# Sliding Intermittent Control for BAM Neural Networks with Delays

Authors: Jianqiang Hu; Jinling Liang; Hamid Reza Karimi; Jinde Cao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper addresses the exponential stability problem for a class of delayed bidirectional associative memory (BAM) neural networks with delays. A sliding intermittent controller which takes the advantages of the periodically intermittent control idea and the impulsive control scheme is proposed and employed to the delayed BAM system. With the adjustable parameter taking different particular values, such a sliding intermittent control method can comprise several kinds of control schemes as special cases, such as the continuous feedback control, the impulsive control, the periodically intermittent control, and the semi-impulsive control. By using analysis techniques and the Lyapunov function methods, some sufficient criteria are derived for the closed-loop delayed BAM neural networks to be globally exponentially stable. Finally, two illustrative examples are given to show the effectiveness of the proposed control scheme and the obtained theoretical results.


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## Body

## 1. Introduction

Since the bidirectional associative memory (BAM) neural networks were first proposed by Kosko [1] which are well known as the extension of the unidirectional autoassociators such as the Hopfield neural networks, they have been widely studied due to their extensive applications such as pattern recognition, signal or image processing, solving optimization problems, and automatic control [2–7]. Later, constant delays are introduced in [8] to the BAM neural networks, and it is proved that the delayed versions of the neural networks are significant for handling certain motion-related optimization problems [9]. For more results concerning the dynamical behaviors of the BAM neural networks with delays, we refer to [10, 11].In practice, most of the neural networks are unstable or convergent with a rate far less than the requirement. Under such cases we need to try to stabilize them or speed up the convergence rate of the neural system in order to make the system work more efficiently. Therefore, the designing of appropriate control input becomes extremely urgent. When it comes to the problem of stabilizing a nonlinear system, it is natural to consider the feedback strategies. There are two basic kinds of feedback control: the state feedback control and the output feedback control. When referring to the control methods, different kinds of schemes have been utilized to stabilize the nonlinear system such as static feedback control [12], delayed feedback control [13], adaptive control [14], fuzzy control [15], sampled control [16], sliding mode control [17], and random control [18, 19]. In terms of the control time, the controllers are classified with continuous control and discontinuous control. Compared with the continuous control, the discontinuous control including the impulsive control [20, 21] and the intermittent control [22, 23] has attracted much more attention, and it is very effective, practical, and applicable in many areas, especially in secure communication [24].In the literature, the impulsive neural networks have been extensively studied from the following two aspects: either the system is subject to the impulsive state displacements at fixed time instants or the system is imposed by external impulsive control [25]. The main idea of periodically intermittent control [26] is that when the system signal becomes weak to a low level, the external control will be imposed to supplement the loss of signal; after some period of time the external control is stopped; in the next control period the external control is needed again. Compared with the method of periodically intermittent control, the system with the impulsive control is activated only at some isolated time moments. Both the impulsive control and the intermittent control have their own benefits and disadvantages. The main difference between these two control techniques lies in the length of control period; the former has zero duration, while the later has a nonzero control width. Meanwhile, the cost of the intermittent control is much higher.The aim of this paper is to design a sliding intermittent controller by combining the advantages of both the impulsive control and the periodically intermittent control. More specifically, in one control period, we will impose the continuous state feedback control at the preceding control width and the impulsive control in the latter control width. The sliding intermittent control method is very flexible and could achieve the expected control performance. The sketch of such a controller is shown in Figure1. Motivated by the name of the slide rheostat in the physical electronic circuitry, we named such a joint controller as the sliding intermittent controller.Figure 1
Sketch map of the sliding intermittent control.In this paper, we will investigate the exponential stability problem of the delayed BAM neural networks under the proposed sliding intermittent control. The closed-loop neural system becomes a switched network where the switching rules are dependent on the time index. To the best of the authors’ knowledge, this is the first time in the literature to consider such a joint controller. The rest of the paper is organized as follows. In Section2, the model discussed in this paper and the novel sliding intermittent control idea are introduced, and some preliminaries are also given. In Section 3, several sufficient criteria are established to ensure the delayed BAM neural networks to be exponentially stable under the sliding intermittent control scheme. Meanwhile, several particular cases are discussed. In Section 4, two illustrative examples are given to demonstrate the effectiveness of the proposed results. And finally, the paper is concluded in Section 5.


## 2. Problem Formulation and Some Preliminaries

The delayed bidirectional associative memory (BAM) neural networks have been investigated in [8, 9] as follows:
(1)w˙i(t)=-aiwi(t)+∑j=1mpijf~j(zj(t-τij))+ci,i∈ℐn,z˙j(t)=-bjzj(t)+∑i=1nqjig~i(wi(t-σji))+dj,j∈ℐm,
where the index setℐn={1,2,…,n}, ℐm={1,2,…,m};wi(t), zj(t)∈ℝare the activations of theith neuron and thejth neuron, respectively;ai, bjare positive constants denoting the rates with which the cellsiandjreset their potential to the resting states when isolated from the other cells and inputs; time delaysτijandσjiare nonnegative constants corresponding to the finite speeds of axonal signal transmission withτ*=maxi,j{τij,σji};pijandqjiare the delayed connection weights denoting the strengths of connectivity between the cellsjandi;cianddjdenote, respectively, theith and thejth components of an external input source introduced from the network outside to the cellsiandj;f~jandg~iare bounded nonlinear activation functions, and throughout this paper they are assumed to satisfy the following conditions:
(2)|f~j(ν1)-f~j(ν2)|≤Ljf|ν1-ν2|,|g~i(ν1)-g~i(ν2)|≤Lig|ν1-ν2|,
whereν1, ν2∈ℝandi∈ℐn, j∈ℐm; positive scalarsLjf, Ligare known.Remark 1.
The above conditions are general in the literature to study the existence and uniqueness of the equilibrium for the delayed BAM neural networks (1) without assuming the activation functions to be monotonic or differentiable [27–29].
Letting(w*,z*)withw*=(w1*,w2*,…,wn*)T∈ℝn and z*=(z1*,z2*,…,zm*)T∈ℝmbe an equilibrium of the system (1) and denotingxi(t)=wi(t)-wi*,yj(t)=zj(t)-zj*, the equilibrium of the network (1) can be transformed to the origin of the following system:
(3)x˙i(t)=-aixi(t)+∑j=1mpijfj(yj(t-τij)),i∈ℐn,y˙j(t)=-bjyj(t)+∑i=1nqjigi(xi(t-σji)),j∈ℐm,
wherefj(yj)=f~j(yj+zj*)-f~j(zj*)andgi(xi)=g~i(xi+wi*)-g~i(wi*).It is generally known that, under some specific cases such as the abrupt changes of system parameters or the occurrence of time delays, the network may present unstable dynamics such as bifurcation, oscillation, divergence, or instability. In this paper, we will focus on the unstable delayed BAM neural system (3). To stabilize the network (3), the novel sliding intermittent control scheme is imposed as follows:
(4)x˙i(t)=-aixi(t)+∑j=1mpijfj(yj(t-τij))+ui(1)(t),t∈[lT,(l+θ)T),x˙i(t)=-aixi(t)+∑j=1mpijfj(yj(t-τij)),t∈[(l+θ)T,(l+1)T),t≠tk,Δxi(t)=Ik(1)(xi(t)),t=tk,y˙j(t)=-bjyj(t)+∑i=1nqjigi(xi(t-σji))+uj(2)(t),t∈[lT,(l+θ)T),y˙j(t)=-bjyj(t)+∑i=1nqjigi(xi(t-σji)),t∈[(l+θ)T,(l+1)T),t≠tk,Δyj(t)=Ik(2)(yj(t)),t=tk,
wherei∈ℐn,j∈ℐm;l∈ℕ0+≜{0,1,2,…,} and k∈ℕ+≜{1,2,…}; constantθ∈[0,1]andTare scalars denoting the control width and the control period, respectively;Δxi(tk)=xi(tk+)-xi(tk-), Δyj(tk)=yj(tk+)-yj(tk-)andxi(tk+)=limh→0+xi(tk+h), xi(tk)=xi(tk-)=limh→0-xi(tk+h),yj(tk+)=limh→0+yj(tk+h), and yj(tk)=yj(tk-)=limh→0-yj(tk+h). The impulsive sequence{tk}k=1∞satisfies
(5)0<t1<t2<⋯<tk<⋯,limk→∞tk=∞,tk∈⋃l∈ℕ0+((l+θ)T,(l+1)T].ui(1)(t), uj(2)(t)are controllers imposed on the system (3) whent∈[lT,(l+θ)T), andIk(1)(·), Ik(2)(·)are the impulsive operators imposed at the impulsive moments{tk}k=1∞. Here we use the linear state feedback and linear impulsive control strategies; that is,
(6)ui(1)(t)=ki(1)xi(t),uj(2)(t)=kj(2)yj(t),Ik(1)(xi(tk))=hik(1)xi(tk),Ik(2)(yj(tk))=hjk(2)yj(tk),
whereki(1), kj(2), hik(1), and hjk(2)are gain constants.Remark 2.
The sliding intermittent control idea is illustrated in Figure1 with the adjustable parameterθ(i.e., the control widthθshown in Figure 1). The closed-loop system (4) can be the continuous controlled neural networks (θ=1), the impulsive controlled neural networks (θ=0), or the hybrid controlled neural networks (0<θ<1).
The system (4) is supplemented with initial function given byxi(s)=φi(s),yj(s)=ψj(s),-τ*≤s≤0, whereφ(s)=(φ1(s),φ2(s),…,φn(s))T∈C([-τ*,0],ℝn),ψ(s)=(ψ1(s),ψ2(s),…,ψm(s))T∈C([-τ*,0],ℝm), andC([-τ*,0],ℝn)represents the set of alln-dimensional continuous functions defined on the interval[-τ*,0]. Obviously, the solution of (4) is piecewise left-hand continuous with possible discontinuity att=tkfork∈ℕ+.The following definition and lemmas are introduced before we give the main results of the paper.Definition 3.
The system (4) is said to be globally exponentially stable if there exist constantsα>0, β>0such that for allt>0,
(7)∑i=1n|xi(t)|+∑j=1m|yj(t)|≤βe-αtsup-τ*≤s≤0(∑i=1n|φi(s)|+∑j=1m|ψj(s)|)
holds for allφ(·)∈C([-τ*,0],ℝn)andψ(·)∈C([-τ*,0],ℝm).Lemma 4 (see [30]).
LetV(·):[t0-τ,∞)→[0,∞)be a continuous function such that
(8)V˙(t)≤-aV(t)+bV-(t)
is satisfied fort≥t0. Ifa>b>0, then
(9)V(t)≤V-(t0)e-λ(t-t0),t≥t0,
whereV-(t)=supt-τ≤s≤tV(s),λ>0is the unique positive real root of the equation-a+λ+beλτ=0.Lemma 5 (see [31]).
Letq≥0,τ>0,μk>0(k=1,2,…), andpbe constants, and assume thatV(t)is a piecewise continuous nonnegative function satisfying
(10)D+V(t)≤pV(t)+qV-(t),t≥t0,t≠tkV(tk+)≤μkV(tk),k=1,2,….
If there exists constantβsuch that
(11)lnμktk-tk-1≤β,p+dq+β<0,
hold fork=1,2,…, then
(12)V(t)≤dV-(t0)e-λ(t-t0),
whereV-(t)=supt-τ≤s≤tV(s),d=supk{eβ(tk-tk-1),e-β(tk-tk-1)}<∞, andλis the unique positive root of the equationλ+p+dqeλτ+β=0.


## 3. Main Results

In this section, some sufficient criteria will be given based on the sliding intermittent control scheme. First, the global exponential stability of the closed-loop hybrid neural networks (4) is analyzed, and then several criteria are obtained by setting different parameters in the sliding intermittent controller.Theorem 6.
Assume the upper bound delayτ*<min{θT,(1-θ)T}and the external imposed impulsive strengths satisfyhik(1), hjk(2)≠-1. Under the sliding intermittent control, the closed-loop control system (4) is globally exponentially stable if there exist constantsβandki(1), kj(2)such that the following conditions hold:
(i)(13)ai-ki(1)>Lig∑j=1m|qji|,i∈ℐn,bj-kj(2)>Ljf∑i=1n|pij|,j∈ℐm,
(ii)(14)lnμk+1tk+1-tk≤β,ρ+db~1+β<0,k∈ℕ+∖{il∣l=0,1,2,…},
(iii)(15)a~1>b~1,ϱ≜η+λ1(θ-γ)-λ2(θ+γ)-lndT>0,
wherea~1=mini,j{ai-ki(1)-η,bj-kj(2)-η};b~1=maxi{Lig∑j=1m‍|qji|eησji}+maxj{Ljf∑i=1n‍|pij|eητij};μk=maxi,j{|1+hik(1)|,|1+hjk(2)|};ρ=maxi,j{η-ai,η-bj};d=supk{eβ(tk+1-tk),e-β(tk+1-tk)};tilis used to denote the last impulsive moment on the interval[(l+θ)T,(l+1)T)withtil<(l+1)T-τ*;γ=τ*/T;0<η≤η*;η*=mini,j{θi*,ϑj*∣Fi(θi*)=0,Gj(ϑj*)=0}with functionsFi(·),Gj(·)defined as
(16)Fi(θi)=ai-ki(1)-θi-∑j=1m|qji|Ligeθiσji,θi∈[0,∞),Gj(ϑj)=bj-kj(2)-ϑj-∑i=1n|pij|Ljfeϑjτij,ϑj∈[0,∞),
andλ1, λ2are, respectively, the unique positive real root of the equations-a~1+λ1+b~1eλ1τ*=0andλ2+ρ+db~1eλ2τ*+β=0.
More specifically, we have the following inequality:(17)∑i=1n|xi(t)|+∑j=1m|yj(t)|≤Me-ϱtsup-τ*≤s≤0(∑i=1n|φi(s)|+∑j=1m|ψj(s)|),t>0,
withM=max{eλ1(θ-γ)θT,eλ2(1-θ-γ)θT+(1-θ)lnd}.Proof.
For the functionsFi(·)andGj(·)defined in (16), from the condition (13), it is clear that
(18)Fi(0)=ai-ki(1)-∑j=1m|qji|Lig>0,Gj(0)=bj-kj(2)-∑i=1n|pij|Ljf>0.
SinceFi(·)andGj(·)are continuous on[0,∞) and limθi→∞Fi(θi)=-∞,limϑj→∞Gj(ϑj)=-∞, there must exist constantsθi*>0andϑj*>0such thatFi(θi*)=0 andGj(ϑj*)=0. By settingη*=min{θ1*,θ2*,…,θn*,ϑ1*,ϑ2*,…,ϑm*}, one obtains that, for any0<η≤η*,
(19)Fi(η)=ai-ki(1)-η-∑j=1m|qji|Ligeησji≥0,Gj(η)=bj-kj(2)-η-∑i=1n|pij|Ljfeητij≥0.
Now consider the Lyapunov function defined as follows:
(20)V(t)=∑i=1nui(t)+∑j=1mvj(t),t≥0,
whereui(t)=eηt|xi(t)|andvj(t)=eηt|yj(t)|. Obviously,V(t)is a positive definite function fort≥0.
(
1
) Whent∈[lT,(l+θ)T),l∈ℕ0+, one is easy to have
(21)D+ui(t)≤-(ai-ki(1)-η)ui(t)+∑j=1m|pij|Ljfeητijvj(t-τij),D+vj(t)≤-(bj-kj(2)-η)vj(t)+∑i=1n|qji|Ligeησjiui(t-σji).
Calculating the upper right Dini derivative ofV(t)along the solutions of network (4), from the above inequality, we get
(22)D+V(t)≤-∑i=1n(ai-ki(1)-η)ui(t)-∑j=1m(bj-kj(2)-η)vj(t)+∑i=1n∑j=1m|pij|Ljfeητijvj(t-τij)+∑j=1m∑i=1n|qji|Ligeησjiui(t-σji)≤-a~1V(t)+b~1(supt-τ*≤s≤tV(s)),
wherea~1=mini,j{ai-ki(1)-η,bj-kj(2)-η} andb~1=maxi{Lig∑j=1m‍|qji|eησji}+maxj{Ljf∑i=1n‍|pij|eητij}. By Lemma 4, one has
(23)V(t)≤e-λ1(t-lT)(suplT-τ*≤s≤lTV(s))=e-λ1(t-lT)V-(lT),t∈[lT,(l+θ)T),
whereλ1is the unique positive real root of the equation-a~1+λ1+b~1eλ1τ*=0.
(
2
) Whent∈[(l+θ)T,(l+1)T) andt≠tk, l∈ℕ0+, k∈ℕ+, it is easy to have
(24)D+ui(t)≤-(ai-η)ui(t)+∑j=1m|pij|Ljfeητijvj(t-τij),D+vj(t)≤-(bj-η)vj(t)+∑i=1n|qji|Ligeησjiui(t-σji).
Without loss of generality, supposet∈(tk-1,tk], k≤il, wheretilis assumed to be the last impulsive moment on the interval[(l+θ)T,(l+1)T). It follows from the inequality (24) that
(25)D+V(t)≤∑i=1n(η-ai)ui(t)+∑j=1m(η-bj)vj(t)+∑i=1n(Lig∑j=1meησji|qji|)ui(t-σji)+∑j=1m(Ljf∑i=1neητij|pij|)vj(t-τij)≤ρV(t)+b~1(supt-τ*≤s≤tV(s)),
whereρ=maxi,j{η-ai,η-bj};b~1=maxi{Lig∑j=1m‍eησji|qji|}+maxj{Ljf∑i=1n‍eητij|pij|}.
Whent∈[(l+θ)T,(l+1)T)andt=tk,
(26)xi(tk+)=xi(tk)+Ik(1)(xi(tk))=(1+hik(1))xi(tk),yj(tk+)=yj(tk)+Ik(2)(yj(tk))=(1+hjk(2))yj(tk).
Considering the condition thathik(1), hjk(2)≠-1, we haveμk>0 and
(27)ui(tk+)=eηtk+|xi(tk+)|≤|1+hik(1)|ui(tk)≤μkui(tk),i∈ℐn,vj(tk+)=eηtk+|yj(tk+)|≤|1+hjk(2)|vj(tk)≤μkvj(tk),j∈ℐm,
which infers that
(28)V(tk+)=∑i=1nui(tk+)+∑j=1mvj(tk+)≤μkV(tk).
From condition (14) and Lemma 5, one has
(29)V(t)≤de-λ2(t-(l+θ)T)V-((l+θ)T),t∈[(l+θ)T,(l+1)T),
whered=supk{eβ(tk-tk-1),e-β(tk-tk-1)} and λ2is the unique positive real root of the equationλ2+ρ+db~1eλ2τ*+β=0.
(
3
) Now, we are ready to estimateV(t)based on the inequalities (23) and (29) with the method of induction.Whent∈[0,θT), one obtains
(30)V(t)≤e-λ1tV-(0).Whent∈[θT,T), one can derive
(31)V(t)≤de-λ2(t-θT)V-(θT)≤de-[λ2(t-θT)+λ1(θT-τ*)]V-(0).Whent∈[T,(1+θ)T), we have
(32)V(t)≤e-λ1(t-T)V-(T)≤de-[λ2(T-τ*-θT)+λ1(t-(1-θ)T-τ*)]V-(0).Whent∈[(1+θ)T,2T), one can derive
(33)V(t)≤de-λ2(t-(1+θ)T)V-((1+θ)T)≤d2e-[λ2(t-2θT-τ*)+2λ1(θT-τ*)]V-(0).
By induction, one can derive the following estimation ofV(t)for any integerl∈ℕ0+:whent∈[lT,(l+θ)T),
(34)V(t)≤dle-[lλ2(T-θT-τ*)+λ1(t-l(T-θT+τ*))]V-(0),and whent∈[(l+θ)T,(l+1)T),
(35)V(t)≤dl+1e-[λ2(t-(l+1)θT-lτ*)+(l+1)λ1(θT-τ*)]V-(0).
By settingγ=τ*/Tand substituting it to the above two inequalities, one has, fort∈[lT,(l+θ)T),
(36)V(t)≤dle-[λ2lT(1-θ-γ)+λ1(t-lT)+λ1(θ-γ)lT]V-(0)≤dle[λ2(θ+γ)lT-λ1(θ-γ)lT]V-(0)≤dle[λ2(θ+γ)t+λ1(θ-γ)(-t+θT)]V-(0)=dle-[λ1(θ-γ)-λ2(θ+γ)]teλ1(θ-γ)θTV-(0)≤e(lnd/T)te-[λ1(θ-γ)-λ2(θ+γ)]teλ1(θ-γ)θTV-(0)≤e-[λ1(θ-γ)-λ2(θ+γ)-((lnd)/T)]tMV-(0),
and, fort∈[(l+θ)T,(l+1)T),
(37)V(t)≤dl+1e-[λ2(t-(l+1)θT-lγT)+λ1(θ-γ)(l+1)T]V-(0)≤dl+1e[λ2θT+λ2lT(θ+γ)-λ1(θ-γ)t]V-(0)≤dl+1e[λ2θT+λ2(t-θT)(θ+γ)-λ1(θ-γ)t]V-(0)=dl+1e[λ2(θ+γ)t-λ1(θ-γ)t]+λ2θT(1-θ-γ)V-(0)≤e((t/T)-θ+1)lnde-[λ1(θ-γ)-λ2(θ+γ)]teλ2θT(1-θ-γ)V-(0)≤e-[λ1(θ-γ)-λ2(θ+γ)-((lnd)/T)]tMV-(0).
Therefore, we have
(38)V(t)≤e-[λ1(θ-γ)-λ2(θ+γ)-((lnd)/T)]tMV-(0),t>0,
which means that, for anyt>0,
(39)∑i=1n|xi(t)|+∑j=1m|yj(t)|≤e-[η+λ1(θ-γ)-λ2(θ+γ)-((lnd)/T)]tMV-(0),
whereM=max{eλ1(θ-γ)θT,eλ2(1-θ-γ)θT+(1-θ)lnd}. And the conclusion that the origin of network (4) is exponentially stable. The proof is complete.Remark 7.
In the above Theorem6, the control widthθ∈(0,1)which does not include the boundary cases. If the parameterθ→0, the controlled system approximates to impulsive neural networks. If the parameterθ→1, the controlled system approximates to continuous neural networks. And the controlled system can be handled, respectively, by the methods of the impulsive system and the continuous system. In order to avoid such boundary cases, usually we can takeθ=0.5; that is, the periodically intermittent controller and the impulsive controller play a significant role in the process of control; this results a switched neural networks.Remark 8.
Most of the literatures [28, 32] concerning the global exponential stability of the delayed BAM neural networks with impulses have focused on the stable system. Namely, without the impulsive disturbance, the original neural networks are stable, and under the impulsive disturbance the system can still be kept stable with particular conditions. While, in this article, the impulses can be viewed as either control input or impulsive disturbance, at the same time, the original system is not required to be stable initially.
If adjustable parameterθ=1, the controlled system turns out to be the following model; that is, the original unstable system (3) is controlled with the continuous feedback controller. Under such case, the conditions in the following proposition will guarantee the closed-loop system to be globally exponentially stable:
(40)x˙i(t)=-aixi(t)+∑j=1mpijfj(yj(t-τij))+ui(1)(t),i∈ℐn,y˙j(t)=-bjyj(t)+∑i=1nqjigi(xi(t-σji))+uj(2)(t),j∈ℐm.Proposition 9.
Under the continuous feedback control scheme, the origin of the closed-loop control system (40) is globally exponentially stable if there exist constantski(1)andkj(2)such that
(41)ai-ki(1)>Lig∑j=1m|qji|,i∈ℐn,bj-kj(2)>Ljf∑i=1n|pij|,j∈ℐm.
Furthermore, one has the following inequality:
(42)∑i=1n|xi(t)|+∑j=1m|yj(t)|≤(b~1τ*+1)e-ηtsup-τ*≤s≤0(∑j=1m‍|ψj(s)|∑i=1n|φi(s)|+∑j=1m|ψj(s)|),t>0,
where the parametersηandb~iare consistent with the ones in Theorem 6.Proof.
Consider the Lyapunov function defined as follows:(43)W(t)=V(t)+∑i=1n∑j=1m(|pij|Ljfeητij∫t-τijt‍vj(s)ds+|qji|Ligeησji∫t-σjit‍ui(s)ds),t≥0.
The upper right Dini derivative ofW(t)with respect to timetalong the solutions of the network (40) can be calculated as follows:
(44)D+W(t)=∑i=1nD+ui(t)+∑j=1mD+vj(t)+∑i=1n∑j=1m|pij|Ljfeητij[vj(t)-vj(t-τij)]+∑j=1m∑i=1n|qji|Ligeησji[ui(t)-ui(t-σji)]≤-∑i=1n(∑j=1m‍ai-ki(1)-η-∑j=1m|qji|Ligeησji)ui(t)-∑j=1m(bj-kj(2)-η-∑i=1n|pij|Ljfeητij)vj(t)=-∑i=1nFi(η)ui(t)-∑j=1mGj(η)vj(t)≤0
which means thatW(t)≤W(0). Hence we have ∑i=1n|xi(t)|+∑j=1m|yj(t)|≤(b~1τ*+1)e-ηtV-(0), and this completes the proof.If the impulsive strengthshik(1)=hjk(2)≡0, namely, there are no impulsive controls on the latter control interval in each control period, which means the closed-loop system is only subject to the continuous feedback control in the preceding control width of each control period. Such a case is then reduced to the pure periodically intermittent control, and the neural network system (4) turns into the following controlled neural network (45). The conditions in the following proposition will guarantee the closed-loop system to be globally exponentially stable. In order to obtain the main result, the following lemma is given firstly
(45)x˙i(t)=-aixi(t)+∑j=1mpijfj(yj(t-τij))+ui(1)(t),lT≤t<(l+θ)T,x˙i(t)=-aixi(t)+∑j=1mpijfj(yj(t-τij)),(l+θ)T≤t<(l+1)T,y˙j(t)=-bjyj(t)+∑i=1nqjigi(xi(t-σji))+uj(2)(t),lT≤t<(l+θ)T,y˙j(t)=-bjyj(t)+∑i=1nqjigi(xi(t-σji)),(l+θ)T≤t<(l+1)T.Lemma 10 (see [23]).
LetV(·):[t0-τ,∞)→[0,∞)be a continuous function such that
(46)V˙(t)≤aV(t)+b(supt-τ≤s≤tV(s))
is satisfied fort≥t0. Ifa>0,b>0, then
(47)V(t)≤supt-τ≤s≤tV(s)≤(supt0-τ≤s≤t0V(s))e(a+b)(t-t0),t≥t0.Proposition 11.
Assuming the upper bound delayτ*<min{θT,(1-θ)T}, under the periodically intermittent control, the closed-loop control system (45) is globally exponentially stable if the control gainski(1) andkj(2)satisfy the following conditions:(i)
(48)ai-ki(1)>Lig∑j=1m|qji|,i∈ℐn,bj-kj(2)>Ljf∑i=1n|pij|,j∈ℐm,(ii)
(49)a~1>b~1,η+λ1(θ-γ)-(ρ~+b~1)(1-θ)>0.Moreover, we have
(50)∑i=1n|xi(t)|+∑j=1m|yj(t)|≤M-e-[η+λ1(θ-γ)-(ρ~+b~1)(1-θ)]t×sup-τ*≤s≤0(∑i=1n|φi(s)|+∑j=1m|ψj(s)|),t>0,
whereρ~=maxi,j{η-ai,η-bj,κ},andκis any positive constant; M~=eλ1θT(θ-γ), and the parametersa~1, b~1, η, λ1, and γare consistent with those in Theorem 6.Proof.
Considering the same Lyapunov function as that in Theorem6, by similar analytical technique, one can get the following results on the control period and the control width as follows.(1)
Whent∈[lT,(l+θ)T),l∈ℕ0+, we haveD+V(t)≤-a~1V(t)+b~1V-(t). By Lemma 4, one obtains that, for anyt∈[lT,(l+θ)T), V(t)≤e-λ1(t-lT)V-(lT).(2)
Whent∈[(l+θ)T,(l+1)T),l∈ℕ0+, we haveD+V(t)≤ρ~V(t)+b~1V-(t). By Lemma 10, it is derived that, for anyt∈[(l+θ)T, (l+1)T),V(t)≤e(ρ~+b~1)(t-(l+θ)T)V-((l+θ)T).
From the above inequality relationships, by similar estimation procedure, we can get the following conclusion:(51)V(t)≤e-λ1(t-lT)+l(ρ~+b~1)(1-θ)T-λ1l(θ-τ*)V-(0),t∈[lT,(l+θ)T),V(t)≤e-λ1(l+1)(θT-τ*)+l(ρ~+b~1)(1-θ)T+(ρ~+b~1)(t-lT-θT)V-(0),t∈[(l+θ)T,(l+1)T).
By the notational expressionτ*=γT, one can further obtain
(52)V(t)≤e[-λ1(θ-γ)+(ρ~+b~1)(1-θ)]teλ1θT(θ-γ)V-(0),t∈[lT,(l+θ)T),V(t)≤e[-λ1(θ-γ)+(ρ~+b~1)(1-θ)]tV-(0),t∈[(l+θ)T,(l+1)T).
Hence it follows that, for anyt>0,
(53)∑i=1n|xi(t)|+∑j=1m|yj(t)|≤M-e-[η+λ1(θ-γ)-(ρ~+b~1)(1-θ)]t×sup-τ*≤s≤0(∑i=1n|φi(s)|+∑j=1m|ψj(s)|),
and this completes the proof.If the adjustable parameterθ=0, that is, the original unstable system (3) is subjected to the impulsive controller, then system (4) becomes the following impulsive neural networks (54). Under such a case, it would be natural to assume that the frequency of the impulses should not be too low; that is, some restrictions on the impulsive periods and the impulsive strengths are needed. The conditions in the following proposition will guarantee the closed-loop system (54) to be globally exponentially stable:
(54)x˙i(t)=-aixi(t)+∑j=1mpijfj(yj(t-τij)),t≥0,t≠tk,k∈ℕ+,Δxi(tk)=Ik(1)(xi(tk)),k∈ℕ+,i∈ℐn,y˙j(t)=-bjyj(t)+∑i=1nqjigi(xi(t-σji)),t≥0,t≠tk,k∈ℕ+,Δyj(tk)=Ik(2)(yj(tk)),k∈ℕ+,j∈ℐm.Proposition 12.
Assume the external imposed impulsive strengthshik(1),hjk(2)≠-1. The origin of the closed-loop control system (54) is globally exponentially stable if there exists constantβsuch that, fork=1,2,…, the following conditions hold:
(55)lnμktk-tk-1≤β,ρ+db~1+β<0.
More specifically, we have
(56)∑i=1n|xi(t)|+∑j=1m|yj(t)|≤de-(η+λ2)tsup-τ*≤s≤0(∑i=1n|φi(s)|+∑j=1m|ψj(s)|),t>0,
where the parametersμk, ρ, d, b~1, andηare consistent with those in Theorem 6.Proof.
Considering the same Lyapunov function as that in Theorem6, by similar analytical technique, one can get the following results on the impulsive interval and the impulse moments as follows.(1)
Whent∈(tk-1,tk], the upper right Dini derivative ofV(t)along the solution of (54) is depicted asD+V(t)≤ρV(t)+b~1(supt-τ*≤s≤tV(s)).(2)
Whent=tk,k∈ℕ+,V(tk+)≤μkV(tk).
From Lemma5 and the conditions in (55), it follows that
(57)V(t)≤d(sup0-τ*≤s≤0V(s))e-λ2(t-0),
which means
(58)∑i=1n|xi(t)|+∑j=1m|yj(t)|≤e-ηtde-λ2tV-(0)≤de-(η+λ2)tV-(0),t>0
and the proof is completed.If the continuous feedback control gainski(1)=kj(2)≡0, that is, there is no feedback on the preceding control interval in a control period, and only impulsive control is imposed on the latter control interval, this means the system is only under the piecewise impulsive (not the uniformly distributed) control. As for such case, the impulsive system is reduced to the following one:
(59)x˙i(t)=-aixi(t)+∑j=1mpijfj(yj(t-τij)),t∈[lT,(l+θ)T)∪{[(l+θ)T,(l+1)T)∖{t≠tk}k=1∞},Δxi(t)=Ik(1)(xi(t)),t=tk,y˙j(t)=-bjyj(t)+∑i=1nqjigi(xi(t-σji)),t∈[lT,(l+θ)T)∪{[(l+θ)T,(l+1)T)∖{t≠tk}k=1∞},Δyj(t)=Ik(2)(yj(t)),t=tk.
From (59), it is noticed that the occurrence of the impulses is not uniformly distributed since the impulses never occur on the interval[lT,(l+θ)T),l∈ℕ0+, whereas they frequently occur on the interval((l+θ)T,(l+1)T),l∈ℕ0+. By observing the proof of Proposition 12, the result is somewhat more conservative especially when the control periodTis very large and the control widthθapproaches to one. In order to describe the conservatism for such case, we will utilize the notation of average impulsive interval proposed in [21] to characterize the occurrence frequency of the impulses. The definition of average impulsive interval and the corresponding impulsive differential inequality are given firstly.Definition 13 ([21] average impulsive interval).
The average impulsive interval of the impulsive sequence{tk}k=1∞is equal toTaif there exist positive integerN0and positive numberTasuch that
(60)N(κ2,κ1)≥κ2-κ1Ta-N0,∀κ2≥κ1≥0,
whereN(κ2,κ1)denotes the number of impulsive times of the impulsive sequence{tk}k=1∞on the interval(κ1,κ2).Lemma 14 (see Lakshmikantham et al., Theorem1.4.1, [33, 34]).
LetPC(ℝ+,ℝ)(PC1(ℝ+,ℝ)) denote the set of piecewise continuous (piecewise continuously differentiable) functions with first kind of discontinuities attk,k=1,2,…, fromℝ+toℝ. If the following conditions hold,
(61)m(t)∈PC1(ℝ+,ℝ)isleftcontinuousattk,k=1,2,…,D+m(t)≤p(t)m(t)+q(t),t≥t0,t≠tk,m(tk+)≤dkm(tk)+bk,k=1,2,…,
wherep(t), q(t)∈PC(ℝ+,ℝ),dk≥0andbkare real constants, then
(62)m(t)≤m(t0)∏t0<tk<tdkexp(∫t0tp(s)ds)+∑t0<tk<t(∏t0<tj<tdjexp(∫tktp(s)ds))bk+∫t0t∏s<tk<tdkexp(∫stp(ξ)dξ)q(s)ds,t≥t0.Lemma 15.
Letq≥0,τ>0,0<μ<1, andpbe constants, and assume thatV(t)is a piecewise continuous nonnegative function satisfying
(63)D+V(t)≤pV(t)+qV-(t),t≥t0,t≠tk,V(tk+)≤μV(tk),k=1,2,…,
and the average impulsive interval of the impulsive sequence{tk}k=1∞is equal toTa. If the following inequality holds,
(64)p+lnμTa+μ-N0q<0,
then one has
(65)V(t)≤μ-N0V-(t0)e-λ(t-t0),
whereV-(t)=supt-τ≤s≤tV(s),λis the unique positive root of the equationλ + p + (lnμ/Ta) + μ-N0qeλτ = 0.Proof.
By Lemma14 and the definition of average impulsive interval, it follows from (63) that, fort≥t0,
(66)V(t)≤V-(t0)μN(t,t0)ep(t-t0)+∫t0tep(t-s)μN(t,s)qV-(s)ds≤V-(t0)μ(((t-t0)/Ta)-N0)ep(t-t0)+∫t0tep(t-s)μ(((t-s)/Ta)-N0)qV-(s)ds≤μ-N0(∫t0t‍V-(t0)e[p+(lnμ/Ta)](t-t0)+∫t0te[p+(lnμ/Ta)](t-s)qV-(s)ds).
Denoteϕ(λ)=λ+p+(lnμ/Ta)+μ-N0qeλτ. Sinceϕ(0)=p+(lnμ/Ta)+μ-N0q<0, andlimλ→∞ϕ(λ)=+∞andϕ′(λ)=1+μ-N0qτeλτ>0, one knows thatϕ(λ)=0has a unique positive root.
Next, it is claimed that, for allt>t0,
(67)V(t)<μ-N0(supt0-τ≤s≤t0V(s))e-λ(t-t0).
Whent∈[t0-τ,t0],
(68)V(t)<V-(t0)<μ-N0V-(t0)e-λ(t-t0).
Supposing (67) is not always true fort>t0, there must exist one time pointt*>t0such that
(69)V(t*)=μ-N0V-(t0)e-λ(t*-t0),V(t)<μ-N0V-(t0)e-λ(t-t0),t0-τ≤t<t*.
From inequalities (66) and (69), one can obtain that
(70)V(t*)≤μ-N0(∫t0t*‍V-(t0)e[p+(lnμ/Ta)](t*-t0)+∫t0t*e[p+(lnμ/Ta)](t*-s)qV-(s)ds)<μ-N0(∫t0t*‍V-(t0)e[p+(lnμ/Ta)](t*-t0)+∫t0t*e[p+(lnμ/Ta)](t*-s)qμ-N0V-(t0)e-λ(s-τ-t0)ds)=μ-N0V-(t0)e[p+(lnμ/Ta)](t*-t0)+qμ-2N0V-(t0)eλτ∫t0t*e[p+(lnμ/Ta)](t*-s)e-λ(s-t0)ds.
Considering the equalityψ(λ)=λ+p+(lnμ/Ta)+μ-N0qeλτ=0, one has
(71)V(t*)<μ-N0V-(t0)e-λ(t*-t0)×(∫t0t*‍e[-μ-N0qeλτ](t*-t0)+qμ-N0eλτ∫t0t*e[-μ-N0qeλτ](t*-s)ds)=μ-N0V-(t0)e-λ(t*-t0)e-[μ-N0qeλτ]t*×(e[μ-N0qeλτ]t0+e[μ-N0qeλτ]t*-e[μ-N0qeλτ]t0)=μ-N0V-(t0)e-λ(t*-t0)
which contradicts with the first inequality of (69). Therefore, the inequality (67) holds, and this completes the proof.The following proposition will guarantee the system (59) to be globally exponentially stable.Proposition 16.
Assume the external imposed impulsive strengthshik(1), hjk(2)∈(-2,-1)∪(-1,0), and the average impulsive interval of the impulsive sequence{tk}k=1∞is equal toTa. The impulsive system (59) is globally exponentially stable if the following condition holds:
(72)ρ+lnμTa+μ-N0b~1<0.
Moreover, we have
(73)∑i=1n|xi(t)|+∑j=1m|yj(t)|≤μ-N0e-(η+λ)tsup-τ*≤s≤0(∑i=1n|φi(s)|+∑j=1m|ψj(s)|),t>0,
whereμ=supk{μk}∈(0,1), andμk, ρ, and b~1are defined as in Theorem 6 andλis the unique positive root of the equationλ+p+(lnμ/Ta)+μ-N0qeλτ=0.Proof.
Considering the same Lyapunov function as that in Theorem6, the following results on the impulsive interval and the impulse moments can be obtained.(1)
Whent∈[lT,(l+θ)T)∖{tk}k=1∞,l∈ℕ0+, the upper right Dini derivative ofV(t)along the solutions of (59) is depicted asD+V(t)≤ρV(t)+b~1V-(t).(2)
Whent=tk, k∈ℕ+,V(tk+)≤μV(tk).
From the above two inequality relationships, and Lemma15, one has
(74)∑i=1n|xi(t)|+∑j=1m|yj(t)|≤μ-N0e-(η+λ)t(supt0-τ≤s≤t0V(s)),t>0,
whereλis the unique positive root of the equationλ+p+(lnμ/Ta)+μ-N0qeλτ=0. The proof is completed.Remark 17.
It should be pointed out that, in the preceding control width of the control period, other kinds of continuous controllers can also be used to achieve the same performance. For example, in [35], the adaptive control scheme has been employed in the control width instead of the continuous state feedback, where the adjusting gains can be designed based on different norms. We can borrow such an idea to the sliding intermittent control design. Moreover, the sliding width parameter can be{θi}rather than the fixed widthθ, and the period can be{Ti}rather than constantT. By doing so, we might obtain more general conditions. On the other hand, the phenomena of stochastic nonlinearities are extremely ubiquitous in practical controlled systems [36–39]; hence it is more reasonable to consider neural networks with random nonlinearities, and this will be our future works.


## 4. Illustrative Example

In this section, we present some examples to illustrate the applicability and efficiency of the proposed control scheme.Example 1.
Considering the following extensively studied BAM neural system,(75)x˙(t)=-ax(t)+pf(y(t-τ)),y˙(t)=-by(t)+qg(x(t-σ))
witha=1.922,p=9.8501,b=1.1631,q=8.2311,τ=σ=3, andf(x)=g(x)=1/(1+e-x)-1/2. Obviously,Lf=Lg=0.25. The initial condition is given asx(t)=-0.43, y(t)=0.42, t∈[-3,0]. With the above system parameters, the phase diagram of system (75) is given in Figure 2. Obviously, the origin of system (75) is not stable. We will design the sliding intermittent controller to stabilize it.Figure 2
The phase diagram of the original system (75).Applying the sliding intermittent controller to the unstable system (75), one can derive the following system:
(76)x˙(t)=-ax(t)+pf(y(t-τ))+k1x(t),t∈[lT,(l+θ)T)x˙(t)=-ax(t)+pf(y(t-τ)),t∈[(l+θ)T,(l+1)T),t≠tk,Δx(tk)=h1x(tk),t=tk,y˙(t)=-by(t)+qg(x(t-σ))+k2y(t),t∈[lT,(l+θ)T),y˙(t)=-by(t)+qg(x(t-σ)),t∈[(l+θ)T,(l+1)T),t≠tk,Δy(tk)=h2y(tk),t=tk.In the following, we will give the convergence results by simulating the system (76). Firstly, by setting the continuous feedback gainsk1=-4.8512,k2=-4.6378and the impulsive strengthsh1=h2=-0.15. With the above parameters setting, calculations show thatη*=0.2679andμ=0.85. By setting the control periodT=10, we have control width0.3<θ<0.7andγ=0.3. Here we take the parameterθ=0.5. If we utilize uniform distributed impulsive sequences (tk-tk-1=0.01) in the latter control width of the control period, it is easy to check that whenη=0.0221, we haveρ=-1.1410,a~1=5.7788, andb~1=4.8302. By choosingβ=-4.8519, one hasd=1.1765,lnμ/(tk-tk-1)-β=-11.4000, ρ+db~1+β=-0.3104,λ1=0.0565,λ2=0.0168, and η+λ1(θ-γ)-λ2(θ+γ)-lnd/T=0.0037, which means all the conditions in Theorem 6 hold. The simulation result under the sliding intermittent control is given in Figure 3.Figure 3
Simulation of the system (75) under the sliding intermittent control.Next, the corresponding convergence results under the propositions obtained in this article will be illustrated.If we utilize the continuous feedback control, by taking the same continuous feedback gains as the above sliding intermittent control, the conditions in Proposition9 hold, and the simulation result under the continuous feedback control is given in Figure 4.Figure 4
Simulation of the system (75) under the continuous feedback control.If we use pure periodically intermittent control, by setting the continuous feedback gainsk1=-6.8512,k2=-6.6378, we getη*=0.3682,ρ~=κ=0.0001. It is easy to know whenη=0.0002, we havea~1=7.8007,b~1=4.5230, and λ1=0.1742. In Proposition 11, by takingT=100, the relationships among the upper bound delay, the control width, and the control period infer that0.03<θ<0.97, while the last inequality in Proposition 11 meansθ>0.9640. Here we set theθ=0.9650, and the simulation result under the periodically intermittent control is given in Figure 5.Figure 5
Simulation of the system (75) under the periodically intermittent control.If we use full impulsive control with uniform distributed impulsive sequences (tk-tk-1=0.04) and the impulsive strengthsμk=0.85, k=1,2,…, it is easy to check whenβ=-3.1262,η=0.0002, the parametersd=1.1332,ρ=-1.1629, and b~1=4.5230, all the conditions in Proposition 12 satisfied. If we use the semi-impulsive control scheme and set the impulsive sequences satisfying the average impulsive interval with Ta=0.038andN0=1; it is easy to checkρ+(lnμ/Ta)+μ-N0b~1=-0.1185<0, which infers that the condition in Proposition 16 holds. The simulation results for system (75) with the full impulsive control and the semi-impulsive control are given in Figures 6 and 7.Figure 6
Simulation of the system (75) under the full impulsive control.Figure 7
Simulation of the system (75) under the semi impulsive control.Remark 18.
From the above verifying process, it can be found that the sliding intermittent control is much better than the pure periodically intermittent control. More specifically, in the periodically intermittent control, the control periodT=100and the control widthθ=0.965, while in the sliding intermittent control, the control periodT=10and the control widthθ=0.5. As for the impulsive control, the full impulsive control is better than the semi-impulsive control in that the earlier converges faster. When dealing with the nonuniformly distributed impulsive sequence, the result derived in Proposition 16 is less conservative.Example 2.
Consider the following unstable delayed BAM neural network, and we will show that the sliding intermittent control benefits the stabilization of the unstable system:(77)x˙i(t)=-aixi(t)+∑j=13pijfj(yj(t-τij)),i∈ℐ3,y˙j(t)=-bjyj(t)+∑i=14qjigi(xi(t-σji)),j∈ℐ4,
whereA=diag{3.1220,2.3156,2.2683},B=diag{2.9631,2.3456,2.6341,3.0726},
(78)P=[7.85012.30703.22804.71913.24636.05895.37512.26092.01591.78032.66015.7647],Q=[4.23314.37412.24594.88302.32591.28572.40221.33773.79302.13512.67593.4719].The activations functionsf(x)=g(x)=1/(1+e-x)-1/2withLf=Lg=0.25and the time delaysτ=σ=3. The system has an unstable equilibrium0under the above parameters with the initial functionsx(t)=[-0.73,-0.79,-0.82]T, y(t)=[0.72,0.87,0.83,0.76]T, t∈[-3,0].In the following, we will check the convergence results for system (77) under the sliding intermittent controller. Setting the continuous feedback gainsk(1)=diag{-8.0350,-10.4875,-13.1075},k(2)=diag{-11.1950,-8.4125,-5.7425,-4.1475}, the impulsive strengths are identical withhik(1)=hjk(2)=-0.25, i∈ℐn, j∈ℐm, k∈ℕ+, and the control periodT=15; some calculations show thatη*=0.2604,μ=0.75. We can get the control width0.2<θ<0.8andγ=0.2; here we take the parameterθ=0.4. If we utilize the uniform distributed impulsive sequences (tk-tk-1=0.02) in the latter control width of the control period, it is easy to know whenη=0.1250, we haveρ=-2.1433,a~1=7.0951, and b~1=4.9664. By choosingβ=-4.6841, one haslnμ/(tk-tk-1)-β=-9.7000,d=1.333,ρ+db~1+β=-0.2055,λ1=0.1135,λ2=0.1924, and η + λ1(θ-γ) − λ2(θ+γ) − lnd/T = 0.0131, which mean all the conditions in Theorem 6 hold. The simulation result under the sliding intermittent control is given in Figure 8.Figure 8
Simulation of the system (77) under the sliding intermittent controller.


## 5. Conclusions

In this paper, a sliding intermittent controller has been proposed by unifying the periodically intermittent control with the impulsive control together with continuous feedback control. More specifically, the continuous feedback control is employed as the preceding control width, and the impulsive control is resorted in the latter control width. Furthermore, the adjustable parameterθ∈[0,1]is very flexible in that the continuous feedback control (θ=1), the impulsive control (θ=0), the periodically intermittent control, and the semi-impulsive control (0<θ<1) are all possible cases. Based on the analysis technique and the Lyapunov function approach, the conditions have been constructed for the exponential stability of the delayed BAM neural networks under the proposed control schemes. Finally, numerical simulations are used to illustrate the effectiveness of the control technique.



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*Source: 615947-2013-06-25.xml*