# Finite-Time Stabilization of Memristive Cohen-Grossberg Neural Networks with Time-Varying Delay

Authors: Ge Li; Yaqiong Li; Zhaohui Yuan
Journal: Complexity
 (2018)

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## Abstract

In this paper, the finite-time stabilization problem for memristive Cohen-Grossberg neural networks with time-varying delay is discussed. By using the novel fixed point theory of set-valued maps, we establish the existence theorem of equilibrium point. In order to realize the finite-time stabilization, two different kinds of discontinuous state feedback controllers whether including time-varying delay are designed. Based on the extended Filippov framework and two different kinds of methods whether using finite-time stability theory, some novel sufficient conditions and the upper bound of the settling time for finite-time stabilization are proposed. Finally, two numerical examples are given to demonstrate the validity of theoretical results.


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## Body

## 1. Introduction

Memristor, first hypothesized by Chua in 1971 [1], was realized by Hewlett-Packard (HP) lab in 2008 [2]. The new circuit possesses memory characteristic as biological neurons, which is distinct from other circuit elements such as resistor and capacitor. As a result, memristor is widely used in artificial neural networks for emulating human brain [3]. When using memristor to replace other resistors, researchers can construct a new kind of neural networks named memristor-based neural networks (MNNs). Nowadays, MNNs are widely used in the area of science and technology, such as artificial intelligence, signal processing, and associate memory [3–6].Due to the state-dependent switching feature, MNNs can be viewed as a differential equation with discontinuous right-hand side. Thus, the differential inclusion theory is a new kind of effective methods to study the dynamical behaviors of various MNNs. There are already many theoretical results about MNNs, such as stability and synchronization [7–12]. In [8], the authors proposed a new kind of complex-valued memristor-based neural networks with time-varying delays and studied their exponential stability by using some analytic techniques and constructing a Lyapunov functional. However, due to the switching feature of memristor, the dynamical behaviors of MNNs are always unstable such as oscillation and chaos. Thereby, it is necessary and challengeable to investigate the stabilization problem of unstable MNNs by designing suitable controllers.In most researches about the stabilization of MNNs, the converge time of system states is sufficiently large such as exponential stabilization and asymptotic stabilization. However, researchers always hope the converge time can be shortened in practice. It is worth noting that the finite-time stability theorem is a very effective method to realize this goal. The finite-time stabilization (FTS) indicates the system states can keep in a certain range of equilibrium point after a finite time. Besides the faster converge speed, FTS also has other desirable advantages when compared with the infinite-time stabilization, such as better robustness and disturbance rejection properties [13, 14]. In recent years, the finite-time synchronization and FTS of discontinuous neural networks have received great attention [15–19]. In [16], the finite-time synchronization of time-varying delayed neural networks (DNNs) with discontinuous activations was studied by using the famous finite-time stability theorem. In [17], the authors studied the finite-time projective synchronization issue of memristor-based delay fractional-order neural networks by using the definition of finite-time projective synchronization. On the other hand, the discontinuous controllers were creatively designed to realize the FTS of neural networks with discontinuous right-hand side [18, 19]. However, there are a few papers concerning the finite-time stabilization of MNNs [20–23]. In [20], the authors investigated the FTS of MNNs by designing the discontinuous state feedback controller, but they ignored time delay. In fact, time delay is always inevitable in most neural networks due to the finite transmitting speed between neuron signals. In [23], the authors mainly studied the finite-time stability and synchronization problems of memristor-based fractional-order fuzzy cellular neural network with time delay. For realizing the FTS of discontinuous neural networks, it is worth noting that discontinuous controllers are always more effective than conventional continuous controllers. This is mainly because the uncertain differences of Filippov solutions can be well handled by using discontinuous controllers. On the other hand, Cohen-Grossberg neural networks that was introduced by Cohen and Grossberg in 1983 is also an important kind of neural networks [24]. There always exist many results about the dynamical behaviors of Cohen-Grossberg neural networks [25–28]. In [26], the global and local finite-time synchronization for a class of memristor-based Cohen-Grossberg neural networks with time-varying delays are discussed. In [28], the authors studied the finite-time stability and synchronization problem of a class of memristor-based fractional-order Cohen-Grossberg neural network with time delay by using differential inclusion theory and Gronwalls inequality. However, the time delay is constant and the activation functions are bounded in [28]. To the best of our knowledge, there are few papers concerning the FTS of memristive Cohen-Grossberg neural networks with time-varying delay.The main contributions of this paper have three aspects. First, the existence of equilibrium point is studied by using the novel fixed point theory of set-valued maps. Second, there are few results about the finite-time stabilization of neural networks concerning the factor of memristor, amplification function, and time-varying delay. Third, two discontinuous controllers whether including time-varying delay and two analysis methods whether using finite-time stability theory are considered.The structure of this paper is organized as follows. In Section2, the delayed memristive Cohen-Grossberg neural networks and some preliminary knowledge are presented. In Section 3, the existence theorem of equilibrium point is obtained. In Section 4, main theorems and some sufficient conditions about finite-time stabilization are provided. In Section 5, we illustrate the effectiveness of theoretical results by using numerical examples. Finally, conclusions and future research topics are presented in Section 6.Notations.
ℝ and ℝn denote the space of real number and the n-dimensional Euclidean space, respectively. Let X⊆ℝn, PkcX denotes the collection of all nonempty, compact, and convex subset of X. For any column vector x=x1,x2,…,xnT∈ℝn, the norm of x is defined as x1=∑i=1nxi. Given vectors x=x1,x2,…,xnT and y=y1,y2,…,ynT∈ℝn, x,y=xTy=∑i=1nxiyi denotes the inner product of x and y, and the superscript T represents the transposition of a vector or matrix. Finally, sign· stands for the sign function.


## 2. System Description and Preliminaries

In this section, the memristive Cohen-Grossberg neural networks with time-varying delay and some important preliminaries are introduced. The neural network system considered in this paper is described as follows [26]:
(1)v̇it=−aivitbivit−∑j=1ncijvitfjvjt−∑j=1nwijvitgjvjt−τt−Ii,i∈ℕ,where ℕ=1,2,…,n; vit represents the voltage of capacitor Ci; aivit and bivit denote the amplification function and self-inhibition function, respectively; τt represents the time-varying delay and 0≤τt≤τ, τ̇t≤τ^<1; fjvjt and gjvjt−τt are the activation functions without and with time-varying delay, respectively; Ii represents the external input to the ith neuron; and cijvit and wijvit represent memristive connection weights satisfying
(2)cijvit=WijCi×SGNij,wijvit=W~ijCi×SGNij,in which
(3)SGNij=1,ifi≠j,−1,ifi=j.Wij and W~ij stand for the memductances of memristors Mij and M~ij, respectively. Mij denotes the memristor between the feedback function fjvjt and vit. M~ij denotes the memristor between the time-delayed feedback function gjvjt−τt and vit. According to the current-voltage characteristic of memristor, cijvit and wijvit satisfy the following properties:
(4)cijvit=c^ij,vit<γi,cˇij,vit≥γi,wijvit=w^ij,vit<γi,wˇij,vit≥γi,where the switching jumps γi>0, and c^ij, cˇij, w^ij, and wˇij are constants, i, j∈ℕ.Throughout this paper, we need the following assumptions:Assumption 1.
aiv is continuous and bounded. There exist positive constants a¯i and a¯i such that 0<a¯i<aiv<a¯i,v∈ℝ, i∈ℕ.Assumption 2.
biv is differentiable, bi0=0 and there exists a positive constant li such that 0<li≤bi′v,v∈ℝ, i∈ℕ.Assumption 3.
Fori∈ℕ and any two different x, y∈ℝ, the activation functions fi and gi satisfy Lipschitz conditions, i.e., there exist positive constants Li and Ki such that
(5)fix−fiy≤Lix−y,gix−giy≤Kix−y.Assumption 4.
Fori∈ℕ, there exist positive constants λi, βi, δi, and θi such that the activation functions fi and gi satisfy the following conditions
(6)fix≤λix+βi,gix≤δix+θi,∀x∈ℝ.By taking the equivalent transformation method [26, 27], the amplification function in the delayed memristive Cohen-Grossberg neural networks (1) can be well handled. Let us choose a transformation function hiv such that
(7)ddvhiv=1aiv,hi0=0,v∈ℝ,i∈ℕ.According to Assumption1, 1/aiv is continuous and positive. Thus, hiv is a strictly increasing function. Denoting xit=hivit, thus ẋit=v̇it/aivit and vit=hi−1xit.Taking the above variable transformations, it follows from (1) that
(8)ẋit=−bihi−1xit+∑j=1ncijhi−1xitfjhj−1xjt+∑j=1nwijhi−1xitgjhj−1xjt−τt+Ii,i∈ℕ.Next, we introduce some important definitions and lemmas for subsequent analysis. For more relevant details, we can refer to [29]. For τ>0, C−τ,0,ℝn represents the collection of all continuous functions ϕ:−τ,0⟶ℝn and denotes the Banach space equipped with the norm ϕC=sup−τ<t<0ϕt. If T∈0,+∞, for any continuous function xt:−τ,T⟶ℝn, xt can be defined as xts=xt+s,s∈−τ,0 for all t∈0,T. Then, xt∈C−τ,0,ℝn.Definition 1 ([20]).
Let setX⊆ℝn, if for each point x of X, there corresponds a nonempty set Fx⊆ℝn, then x⟶Fx is said to be a set-valued map from X⟶ℝn. A set-valued map F with nonempty values is said to be upper semicontinuous (USC) at x0∈X, if for any open set N containing Fx0, there exists a neighborhood M of x0, such that FM⊆N.Definition 2 ([30]).
Consider a differential equation with discontinuous right-hand sidedx/dt=ft,xt. Let us define a Filippov set-valued map Ft,xt (i.e., Filippov regularization) as follows:
(9)ℱt,xt=⋂δ~>0⋂measN=0co¯ft,ℬxt,δ~\N,where measN represents the Lebesgue measure of set N; co¯E represents the closure of the convex hull of some set E; intersection is taken over all sets N of Lebesgue measure zero and over all δ~>0; Bxt,δ~≔yt∈C−τ,0,ℝn:yt−xtC<δ~. A vector-valued function xt defined on a nongenerate internal I⊆ℝ is said to be a Filippov solution of the discontinuous differential equation, if it is absolutely continuous on any compact subinterval t1,t2 of I, and also satisfies the differential inclusion dx/dt∈Ft,xt, for almost all t∈I.Definition 3 ([30]).
A vector-valued functionvt=v1t,v2t,…,vntT:−τ,T⟶ℝn, T∈0,+∞ is a Filippov solution of system (1) if
(i)
vt is continuous on −τ,T and absolutely continuous on any compact subinterval of 0,T
(ii)
for almost allt∈0,T, vt satisfies the following differential inclusion:
(10)v̇it∈−aivitbivit−∑j=1nco¯cijvitfjvjt−∑j=1nco¯wijvitgjvjt−τt−Ii≜ℱit,vt,i∈ℕ,where
(11)co¯cijvit=c^ij,vit<γi,co¯c¯ij,c¯ij,vit=γi,cˇij,vit>γi,co¯wijvit=w^ij,vit<γi,co¯w¯ij,w¯ij,vit=γi,wˇij,vit>γi,c¯ij=minc^ij,cˇij,c¯ij=maxc^ij,cˇij,w¯ij=minw^ij,wˇij,w¯ij=maxw^ij,wˇij.
Obviously, the set-valued mapt,v⟶F1t,vt,F2t,vt,…,Fnt,vtT is USC and measurable. By using the measurable selection theorem [29], there exist measurable functions γijvit∈co¯cijvit, ηijvit∈co¯wijvit, for a.e. t≥0, such that
(12)v̇it=−aivitbivit−∑j=1nγijvitfjvjt−∑j=1nηijvitgjvjt−τt−Ii.Based on Definition3, we can obtain that the Filippov solution xt=x1t,x2t,…,xntT of system (8) satisfies
(13)ẋit∈−bihi−1xit+∑j=1nco¯cijhi−1xitfjhj−1xjt+∑j=1nco¯wijhi−1xitgjhj−1xjt−τt+Ii,i∈ℕ,where
(14)co¯cijhi−1xit=c^ij,hi−1xit<γi,co¯c¯ij,c¯ij,hi−1xit=γi,cˇij,hi−1xit>γi,co¯wijhi−1xit=w^ij,hi−1xit<γi,co¯w¯ij,w¯ij,hi−1xit=γi,wˇij,hi−1xit>γi.Equivalently, there exist(15)γijhi−1xit∈co¯cijhi−1xit,ηijhi−1xit∈co¯wijhi−1xit,for a.e. t≥0, such that
(16)ẋit=−bihi−1xit+∑j=1nγijhi−1xitfjhj−1xjt+∑j=1nηijhi−1xitgjhj−1xjt−τt+Ii.Definition 4 ([20]).
A constant vectorx∗=x1∗,x2∗,…,xn∗T∈ℝn is said to be an equilibrium point of system (8), if and only if,
(17)0∈−bihi−1xi∗+∑j=1nco¯cijhi−1xi∗fjhj−1xj∗+∑j=1nco¯wijhi−1xi∗gjhj−1xj∗+Ii,i∈ℕ,or, equivalently, there exist γij∗∈co¯cijhi−1xi∗ and ηij∗∈co¯wijhi−1xi∗ such that
(18)0=−bihi−1xi∗+∑j=1nγij∗fjhj−1xj∗+∑j=1nηij∗gjhj−1xj∗+Ii.Lemma 1 (Kakutani’s fixed point theorem [20]).
LetΩ be a compact convex subset of a Banach space X, if the set-valued map ϕ:Ω⟶PkcΩ is an upper semicontinuous convex compact map, then ϕ has a fixed point in Ω, i.e., there exists x∈Ω such that x∈ϕx.Lemma 2 (Chain rule [20]).
Suppose thatVy:ℝn⟶ℝ is C-regular, and yt:0,+∞⟶ℝn is absolutely continuous on any compact subinterval of 0,+∞. Then, yt and Vyt:0,+∞⟶ℝ are differential for a.e. t∈0,+∞ and
(19)dVytdt=ζt,dytdt,∀ζt∈∂Vyt.
Here,∂Vy represents the Clarke’s generalized gradient of V at point y∈ℝn.Lemma 3 ([31, 32]).
Suppose thatVy:ℝn⟶ℝ is C-regular, and that yt:0,+∞⟶ℝn is absolutely continuous on any compact interval of 0,+∞. If there exists a continuous function γ:0,+∞⟶ℝ, with γρ>0 for ρ∈0,+∞, such that
(20)dVytdt≤−γVyt,fora.e.t≥0,and
(21)∫0Vy01γρdρ=t∗<+∞,then, we have Vyt=0 for t≥t∗. Especially, we have the following conclusions.
(i)
Ifγρ=K1ρ+K2ρσ~, for all ρ∈0,+∞, where σ~∈0,1 and K1, K2>0, then the settling time is estimated by
(22)t∗=1K11−σ~lnK1V1−σ~y0+K2K2.
(ii)
Ifγρ=Kρσ~ and K>0, then the settling time is estimated by
(23)t∗=V1−σ~y0K1−σ~.Lemma 4 (Young inequality [33]).
Assume thata>0, b>0, p>1, 1/p+1/q=1, then the following inequality holds
(24)ab≤1pap+1qbq.Lemma 5 (Hardy inequality [20]).
Ifa1,a2,…,an are positive numbers and 0<τ1<τ2, then
(25)∑i=1naiτ21/τ2≤∑i=1naiτ11/τ1.


## 3. Existence Analysis of Equilibrium Point

In this section, we discuss the existence of equilibrium point of system (1) using the Kakutani’s fixed point theorem of set-valued maps. For further analysis, the following Assumption is needed.Assumption 5.
mini∈ℕlia¯i−2maxj∈ℕ∑i=1ncij+λja¯j,∑i=1nwij+δja¯j>0, where cij+=maxc^ij,cˇij and wij+=maxw^ij,wˇij.Theorem 1.
Under the Assumptions1, 2, 4, and 5, the discontinuous differential system (1) has at least one equilibrium point.Proof.
From Assumption2, we can obtain biv is a monotone increasing function. On the other hand, hi−1v is also a monotone increasing function due to the monotone increasing property of hiv. Thus, if x∗=x1∗,x2∗,…,xn∗T is an equilibrium point of system (8), it follows from (17) that
(26)xi∗∈hibi−1∑j=1nco¯cijhi−1xi∗fjhj−1xj∗+∑j=1nco¯wijhi−1xi∗gjhj−1xj∗+Ii,i∈ℕ.
Let us denoteX=ℝn and x1 be the norm of x, ∀x∈X. Obviously, X is a Banach space with the norm ·1. Next, we define a compact convex subset Ω⊆X as follows:
(27)Ω=x1,x2,…,xnT∈X:x1≤r,where
(28)r=∑j=1n∑i=1ncij+βj+wij+θj+∑i=1nIimini∈ℕlia¯i−2maxj∈ℕ∑i=1ncij+λja¯j,∑i=1nwij+δja¯j.
Now, let us define a set-valued mapφ:X⟶PkcX as follows:
(29)φx=φ1x,φ2x,…,φnxT,∀x∈X,where
(30)φix=hibi−1∑j=1nco¯cijhi−1xifjhj−1xj+∑j=1nco¯wijhi−1xigjhj−1xj+Ii,i∈ℕ.
Obviously, the set-valued mapφx is USC with nonempty compact convex values. In other words, φx is an upper semicontinuous convex compact map.
For anyx=x1,x2,…,xnT∈Ω and v¯=v¯1,v¯2,…,v¯nT∈φx, then there exist γij∈co¯cijhi−1xi and ηij∈co¯wijhi−1xi such that
(31)v¯i=hibi−1∑j=1nγijfjhj−1xj+∑j=1nηijgjhj−1xj+Ii,i∈ℕ.
Becausebi· and hi−1· are all monotone increasing functions, it follows that
(32)bihi−1v¯i=∑j=1nγijfjhj−1xj+∑j=1nηijgjhj−1xj+Ii,i∈ℕ.
Thus,(33)bihi−1v¯i≤∑j=1nγijfjhj−1xj+∑j=1nηijgjhj−1xj+Ii,i∈ℕ.
Sincebi· and hi−1· are differentiable, and bi0=hi0=0, we can obtain
(34)bihi−1v¯i=bihi−1v¯i−bihi−10=bi′ϱ1hi−1v¯i−hi−10=bi′ϱ1hi−1ϱ2′v¯i−0≥lia¯iv¯i,where ϱ1 is between hi−10 and hi−1v¯i, ϱ2 is between 0 and v¯i.
Under Assumption4, we can get
(35)γijfjhj−1xj=γijfjhj−1xj≤cij+λjhj−1xj+βj=cij+λjhj−1xj−hj−10+cij+βj=cij+λjhj−1ϱ3′xj−0+cij+βj≤cij+λja¯jxj+cij+βj,where ϱ3 is between 0 and xj. Similarly,
(36)ηijgjhj−1xj≤wij+δja¯jxj+wij+θj.
It follows from (32)–(36) that
(37)lia¯iv¯i≤∑j=1ncij+λja¯jxj+∑j=1nwij+δja¯jxj+∑j=1ncij+βj+wij+θj+Ii.
By summing up the above inequity, we can get(38)∑i=1nlia¯iv¯i≤∑j=1n∑i=1ncij+λja¯jxj+∑j=1n∑i=1nwij+δja¯jxj+∑j=1n∑i=1ncij+βj+wij+θj+∑i=1nIi.
Under Assumption5, it follows that
(39)mini∈ℕlia¯i∑i=1nv¯i≤2maxj∈ℕ∑i=1ncij+λja¯j,∑i=1nwij+δja¯j∑j=1nxj+∑j=1n∑i=1ncij+βj+wij+θj+∑i=1nIi≤2maxj∈ℕ∑i=1ncij+λja¯j,∑i=1nwij+δja¯jr+∑j=1n∑i=1ncij+βj+wij+θj+∑i=1nIi=mini∈ℕlia¯ir.
It is easy to obtain∑i=1nv¯i≤r, i.e., v¯1≤r.
Thus,v¯∈Ω and φx∈PkcΩ for any x∈Ω. Up to now, φx satisfies all the conditions in Lemma 1. Therefore, the set-valued map φx exists at least one fixed point x∗∈Ω, i.e., x∗∈φx∗. Then, it is easy to obtain that the discontinuous neural networks (1) exists at least one equilibrium point v∗=h1−1x1∗,h2−1x2∗,…,hn−1xn∗T. The proof is completed.Remark 1.
The novel Kakutani’s fixed point theorem of set-valued maps is used to discuss the existence of equilibrium point for memristive Cohen-Grossberg neural networks with time-varying delay [20, 32]. Compared with [20, 32], the factor of amplification function are considered and the activation functions can be unbounded in the neural networks (1). Thus, the neural networks in this paper is more general. On the other hand, the method by using the equivalent transformation system (8) is effective and novel. In one word, the method of proof and theoretical results in Theorem 1 are novel and extend the scope of the previous work.


## 4. Finite-Time Stabilization

In this section, we will discuss the FTS of system (1), i.e., stabilize the system states to the equilibrium point in finite time. In order to realize this goal, two kinds of discontinuous controllers are designed. By adding the discontinuous controller uit, we obtain the following discontinuous controlled system from (1)
(40)v̇it=−aivitbivit−∑j=1ncijvitfjvjt−∑j=1nwijvitgjvjt−τt−Ii−uit,i∈ℕ.By taking the variable transformationyt=xt−x∗=y1t,y2t,…,yntT, it follows from (40) that
(41)ẏit=−bihi−1xit−bihi−1xi∗+∑j=1nγijhi−1xitfjhj−1xjt−γij∗fjhj−1xj∗+∑j=1nηijhi−1xitgjhj−1xjt−τt−ηij∗gjhj−1xj∗+uit,i∈ℕ.For further analysis, the following important inequalities are needed. Letet=e1t,e2t,…,entT, where eit=vit−vi∗, i∈ℕ. Since hi−1· is differentiable, we can get
(42)eit=hi−1xit−hi−1xi∗=hi−1ξ¯′xit−xi∗=hi−1ξ¯′yit,where ξ¯ is between xit and xi∗. From (42) and hi−1ξ¯′=aiξ¯>0, it follows that signeit=signyit.Similar to (42), we can get
(43)−eit−τt=−hi−1xit−τt−hi−1xi∗=−hi−1ξ~′xit−τt−xi∗≤−a¯iyit−τt,where ξ~ is between xit−τt and xi∗.Using (42), it follows that
(44)−eitsignyit=−hi−1ξ¯′yitsignyit≤−a¯iyit.Sincebi· and hi−1· are all differentiable, bihi−1· is also differentiable. By using the Lagrange mean value theorem, it follows that
(45)−signyitbihi−1xit−bihi−1xi∗=−signyitbi′ξ1hi−1xit−hi−1xi∗=−signyitbi′ξ1hi−1ξ2′xit−xi∗≤−lia¯iyit,where ξ1 is between hi−1xit and hi−1xi∗, ξ2 is between xit and xi∗.From the proof of Theorem1, we can know that the equilibrium point x∗=x1∗,x2∗,…,xn∗T∈Ω. Therefore, there exist positive constants Mi and Ni such that
(46)fihi−1xi∗≤Mi,gihi−1xi∗≤Ni,i∈ℕ.Under the Assumption3 and (46), we can obtain
(47)γijhi−1xitfjhj−1xjt−γij∗fjhj−1xj∗≤γijhi−1xitfjhj−1xjt−γijhi−1xitfjhj−1xj∗+γijhi−1xitfjhj−1xj∗−γij∗fjhj−1xj∗≤cij+Lja¯jyjt+c^ij−cˇijMj.Similar to (47), it follows that
(48)ηijhi−1xitgjhj−1xjt−τt−ηij∗gjhj−1xj∗≤wij+Kja¯jyjt−τt+w^ij−wˇijNj.


### 4.1. Discontinuous State Feedback Controller with Time-Varying Delay

In order to realize the FTS of system (1), the following discontinuous state feedback controller with time-varying delay is designed.
(49)uit=−pieit−qisigneit−risigneit∑j=1nejt−τt,i∈ℕ,where pi, qi, and ri are positive constants to be determined later. In this section, we will use the famous finite-time stability theory to study the FTS of system (1). In the following, we further assumeAssumption 6.
riaˇ≥maxj∈ℕwij+Kja¯j, i∈ℕ, where aˇ=mini∈ℕa¯i.Theorem 2.
Suppose that Assumptions1–6 hold. If p>1, ϖ1=pmini∈ℕlia¯i+pia¯i−n·maxi,j∈ℕcij+Lja¯j>0, and ϖ2=pmini∈ℕqi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj>0. Then, the discontinuous neural networks (1) can be stabilized in a finite time with the discontinuous state feedback controller (49). The settling time for FTS can be estimated by
(50)T∗=pϖ1lnϖ1V1/p0+ϖ2ϖ2.Proof.
Let us consider the Lyapunov function(51)Vt=∑i=1nyitp.
It can be easily verified thatVt is C-regular. Therefore, we can calculate the derivative of Vt along the solution of (41) by using Lemma 2,
(52)dVtdt=∑i=1npyitp−1signyitẏit=∑i=1npyitp−1signyit−bihi−1xit−bihi−1xi∗+∑j=1nγijhi−1xitfjhj−1xjt−γij∗fjhj−1xj∗+∑j=1nηijhi−1xitgjhj−1xjt−τt−ηij∗gjhj−1xj∗−pieit−qisigneit−risigneit∑j=1nejt−τt.
Under Assumption6 and expressions (42)–(48), we can get
(53)dVtdt≤∑i=1npyitp−1−lia¯i+pia¯iyit+∑j=1ncij+Lja¯jyjt+maxj∈ℕwij+Kja¯j∑j=1nyjt−τt−riaˇ∑j=1nyjt−τt+∑j=1nc^ij−cˇijMj+w^ij−wˇijNj−qi≤−mini∈ℕlia¯i+pia¯i·p∑i=1nyitp+maxi,j∈ℕcij+Lja¯j·p∑i=1n∑j=1nyitp−1yjt−pmini∈ℕqi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj∑i=1nyitp−1.
According to Lemma4,
(54)yitp−1yjt≤p−1pyitp+1pyjtp.
Then,(55)p∑i=1n∑j=1nyitp−1yjt≤np−1∑i=1nyitp+n∑j=1nyjtp=np∑i=1nyitp.
By Lemma5, we can deduce the following inequality
(56)∑i=1nyitp−1≥∑i=1nyitpp−1/p.
It follows from (53)–(56) that
(57)dVtdt≤−pmini∈ℕlia¯i+pia¯i−n·maxi,j∈ℕcij+Lja¯j∑i=1nyitp−pmini∈ℕqi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj∑i=1nyitpp−1/p=−ϖ1Vt−ϖ2Vtp−1/p.
Under the conditionsϖ1>0 and ϖ2>0, the FTS of discontinuous neural networks (8) can be realized using Lemma 3. In addition, the settling time can be calculated as follows:
(58)T∗=pϖ1lnϖ1V1/p0+ϖ2ϖ2.
In other words,Vt=0, ∀t≥T∗. It follows that
(59)yt=xt−x∗=0,∀t≥T∗.
It follows from (42) that
(60)eit=hi−1ξ¯′yit≤a¯iyit,i∈ℕ.
Thus,et=0, ∀t≥T∗. Therefore, the FTS of delayed memristive Cohen-Grossberg neural networks (1) with the discontinuous state feedback controller (49) can be realized with the settling time T∗.
The proof is completed.Similar to Theorem2, it is easy to obtain the following corollary by a similar proof.Corollary 1.
Suppose that Assumptions1–6 hold. If p>1, ϖ1=pmini∈ℕlia¯i+pia¯i−n·maxi,j∈ℕcij+Lja¯j=0, and ϖ2=pmini∈ℕqi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj>0. Then, the discontinuous neural networks (1) with the discontinuous state feedback controller (49) can realize FTS. The settling time for FTS can be estimated by
(61)T∗=pV1/p0ϖ2.Remark 2.
When compared with conventional continuous state feedback controllers, the discontinuous state feedback controller (49) contains discontinuous terms −qisigneit and −risigneit·∑j=1nejt−τt. It is easy to obtain that different terms in this controller play different roles. In other words, −risigneit∑j=1nejt−τt can handle the influence of time-varying delay, −pieit and −qisigneit are used to guarantee the FTS. It is worth noting that the expressions (59) and (60) are important to obtain the FTS of the system (1) when we have proved the FTS of the discontinuous system (8).Remark 3.
It is effective to handle the uncertain differences of Filippov solutions by designing discontinuous controllers [16, 20, 27, 32]. From (50) and (61), we can know that the settling time for FTS can be controlled by the tunable constant p. However, this goal was realized by adding the related term in the controllers in [16, 27, 32], such as the term −nieiteitσ−1 in [16]. On the other hand, the neural networks in this paper is more general than [16, 20, 27, 32]. Thus, the discontinuous controller (49) and theoretical results obtained in Theorem 2 are novel.


### 4.2. Discontinuous State Feedback Controller without Time-Varying Delay

In this section, we design the following discontinuous state feedback controller without time-varying delay,(62)uit=−sieit−πisigneit,i∈ℕ,where si and πi are the gain coefficients to be determined later. Besides, the analysis method without using the finite-time stability theory will be presented to investigate the FTS of (1).Based on Assumption1, we need to further assume thatAssumption 7.
a¯i<1, for v∈ℝ, i∈ℕ.Theorem 3.
Suppose Assumptions1–5 and 7 hold. The gain coefficients si and πi satisfy the following conditions,
(63)ϖ^=mini∈ℕlia¯i+sia¯i−nmaxi,j∈ℕcij+Lja¯j+αija¯j1−τ^≥0,and
(64)ϖ∗=nmini∈ℕπi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj>0,where αij=wij+Kja¯j/a¯j. Then, the memristive Cohen-Grossberg neural networks with time-varying delay (1) can be finite-time stabilized via the discontinuous state feedback controller without time-varying delay (62). Furthermore, the settling time can be calculated as follows:
(65)T∗=V0ϖ∗.Proof.
Consider the following Lyapunov function(66)Vt=∑i=1nyit+∑i=1n∑j=1nαij1−τ^∫t−τttejsds.
It is easy to verify thatVt is C-regular. The derivative of Vt along the solution of (41) can be calculated by using Lemma 2,
(67)V̇t=∑i=1nsignyitẏit+∑i=1n∑j=1nαij1−τ^ejt−∑i=1n∑j=1nαij1−τ^ejt−τt1−τ̇t≤∑i=1nsignyit−bihi−1xit−bihi−1xi∗+∑j=1nγijhi−1xitfjhj−1xjt−γij∗fjhj−1xj∗+∑j=1nηijhi−1xitgjhj−1xjt−τt−ηij∗gjhj−1xj∗−sieit−πisigneit+∑i=1n∑j=1nαij1−τ^ejt−∑i=1n∑j=1nαijejt−τt.
By using (42)–(48), we can get
(68)dVtdt≤∑i=1n−lia¯i+sia¯iyit+∑j=1ncij+Lja¯jyjt+∑j=1nwij+Kja¯jyjt−τt+∑j=1nc^ij−cˇijMj+w^ij−wˇijNj−πi+∑i=1n∑j=1nαija¯j1−τ^yjt−∑i=1n∑j=1nαija¯jyjt−τt≤−mini∈ℕlia¯i+sia¯i−nmaxi,j∈ℕcij+Lja¯j+αija¯j1−τ^∑i=1nyit−nmini∈ℕπi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj≤−ϖ∗.
Next, the FTS of discontinuous system (1) can be studied without using the finite-time stability theory. As proposed in [34], we can get Vt=0 for t≥T∗=V0/ϖ∗. By taking a similar analysis in Theorem 2, we can obtain that the FTS of discontinuous neural networks (1) can be realized via the discontinuous controller without time-varying delay (62) with the settling time T∗. The proof is completed.Remark 4.
In this paper, we design two different kinds of state feedback controllers whether including time-varying delay. When compared with the discontinuous controller with time-varying delay (49), the discontinuous controller without time-varying delay (62) is simpler. However, in Theorem 2, we do not need the conditions τ̇t≤τ^<1 and Assumption 7, which are essential conditions in Theorem 3. Thus, the results in Theorem 2 are less conservative. Besides, two different kinds of analysis methods whether using finite-time stability theory are utilized. In practice, the researchers can choose the suitable controller and analysis method according to practical needs. On the other hand, it is worth noting that the expressions (42)–(44) are important to connect the discontinuous systems (1) and (8) in the proof of Theorem 2 and Theorem 3.Remark 5.
There exist a few papers considering the FTS of memristor-based neural networks [20, 23, 28, 35]. However, the authors in [20, 23, 35] did not consider the factor of time-varying delay and amplification function simultaneously. In [28], the finite-time stability and synchronization of a class of memristor-based fractional-order Cohen-Grossberg neural network with time delay was discussed. However, the activation functions are bounded and the time delay is constant in [28]. To the best of our knowledge, there is little work concerning the FTS of memristive Cohen-Grossberg neural networks with time-varying delay. Therefore, the theoretical results obtained in this paper are novel.


## 4.1. Discontinuous State Feedback Controller with Time-Varying Delay

In order to realize the FTS of system (1), the following discontinuous state feedback controller with time-varying delay is designed.
(49)uit=−pieit−qisigneit−risigneit∑j=1nejt−τt,i∈ℕ,where pi, qi, and ri are positive constants to be determined later. In this section, we will use the famous finite-time stability theory to study the FTS of system (1). In the following, we further assumeAssumption 6.
riaˇ≥maxj∈ℕwij+Kja¯j, i∈ℕ, where aˇ=mini∈ℕa¯i.Theorem 2.
Suppose that Assumptions1–6 hold. If p>1, ϖ1=pmini∈ℕlia¯i+pia¯i−n·maxi,j∈ℕcij+Lja¯j>0, and ϖ2=pmini∈ℕqi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj>0. Then, the discontinuous neural networks (1) can be stabilized in a finite time with the discontinuous state feedback controller (49). The settling time for FTS can be estimated by
(50)T∗=pϖ1lnϖ1V1/p0+ϖ2ϖ2.Proof.
Let us consider the Lyapunov function(51)Vt=∑i=1nyitp.
It can be easily verified thatVt is C-regular. Therefore, we can calculate the derivative of Vt along the solution of (41) by using Lemma 2,
(52)dVtdt=∑i=1npyitp−1signyitẏit=∑i=1npyitp−1signyit−bihi−1xit−bihi−1xi∗+∑j=1nγijhi−1xitfjhj−1xjt−γij∗fjhj−1xj∗+∑j=1nηijhi−1xitgjhj−1xjt−τt−ηij∗gjhj−1xj∗−pieit−qisigneit−risigneit∑j=1nejt−τt.
Under Assumption6 and expressions (42)–(48), we can get
(53)dVtdt≤∑i=1npyitp−1−lia¯i+pia¯iyit+∑j=1ncij+Lja¯jyjt+maxj∈ℕwij+Kja¯j∑j=1nyjt−τt−riaˇ∑j=1nyjt−τt+∑j=1nc^ij−cˇijMj+w^ij−wˇijNj−qi≤−mini∈ℕlia¯i+pia¯i·p∑i=1nyitp+maxi,j∈ℕcij+Lja¯j·p∑i=1n∑j=1nyitp−1yjt−pmini∈ℕqi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj∑i=1nyitp−1.
According to Lemma4,
(54)yitp−1yjt≤p−1pyitp+1pyjtp.
Then,(55)p∑i=1n∑j=1nyitp−1yjt≤np−1∑i=1nyitp+n∑j=1nyjtp=np∑i=1nyitp.
By Lemma5, we can deduce the following inequality
(56)∑i=1nyitp−1≥∑i=1nyitpp−1/p.
It follows from (53)–(56) that
(57)dVtdt≤−pmini∈ℕlia¯i+pia¯i−n·maxi,j∈ℕcij+Lja¯j∑i=1nyitp−pmini∈ℕqi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj∑i=1nyitpp−1/p=−ϖ1Vt−ϖ2Vtp−1/p.
Under the conditionsϖ1>0 and ϖ2>0, the FTS of discontinuous neural networks (8) can be realized using Lemma 3. In addition, the settling time can be calculated as follows:
(58)T∗=pϖ1lnϖ1V1/p0+ϖ2ϖ2.
In other words,Vt=0, ∀t≥T∗. It follows that
(59)yt=xt−x∗=0,∀t≥T∗.
It follows from (42) that
(60)eit=hi−1ξ¯′yit≤a¯iyit,i∈ℕ.
Thus,et=0, ∀t≥T∗. Therefore, the FTS of delayed memristive Cohen-Grossberg neural networks (1) with the discontinuous state feedback controller (49) can be realized with the settling time T∗.
The proof is completed.Similar to Theorem2, it is easy to obtain the following corollary by a similar proof.Corollary 1.
Suppose that Assumptions1–6 hold. If p>1, ϖ1=pmini∈ℕlia¯i+pia¯i−n·maxi,j∈ℕcij+Lja¯j=0, and ϖ2=pmini∈ℕqi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj>0. Then, the discontinuous neural networks (1) with the discontinuous state feedback controller (49) can realize FTS. The settling time for FTS can be estimated by
(61)T∗=pV1/p0ϖ2.Remark 2.
When compared with conventional continuous state feedback controllers, the discontinuous state feedback controller (49) contains discontinuous terms −qisigneit and −risigneit·∑j=1nejt−τt. It is easy to obtain that different terms in this controller play different roles. In other words, −risigneit∑j=1nejt−τt can handle the influence of time-varying delay, −pieit and −qisigneit are used to guarantee the FTS. It is worth noting that the expressions (59) and (60) are important to obtain the FTS of the system (1) when we have proved the FTS of the discontinuous system (8).Remark 3.
It is effective to handle the uncertain differences of Filippov solutions by designing discontinuous controllers [16, 20, 27, 32]. From (50) and (61), we can know that the settling time for FTS can be controlled by the tunable constant p. However, this goal was realized by adding the related term in the controllers in [16, 27, 32], such as the term −nieiteitσ−1 in [16]. On the other hand, the neural networks in this paper is more general than [16, 20, 27, 32]. Thus, the discontinuous controller (49) and theoretical results obtained in Theorem 2 are novel.


## 4.2. Discontinuous State Feedback Controller without Time-Varying Delay

In this section, we design the following discontinuous state feedback controller without time-varying delay,(62)uit=−sieit−πisigneit,i∈ℕ,where si and πi are the gain coefficients to be determined later. Besides, the analysis method without using the finite-time stability theory will be presented to investigate the FTS of (1).Based on Assumption1, we need to further assume thatAssumption 7.
a¯i<1, for v∈ℝ, i∈ℕ.Theorem 3.
Suppose Assumptions1–5 and 7 hold. The gain coefficients si and πi satisfy the following conditions,
(63)ϖ^=mini∈ℕlia¯i+sia¯i−nmaxi,j∈ℕcij+Lja¯j+αija¯j1−τ^≥0,and
(64)ϖ∗=nmini∈ℕπi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj>0,where αij=wij+Kja¯j/a¯j. Then, the memristive Cohen-Grossberg neural networks with time-varying delay (1) can be finite-time stabilized via the discontinuous state feedback controller without time-varying delay (62). Furthermore, the settling time can be calculated as follows:
(65)T∗=V0ϖ∗.Proof.
Consider the following Lyapunov function(66)Vt=∑i=1nyit+∑i=1n∑j=1nαij1−τ^∫t−τttejsds.
It is easy to verify thatVt is C-regular. The derivative of Vt along the solution of (41) can be calculated by using Lemma 2,
(67)V̇t=∑i=1nsignyitẏit+∑i=1n∑j=1nαij1−τ^ejt−∑i=1n∑j=1nαij1−τ^ejt−τt1−τ̇t≤∑i=1nsignyit−bihi−1xit−bihi−1xi∗+∑j=1nγijhi−1xitfjhj−1xjt−γij∗fjhj−1xj∗+∑j=1nηijhi−1xitgjhj−1xjt−τt−ηij∗gjhj−1xj∗−sieit−πisigneit+∑i=1n∑j=1nαij1−τ^ejt−∑i=1n∑j=1nαijejt−τt.
By using (42)–(48), we can get
(68)dVtdt≤∑i=1n−lia¯i+sia¯iyit+∑j=1ncij+Lja¯jyjt+∑j=1nwij+Kja¯jyjt−τt+∑j=1nc^ij−cˇijMj+w^ij−wˇijNj−πi+∑i=1n∑j=1nαija¯j1−τ^yjt−∑i=1n∑j=1nαija¯jyjt−τt≤−mini∈ℕlia¯i+sia¯i−nmaxi,j∈ℕcij+Lja¯j+αija¯j1−τ^∑i=1nyit−nmini∈ℕπi−∑j=1nc^ij−cˇijMj+w^ij−wˇijNj≤−ϖ∗.
Next, the FTS of discontinuous system (1) can be studied without using the finite-time stability theory. As proposed in [34], we can get Vt=0 for t≥T∗=V0/ϖ∗. By taking a similar analysis in Theorem 2, we can obtain that the FTS of discontinuous neural networks (1) can be realized via the discontinuous controller without time-varying delay (62) with the settling time T∗. The proof is completed.Remark 4.
In this paper, we design two different kinds of state feedback controllers whether including time-varying delay. When compared with the discontinuous controller with time-varying delay (49), the discontinuous controller without time-varying delay (62) is simpler. However, in Theorem 2, we do not need the conditions τ̇t≤τ^<1 and Assumption 7, which are essential conditions in Theorem 3. Thus, the results in Theorem 2 are less conservative. Besides, two different kinds of analysis methods whether using finite-time stability theory are utilized. In practice, the researchers can choose the suitable controller and analysis method according to practical needs. On the other hand, it is worth noting that the expressions (42)–(44) are important to connect the discontinuous systems (1) and (8) in the proof of Theorem 2 and Theorem 3.Remark 5.
There exist a few papers considering the FTS of memristor-based neural networks [20, 23, 28, 35]. However, the authors in [20, 23, 35] did not consider the factor of time-varying delay and amplification function simultaneously. In [28], the finite-time stability and synchronization of a class of memristor-based fractional-order Cohen-Grossberg neural network with time delay was discussed. However, the activation functions are bounded and the time delay is constant in [28]. To the best of our knowledge, there is little work concerning the FTS of memristive Cohen-Grossberg neural networks with time-varying delay. Therefore, the theoretical results obtained in this paper are novel.


## 5. Numerical Examples

In this section, we take two numerical examples to illustrate the correctness of theoretical results provided in this paper.Example 1.
Consider the 2-D delayed memristive Cohen-Grossberg neural networks as follows:(69)v̇it=−aivitbivit−∑j=12cijvitfjvjt−∑j=12wijvitgjvjt−τt−Ii,where
(70)aivit=1.5+0.5sinvit,fiθ~=tanhθ~,giθ~=2tanhθ~,τt=1,Ii=0,θ~∈ℝ,i=1,2,b1v1t=v1t,b2v2t=v2t,c11v1t=1.c12v1t=−0.5,v1t<0,−0.3,v1t≥0,c21v2t=−0.2,v2t<0,−0.6,v2t≥0,c22v2t=1.2,v2t<0,1.5,v2t≥0,w11v1t=−1.1,v1t<0,−0.8,v1t≥0,w12v1t=1.3,v1t<0,−2,v1t≥0,w21v2t=0.4,v2t<0,0.6,v2t≥0,w22v2t=1,v2t<0,−0.5,v2t≥0.It is obvious that1≤aivit≤2, b1′·=b2′·=1, fiui−fivi≤ui−vi, giui−givi≤2ui−vi, fiui≤1 and giui≤2, λi=δi=0, for any ui, vi∈ℝ, i=1,2. Thus, it is easy to obtain that the system (69) satisfies Assumptions 1–5. Taking initial values v1t=−2.5 and v2t=3 for t∈−1,0, the dynamical behavior of system (69) without any controller can be shown in the case of Figure 1(a).Figure 1
(a) The trajectories of state variablesv1t and v2t in (69) without any controllers. (b) The trajectories of state variables v1t and v2t in (69) with the discontinuous state feedback controller (49).
(a)
(b)Using the discontinuous state feedback controller (49), we take pi=qi=8, ri=10, and p=2, i=1,2. By simple computation, we can get Assumption 6 hold and ϖ1=6, ϖ2=1.2. Thus, all the conditions in Theorem 2 are satisfied. Therefore, the considered discontinuous system (69) can be stabilized in finite time by using the discontinuous state feedback controller (49). We choose the initial values as v1t=−2.5 and v2t=3, for t∈−1,0. The case of Figure 1(b) shows the dynamical behavior of system (69) with the controller (49). The numerical simulations illustrate the correctness of theoretical results in Theorem 2.Example 2.
Consider the following 3-D delayed memristive Cohen-Grossberg neural networks:(71)v̇it=−aivitbivit−∑j=13cijvitfjvjt−∑j=13wijvitgjvjt−τt−Ii,where
(72)aivit=0.85+0.1tanhvit,fiθ~=tanhθ~,giθ~=3tanhθ~,τt=1,Ii=0,θ~∈ℝ,i=1,2,3,b1v1t=v1t,b2v2t=v2t,b3v3t=v3t,c11v1t=0.5,v1t<0,1,v1t≥0,c12v1t=−0.3,v1t<0,−0.1,v1t≥0,c21v2t=1,v2t<0,−1,v2t≥0,c33v3t=−0.6,v3t<0,−0.2,v3t≥0,w11v1t=−1,v1t<0,−0.5,v1t≥0,w12v1t=0.8,v1t<0,−1,v1t≥0,w21v2t=0.1,v2t<0,1,v2t≥0,w22v2t=0.5,v2t<0,−0.8,v2t≥0,w32v3t=0.3,v3t<0,−0.6,v3t≥0,w33v3t=1,v3t<0,−0.6,v3t≥0,c13v1t=c22v2t=c23v2t=c31v3t=0,c32v3t=w13v1t=w23v2t=w31v3t=0.It is straightforward to obtain that(73)0.75≤aivit≤0.95,fiui−fivi≤ui−vi,giui−givi≤3ui−vi,fiui≤1,giui≤3,λi=δi=0,b1′·=b2′·=b3′·=1,ui,vi∈ℝ,i=1,2,3.Thus, it is easy to obtain that Assumptions1–5 and 7 are satisfied. By taking initial values v1t=1, v2t=2, and v3t=−2 for t∈−1,0, the case of Figure 2(a) shows the dynamical behavior of system (71) without any controllers.Figure 2
(a) The trajectories of state variablesv1t, v2t, and v3t in (71) without any controllers. (b) The trajectories of state variables v1t, v2t, and v3t in (71) with the discontinuous state feedback controller (62).
(a)
(b)Using the discontinuous state feedback controller without time-varying delay (62), we take si=20 and πi=10, i=1,2,3. By simple computation, we can get ϖ^>0 and ϖ∗>0. Thus, all the conditions in Theorem 3 are satisfied. Therefore, the finite-time stability of the discontinuous system (71) with the discontinuous controller (62) can be realized. We choose the initial values as v1t=1, v2t=2, and v3t=−2 for t∈−1,0. The case of Figure 2(b) shows the trajectories of the discontinuous system (71) with the discontinuous controller (62). The numerical simulations indicate our theoretical results in Theorem 3 are valid.


## 6. Conclusion

In this paper, the FTS of memristive Cohen-Grossberg neural networks with time-varying delay has been discussed. Based on the novel Kakutani’s fixed point theorem of set-valued maps, we have obtained the existence theorem of equilibrium point without assuming the boundedness of the activation functions. Due to the existence of memristor, amplification function, and time-varying delay, the FTS control of memristive Cohen-Grossberg neural networks with time-varying delay is not an easy work. Thus, we have considered two different kinds of discontinuous state feedback controllers whether including time-varying delay and two different analysis methods whether using finite-time stability theory. Under the extended Filippov framework and some inequality techniques, several sufficient conditions for guaranteeing the FTS have been obtained. In addition, the settling time for FTS are also proposed. Finally, two numerical examples have been used to illustrate the validity of theoretical results. On the other hand, it is a very effective method to investigate the FTS of the discontinuous system (1) by using the equivalent transformation system (8). The approaches in this paper can also be extended to study the fixed-time stabilization and fixed-time synchronization of other types of Cohen-Grossberg neural networks. Besides, it is well known that the factor of impulse has an important effect on the neural networks. In the future research, we hope to study the fixed-time stabilization and fixed-time synchronization problem of other more complex Cohen-Grossberg neural networks, such as concerning the factor of impulse or mixed delays [36–40]. Meanwhile, how to design more simple and effective controllers is also our future research topic.



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*Source: 7160858-2018-11-26.xml*