a novel piecewise Lyapunov–Krasovskii functional (LKF) was constructed by introducing a line-integral type Lyapunov function and some useful terms that take full advantage of the available information about the actual sampling pattern in [49].In this paper, the concepts of average impulsive interval and average impulsive strength are introduced to characterize the features of hybrid impulses. Then, based on the Lyapunov method combined with the utilization of proper mathematical analysis techniques, the stability analysis for neural networks with time-varying delays and hybrid impulses is carried out. Under this circumstance, the classification of stabilizing and destabilizing impulses is not taken into account; just the overall effect of the impulses is taken into consideration. Furthermore, a pinning impulsive controller design procedure for the stabilization of the investigated neural networks is proposed based on the above analysis. In this controller, only a small fraction of variables is impulsively controlled to ensure the global and exponential stability of neural networks. The main contributions of this paper are summarized as follows: (1) a new concept of “average impulsive strength,” which can be used to characterize much wider range of impulsive sequences, is introduced to describe the hybrid impulses investigated in this paper. (2) By virtue of some proper disposing techniques relevant to average impulsive interval and average impulsive strength in the proof process of the main results, less conservative results can thereafter be obtained. (3) In consideration of the advantages of low cost and high efficiency of the pinning controller, the strategy that selects only a small fraction of neurons for impulsive control is adopted to achieve the stabilization of the delayed neural networks.The remainder of this paper is arranged as follows: in Section2, we propose the problem of stability and stabilization of delayed neural networks with hybrid impulses and give some necessary preliminaries. In Section 3, a criterion for determining the stability of delayed neural networks with hybrid impulses is established, and then a pinning impulsive controller is designed to stabilize delayed neural networks. In Section 4, numerical examples are given to illustrate our theoretical results. Finally, Section 5 presents the conclusion.Notation 1.
. The standard notations are used in this paper.ℝ+ and ℝn denote the set of nonnegative real numbers and the n-dimensional Euclidean space. ℕ+ denotes the set of positive integers. The superscript “T” represents the transpose of the matrix or vector. For x∈ℝn,x denotes the Euclidean norm of x. For matrix A∈ℝn,A=λmaxATA, where λmax⋅ represents the largest eigenvalue. diag(⋯) stands for a block-diagonal matrix. Given τ>0,C−τ,0,ℝn denotes the family of continuous functions from −τ,0 to ℝn.


## 2. Preliminaries

In this section, some preliminaries including model formulation, lemmas, and definitions are presented.Consider the following neural network:(1)x˙t=Cxt+Bgxt+Dgxt−τt+I,where xt=x1t,x2t,…,xntT∈ℝn is the state vector associated with the neurons; C=diagc1,c2,…,cn<0 is the self-feedback matrix; B=bijn×n and D=dijn×n are the connection weight matrices; gxt=g1xt,g2xt,…,gnxtT denotes the activation function of the neurons; I∈ℝn signifies constant external input; and τt represents the time-varying delays that satisfies 0≤τt≤τ. For the nonlinear function g⋅, we have the following assumption.Assumption 1.
Assume thatgi⋅i=1,2,…,n are globally Lipschitz continuous functions, i.e., there exist constants li>0i=1,2,…,n such that(2)gix1−gix2≤lix1−x2,for anyx1,x2∈ℝ.
DenoteL=diagl1,l2,…,ln.
Letx∗ be the equilibrium point of (1). For convenience, we can shift the intended equilibrium x∗ to be original by letting y=x−x∗, and then system (1) can be transformed into(3)y˙t=Cyt+Bfyt+Dfyt−τt,where yt=y1t,y2t,…,yntT∈ℝn is the state vector of the transformed system. It follows from (2) that the function fy=gy+x∗−gx∗ satisfies(4)fix1−fix2≤lix1−x2,for anyx1,x2∈ℝ.
In consideration of the time-varying impulse effects, the impulsive delayed neural network can be obtained in the following form:(5)y˙t=Cyt+Bfyt+Dfyt−τt,t≠tk,ytk+=αkytk−,k∈ℕ+,where t1,t2,t3,…, is a sequence of strictly increasing impulsive moments. αk∈ℝ represents the strength of impulses. We assume that yt is right-continuous at t=tk,i.e.,ytk=ytk+. Hence, the solutions of (5) are piecewise right-hand continuous functions which are discontinuous at t=tk for k∈ℕ+. The initial condition of (5) is given by yt=ϕt∈C−τ,0,ℝn.Definition 1.
(average impulsive interval; see [42]). Ta is called the average impulsive interval of the impulsive sequence ζ=t1,t2,…, if(6)T−tTa−N0≤NζT,t≤T−tTa+N0,for allT≥t≥0,where N0 is a positive integer and Ta is a positive number. NζT,t denotes the impulsive times of the impulsive sequence ζ in the time interval t,T.Definition 2.
(average impulsive strength).α is called the average impulsive strength of the sequence ζT,t=tl,tl+1,…,tl+NζT,t−1for allT≥t≥0 if(7)αNζt,0+αNζt,0+1+⋯+αNζT,0−1α−N1≤NζT,t≤αNζt,0+αNζt,0+1+⋯+αNζT,0−1α+N1,where N1 is a positive integer and α is a positive number. NζT,t denotes the impulsive times of the impulsive sequence ζ in the time interval t,T.
Furthermore, inequality (7) implies that(8)α1−N1NζT,t≤αNζt,0+αNζt,0+1+⋯+αNζT,0−1NζT,t≤α1+N1NζT,t.Remark 1.
The concept of average impulsive gain was proposed in [48], where the problem of the globally exponential synchronization of coupled neural networks with hybrid impulses was investigated. However, when the time-varying delays are taken into consideration simultaneously, this idea may fail to be applied to the analysis of the stability of delayed neural networks with hybrid impulses. To deal with the difficulties coming from time-varying delays, a new concept named “average impulsive strength” is introduced in this paper. The conditions of our concept are more strict, so it can ensure the establishment of the inequality in the comparison principle.Definition 3.
Impulsive neural networks (5) are said to be globally exponentially stable if there exist constants M>0,λ>0, and T0>0 such that, for any initial values, yt2≤Me−λt holds for all t≥T0.Lemma 1 (see [42]).
For any vectorsx,y∈ℝn, scale ϵ>0, and positive definite matrix Q∈ℝn×n, the following inequality holds: 2xTy≤εxTQx+ε−1yTQ−1y.Lemma 2 (see [35]).
Let0≤τit≤τ. Ft,u,u¯1,u¯2,…,u¯m:ℝ+×ℝ×⋯×ℝ︷m+1⟶ℝ is nondecreasing in u¯i for each fixed t,u,u¯1,…,u¯i−1,u¯i−2,…,u¯m,i=1,2,…,m, and Iku:ℝ⟶ℝ is nondecreasing in u.
Suppose that(9)D+ut≤Ft,ut,ut−τ1t,…,ut−τmt,utk+≤Ikutk−,k∈ℕ+,D+νt>Ft,νt,νt−τ1t,…,νt−τmt,νtk+≥Ikνtk−,k∈ℕ+.
Then,ut≤νt, for −τ≤t≤0, implies that ut≤νt, for t≥0.


## 3. Main Results

In this section, we will analyze the global exponential stability of delayed neural networks with hybrid impulses including both destabilizing and stabilizing impulses.Theorem 1.
Consider time-varying neural network (5) with hybrid impulses including both destabilizing and stabilizing impulses. Suppose that Assumption 1 holds and that the average impulsive interval of the impulsive sequence ζ=t1,t2,…, is Ta. Then, neural networks (5) with hybrid impulses are globally exponentially stable if the following inequality holds:(10)η1+M0q<0,where η1=p+2lnα/Ta,p=λmaxC+CT+BBT+LTL+DDT,q=λmaxLTL, and M0 is a constant satisfying the following condition: when α≥1,M0=e2N1α2N0; otherwise, M0=e2N1α−2N0.Proof.
Consider the Lyapunov functionVt=yTtyt. Then, the derivative of Vt along the trajectories of system (5) can be obtained as follows:(11)D+Vt=yTtC+CTyt+2yTtBfyt+2yTtDfyt−τt,t∈tk−1,tk,k∈ℕ+.
By Lemma1, one obtains(12)2yTtBfyt≤yTtBBTyt+fTytfyt≤yTtBBTyt+yTtLTLyt,(13)2yTtDft−τt≤yTtDDTyt+fTyt−τtfyt−τt≤yTtDDTyt+yTt−τtLTLyt−τt.
From (11) to (13), it follows that(14)D+Vt≤yTtC+CT+BBT+LTL+DDTyt+yTt−τtLTLyt−τt≤λmaxC+CT+BBT+LTL+DDTVt+λmaxLTLVt−τt=pVt+qVt−τt,t∈tk−1,tk,k∈ℕ+.
Fort=tk, one has(15)Vtk+=yTtk+ytk+=αk2yTtk−ytk−=αk2Vtk−.
For anyε>0, let νt be a unique solution of the following impulsive delay system:(16)ν˙t=pνt+qνt−τt+ε,t≠tk,νtk−=αk2νtk−,t=tkk∈ℕ+,νt=ϕt2,−τ≤t≤0.
According to Lemma2, one gets(17)νt≥Vt,t≥0.
By the formula for the variation of parameters, it follows from (16) that(18)νt=wt,0ν0+∫0twt,sqνs−τs+εds,where wt,s,t>s≥0, is the Cauchy matrix of the linear system(19)z˙t=pzt,t≠tk,ztk+=αk2ztk−,t=tkk∈ℕ+.
According to the representation of the Cauchy matrix, we can obtain the following estimation: ifNζt,s>0, one has(20)wt,s=ept−s∏s<tk≤tαk2≤ept−sαNζs,0+αNζs,0+1+⋯+αNζt−s−1Nζt−s2Nζt−s≤ept−sα1+N1Nζt−s2Nζt−s≤e2N1ept−sα2Nζt−s.
Otherwise, whenNζt,s>0, one also has(21)wt,s=ept−s≤e2N1ept−sα2Nζt,s.
Since the average impulsive interval of the impulsive sequenceζ=t1,t2,…, is equal to Ta, we have(22)t−sTa−N0≤Nζt,s≤t−sTa+N0,for allt≥s≥0.
Ifα≥1, it follows from (20) to (22) that(23)wt,s≤e2N1ept−sα2t−s/Ta+N0≤α2N0e2N1ep+2lnα/Tat−s.
Similarly, whenα<1, it follows from (20) and (22) that(24)wt,s≤e2N1ept−sα2t−s/Ta−N0≤α−2N0e2N1ep+2lnα/Tat−s.
Lettingε⟶0 and summarizing inequalities (23) and (24) give that there exists constant M0=maxα2N0e2N1,α−2N0e2N1 such that(25)wt,s<M0eη1t−s,where η1=p+2lnα/Ta.
Letη=M0sup−τ≤s≤0ϕs2. From (18) and (25), one obtains(26)νt≤ηeη1t+∫0tM0eη1t−sqνs−τs+εds.
Definehv=v+η1+M0qevτ. It follows from (10) that h0<0. Since h+∞=+∞ and h˙v>0, there exists a unique λ>0 such that(27)λ+η1+M0qeλτ=0.
On the contrary, it is obvious from (10) that M0−1η1+q<0. Hence,(28)νt=ϕt2≤η<ηe−λt−εM0−1η1+q,−τ≤t≤0.
Then, we claim(29)νt<ηe−λt−εM0−1η1+q.
If inequality (29) is not true, there exists t∗>0 such that(30)νt∗≥ηe−λt∗−εM0−1η1+q,(31)νt<ηe−λt−εM0−1η1+q,t<t∗.
From (26) to (31), we have(32)νt∗≤ηeη1t∗+∫0t∗M0eη1t∗−sqνs−τs+εds<eη1t∗η−εM0−1η1+q+∫0t∗M0e−η1sqηe−λs−τs−εM0−1η1+q+εds.
It is derived from (27) and (32) that(33)νt∗<eη1t∗η−εM0−1η1+q+ηe−η1+λt∗−η−εM0−1η1+qe−η1t∗+εM0−1η1+q=ηe−λt∗−εM0−1η1+q,which contradicts with (30), and so, (29) holds. Letting ε⟶0, one gets from (17) that(34)Vt≤νt≤ηe−λt.
By Definition3, the solution yt of impulsive neural networks (5) is exponentially stable. The proof is hence completed.Remark 2.
The concepts of average impulse interval [42, 43] and average impulsive strength are employed to characterize the features of hybrid impulses. The classification of stabilizing and destabilizing impulses is not taken into account; just the overall effect of the impulses is taken into consideration. By utilizing the comparison principle [35, 44], Lyapunov theory [35, 42–44], Young’s inequality technique [42, 43, 48], average impulsive interval approach [42, 43], and the concept of average impulsive strength, Theorem 1 presenting conditions of ensuring the global exponential stability of delayed neural networks (5) is derived.
In the following section, a pinning impulsive controller will be designed to globally and exponentially stabilize neural networks (5). In order to drive system (5) into the equilibrium point x∗, the following impulsive controller is constructed for l partial variables l<n:(35)uitk=0,i∉Dtk,∑k=1+∞μyitδt−tk,i∈Dtk,where μ∈−2,−1∪−1,0 is a constant, which means that the impulsive effects can be used to stabilize the delayed neural networks; the index set of l partial variables Dtk which should be impulsively controlled is defined as follows: at time instant tk, for the error of states of the partial variables y1tk,y2tk,…,yntk, one can reorder the states such that yp1tk≥yp2tk≥⋯≥ypntk.
Then, the index set ofl controlled variables Dtk is defined as Dtk = {p1, p2, …, pl}. After adding pinning impulsive controller (35) to the variables Dtk, the controlled delayed neural networks can be rewritten as follows:(36)y˙t=Cyt+Bfyt+Dfyt−τt,t≠tk,yitk+=μyitk−+yitk−,i∈Dtk,ytk+=yitk−,i∉Dtk.
The initial conditions of delayed neural networks (36) are given by(37)yit=ϕit−τ≤t≤0i=1,2,…,n,where ϕit∈C−τ,0,ℝ with C−τ,0,ℝ being the set of continuous functions from −τ,0 to ℝ.Theorem 2.
Letρ=n+lμμ+2/n∈0,1. Suppose that Assumption 1 holds and that the average impulsive interval ζ=t1,t2,…, is Ta. Then, neural networks (36) are globally exponentially stabilized to the state x∗ by pinning controller (35) if the following inequality holds:(38)η2+M1q<0,where η2=p+lnρ/Ta,p=λmaxC+CT+BBT+LTL+DDT,q=λmaxLTL,andM1=ρ−N0.Proof.
Construct a Lyapunov function in the form of(39)Vt=yTtyt=∑i=1nyi2t.
By a similar analysis as Theorem1, for any t∈tk−1,tk, taking the derivative of Vt along the solution of (36) gives that(40)D+Vt≤pVt+qVt−τt.
For anyk∈ℕ+, we have(41)Vtk+=∑i=1nyiTtk+yitk+=∑i∈Dtk1+μ2yiTtk−yitk−+∑i∉DtkyiTtk−yitk−.
For anyk∈ℕ+, let φtk−=minyitk−:i∈Dtk and ψtk−=maxyitk−:i∉Dtk. According to the selection of parameters in set Dtk, we have φtk−≥ψtk−. Since ρ=1+l/n⋅μμ+2∈0,1, we get 1−ρn−l=ρ−1+μ2l. Hence, one has(42)1−ρ∑i∉DtkyiTtk−yitk−≤1−ρN−lψtk−2≤1−ρN−lφtk−2≤lρ−1+μ2φtk−2≤ρ−1+μ2∑i∈DtkyiTtk−yitk−,which follows that(43)1+μ2∑i∈DtkyiTtk−yitk−+∑i∉DtkyiTtk−yitk−≤ρ∑i=1nyiTtk−yitk−.
From (41) to (43), we have(44)Vtk+≤ρVtk−.
For anyε>0, let νt be a unique solution of the following impulsive delayed system:(45)ν˙t=pνt+qνt−τmt+ε,t≠tk,νtk+=ρνtk−,t=tkk∈ℕ+,νt=ϕt2,−τ≤t≤0.
According to Lemma2, we can get(46)νt≥Vt,t≥0.
By the formula for the variation of parameters, it follows from (45) that(47)νt=wt,0ν0+∫0twt,sqνs−τs+εds,where wt,s,t>s≥0, is the Cauchy matrix of the linear system(48)z˙t=pzt,t≠tk,ztk+=ρztk−,t=tkk∈ℕ+.
According to the representation of the Cauchy matrix, since0<ρ<1 and t−s/Ta−N0≤Nζt,s≤t−s/Ta+N0, we can obtain the following estimation:(49)wt,s=ept−s∏s<tk<tρ≤ept−sρNζt,s≤ept−sρt−s/Ta−N0=M1eη2t−s,where M1=ρ−N0andη2=p+lnρ/Ta. Let M2=M1sup−τ≤s≤0ϕs2. Then, it can be derived from (45) and (49) that(50)νt≤M2eη2t+∫0tM1eη2t−sqνs−τs+εds.
Definehv=v+η2+M1qevτ. It follows from (38) that h0<0. Since h+∞=+∞ and h˙v>0, there exists unique λ>0 such that(51)λ+η2+M1qeλτ=0.
On the contrary, it is obvious from (38) that M1−1η2+q<0. Hence,(52)νt=ϕt2<M2e−λt−εM1η2+M1q,−τ≤t≤0.
Then, we claim(53)νt<M2e−λt−εM1η2+M1q.
If inequality (51) is not true, there exists t∗>0 such that(54)νt∗≥M2e−λt∗−εM1η2+M1q,(55)νt<M2e−λt−εM1η2+M1q,t<t∗.
From (50) and (55), we have(56)νt∗≤M2eη2t∗+∫0t∗M1eη2t∗−sqνs−τs+εds<eη2t∗M2−εM1η2+M1q+∫0t∗M1e−η2sqM2e−λs−τs−εM1η2+M1q+εds.
It is derived from (51) and (56) that(57)νt∗<eη2t∗M2−εM1η2+M1q+M2e−η2+λt∗−M2+εM1η2+M1q−εM1η2+M1qe−η2t∗=M2e−λt∗−εM1η2+M1q,which contradicts with (54), and so, (53) holds. Letting ε⟶0, one gets from (17) that(58)Vt≤νt≤M2e−λt.
The proof is completed.


## 4. Numerical Examples

Two numerical examples are presented to demonstrate the validity of the above results in this section.Example 1.
Consider neural networks (5) with the following parameters:(59)C=−300−8,B=0.30.10.20.2,D=0.40.200.2,L=0.1000.2,and fyt=tanh0.8yt,τt=et/1+et, where yt=y1t,y2tT is the state vector of the neural networks. By calculation, we can easily know η1+M0q=−3.5126. Figure 1 represents a hybrid impulsive sequence where Ta=0.2,N0=5,N1=2, and α=1.2. The effect of the hybrid impulsive sequence is destabilizing.
According to Theorem1, neural networks (5) with time-varying impulses will be globally exponentially stable if condition (10) is satisfied. Our numerical simulation draws the conclusion of Figure 2. From Figure 2, it can be observed that the state response of the neural networks tends to be stable quickly, which means the simulation results are consistent with the theory analysis.Figure 1
The hybrid impulsive sequence withTa=0.2,N0=5,N1=2, and α=1.2.Figure 2
The state response of the neural networks in Theorem1.Example 2.
Consider neural networks (5) with(60)C=1000000−4000000−1000000−3000000−2000000−3,B=0.080−0.100000.0200.1−0.30−0.050−0.10.500.40.40.2−0.30.6−0.20.10000.2−0.300.1−0.2000.5−0.2,D=0.20.1000000.10.20.3000.10−0.2040.200.100.10.20.200.10.20.3000.20.10.3000.2,L=100000010000001000000100000010000001,and fyt=tanh0.8yt,τt=et/1+et, where yt=y1t,y2t,…,y6tT is the state vector of the neural networks. The state response of neural networks (5) tends to be unstable without any controller, as is depicted in Figure 3. We select controller (35) with Ta=0.25,N0=3,μ=−1.5,l=3,andn=6. By calculation, we can easily know η2+M1q=−4.570469. According to Theorem 2, the pinning impulsive controller ensures global exponential stability of the considered neural networks. Simulation results approve the declared property (see Figure 4).Figure 3
The state response of neural networks (5) without any controller in Theorem 2.Figure 4
Simulation result for stabilization in Theorem2.


## 5. Conclusion

In this paper, the stability and stabilization problems of delayed neural networks with hybrid impulses have been studied. Based on Lyapunov stability theory combined with the comparison principle and the conception of average impulsive strength, a criterion for the exponential stability of delayed neural networks with hybrid impulses has been proposed. Then, a pinning impulsive controller has been designed to globally and exponentially stabilize the delayed neural networks with hybrid impulses. By revising the proof of the main results, some methods related to the concepts of average impulsive interval and average impulsive strength have been used to make the theoretical results less conservative. The derived stabilization criterion and the convergence rate are closely related with the proportion of the controlled neurons, time delay, impulsive strengths, and average impulsive interval of the neural networks. The validity of the theoretical results has been well explained by simulation results. In the future research, finite-time stabilization, persistent dwell-time, and state constraints will be included.



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