# Robust Fixed-Time Synchronization for Coupled Delayed Neural Networks with Discontinuous Activations Subject to a Quadratic Polynomial Growth

Authors: Lina Yu; Yunfei Ma; Yuntong Yang; Jingchao Zhang; Chunwei Wang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we focus on the robust fixed-time synchronization for discontinuous neural networks (NNs) with delays and hybrid couplings under uncertain disturbances, where the growth of discontinuous activation functions is governed by a quadratic polynomial. New state-feedback controllers, which include integral terms and discontinuous factors, are designed. By Lyapunov–Krasovskii functional method and inequality analysis technique, some sufficient criteria, which ensue that networks can realize the robust fixed-time synchronization, are addressed in terms of linear matrix inequalities (LMIs). Moreover, the upper bound of the settling time, which is independent on the initial values, can be determined to any desired values in advance by the configuration of parameters in the proposed control law. Finally, two examples are provided to illustrate the validity of the theoretical results.


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## Body

## 1. Introduction

In recently years, coupled neural networks (CNNs), as a special sort of complex dynamic networks, have attracted the widespread attention from a lot of scholars due to its potential applications in parallel computation, multiagent cooperative control, cryptography, nuclear magnetic resonance instrument, and other aspects [1–5]. Particularly, the synchronization with respect to CNNs has been extensively studied in many science fields [6–11] and the references therein.Time delay often arises in the transmission of signals in CNNs [12, 13]. In [13], Shao and Zhang considered the delay-dependent stabilization for CNNs with two additive input delays. In [14], He et al. investigated the pinning synchronization for CNNs with hybrid couplings and delays by an adoptive approach. In [15], Wang and Huang discussed the pining synchronization of delayed CNNs with reaction-diffusion effects.Recently, many works are devoted to the synchronization behaviors of NNs with discontinuous activations. For example, in [16], by applying the state-feedback control strategies, the global finite-time synchronization conditions are addressed for delayed NNs with discontinuous activations. In [17], the authors discussed the global synchronization for NNs with time-varying delays and discontinuous right-hand side. The exponential synchronization for discontinuous NNs with delays has been considered in [18]. In [19], the discrete nonfragile control strategy was designed to achieve the synchronization for fractional-order NNs by adjusting the coupling gain.In engineering applications, the synchronization is required to be achieved in a finite time [20, 21]. In [22], the authors pointed out that the finite-time control can demonstrate better disturbance and robustness rejection properties. Therefore, it is meaningful to investigate the global synchronization in finite time for CNNs [23–27].However, it should be noted that the upper bound of the settling time greatly depends on the initial states of CNNs under the finite-time synchronization. In generality, it is difficult to obtain the initial values in the real world. This shows that there exists a constraint to practical applications of CNNs [28]. To solve this problem, the fixed-time synchronization control strategy is considered in [28], where the settling time is independent on the initial conditions of CNNs. At present, as a kind of efficient control strategies, the fixed-time synchronization control has received tremendous attention from many researchers. The global fixed-time synchronization was discussed for semi-Markovian jumping neural networks with time-varying delays and discontinuous activation functions in [29]. The authors discussed the global fixed-time stability of dynamical nonlinear systems and realized the global fixed-time synchronization for CNNs with discontinuous activations in [30]. In [31], the nonsingular fixed-time consensus tracking was considered for second-order multiagent networks. Then, Zheng et al. discussed the fixed-time synchronization of memristive fuzzy BAM cellular neural networks with time-varying delays based on feedback controllers in [32]. Some criterions were obtained for the fixed-time synchronization of Cohen–Grossberg CNNs with and without parameter uncertainties in [33]. By fixed-time controllers, the global synchronization of delayed hybrid CNNs is studied in [34]. In [35, 36], Zhu et al. discussed the synchronization in fixed time with hybrid couplings and delays for semi-Markovian switching complex NNs, and the upper bound of settling time could be determined under the designed controller. The authors considered the global synchronization in fixed time for CNNs with delays and discontinuous/continuous activations and proposed two discontinuous control protocols in [37]. Event-triggered synchronization in fixed time was investigated for semi-Markov switching dynamical complex networks with multiple weights and discontinuous nonlinearity in [38]. It is worth pointing out, to the best of authors’ knowledge, there are few results on the global synchronization for hybrid CNNs with discontinuous activations subject to a nonlinear function growth.Inspired by the above discussion, in this paper, we focus on the global robust fixed-time synchronization for hybrid CNNs with delays and discontinuous activations in the presence of disturbances. Under the designed controllers with integral terms and discontinuous factors, the global synchronization conditions in fixed time are addressed in the terms of LMIs. Moreover, the upper bound of the settling time is estimated explicitly. The primary contributions of this paper are listed as follows:(1)
The neuron activation functions are modeled to be discontinuous and subject to a quadratic polynomial growth.(2)
On the basis of the configuration of parameters in the designed control law, the upper bound of the settling time can be determined to any desired values in advance.(3)
The robust synchronization conditions in fixed time are achieved in the form of LMIs.The rest of this paper is organized as follows. In Section2, some preliminaries and CNNs model are provided. In Section 3, the state-feedback discontinuous controllers are designed, and the global fixed-time synchronization conditions are addressed in the form of LMIs. In Section 4, two numerical simulations verifying the theoretic findings are presented. Conclusion is received in Section 5.Notation.R refers to the set of real numbers. Rn represents the n-dimensional Euclidean space, and Rn×n denotes the set of all n×n real matrices. Given A∈Rn×n, λmaxAλminA stands for the maximal (minimal) eigenvalues of A. A>0A<0 represents A is a positive (negative) definite matrix. Set η=η1,η2,…,ηnT∈Rn, where the superscript T is the transpose operator. Define ηq=∑i=1nηiq1/q, ηα=η1α,η2α,…,ηnαT, and sigαη=η1αsignη1, η2αsignη2,…,ηnαsignηnT, where α and q are positive constants and ⋅ denotes the absolute real value. For a set Φ⊂ℝn, co¯Φ represents the closure of the convex hull of Φ. Let A,B,C, and D be matrices with appropriate dimensions. ⊗ denotes the Kronecker product.


## 2. Preliminaries and System Description

Consider an array of hybrid CNNs described by(1)x˙it=−Qxit+Afxit+Bfxit−τ+c1∑j=1Ndijxjt+c2∑j=1Nhijxjt−τ+Jt+uit+Δit,xit,where i=1,2,…,N, xit=xi1t,xi2t,…,xintT is the state of the ith network at time t; xt=x1Tt,x2Tt,…,xNTtT, Q=diagq1,…,qn, qi>0; A=alrn×n represents the connection weight matrix; B=blrn×n denotes the delayed connection matrix; c1 and c2 are couple strength; fxit=f1xi1t,…,fnxintT intends neuron activation function; τ>0 is delay. uit stands for the control input; Jt=J1t,J2t,…,JntT is an external input; and D=dijN×N and H=hijN×N indicate coupling configuration matrix and delayed coupling configuration matrix, respectively. If there exists an edge from node i to j, then dij=dji>0; otherwise, dij=dji=0i≠j. Laplacian matrix L=lijN×N of a graph corresponding to D is given by lii=∑j=1,j≠iNdij,lij=−dij, i≠j. Δi expresses the uncertain disturbance.In system (1), fi⋅ is made to satisfy the following assumptions: (A1)fi⋅ is continuous expect on a countable set of isolate points ρki, and fi⋅ has at most a limited number of discontinuous points on any compact interval of R; in addition, at the discontinuous points ρki, the finite right limit fiρki+ and left limit fiρki− exist.(A2) LetD^∈Rn be a domain containing the origin. There exist positive real constants νl, ωl, and gl for each l=1,2,…,n, i=1,2,…,N, ∀t≥0, such that(2)supγilt−γ˜lt≤glxilt−ylt2+νlxilt−ylt+ωl,holds for ∀xilt,ylt∈D^, where γ˜lt∈co¯flylt, γil∈co¯flxilt.Based on A1, it follows thatco¯fxi=co¯f1xi1,co¯f2xi2,…,co¯fnxinT, co¯fnxinT, where co¯flxil=minfixil−,fixil+,maxfixil−,fixil+.Under A1, system (1) is a functional differential equation with discontinuous right-hand side [38]. In this paper, analogous to [39, 40], we use the definition of Filippov solutions for system (1).Definition 1.
(see [41]). xit is solution of system (1) in Filippov sense, if the following holds:(i)
xit is continuous on −τ,T, and xit is absolutely continuous on 0,T.(ii)
fora.e. t∈0,T.(3)x˙it∈Fxi=−Qxit+Aco¯fxit+Bco¯fxit−τ+c1∑j=1Ndijxjt+c2∑j=1Nhijxjt−τ+Jt+co¯uit+Δit,xit.Noting that set-valued mapFxi has a nonempty, compact, and convex value and is upper semicontinuous, so it is measurable. By measurable selection theorem [41], there exists measurable function γit=γi1t,…,γintT:−τ,T⟶Rn, ∈co¯fxit and ξit=ξi1t,ξi2t,…,ξintT:−τ,T⟶Rn, ∈co¯uit such that, for a.e., t∈0,T,(4)x˙it=−Qxit+Aγit+Bγit−τ+Jt+ξit+c1∑j=1Ndijxjt+c2∑j=1Nhijxjt−τ+Δit,xit.Definition 2.
(IVP [42]). For any φi=φi1,φi2,…,φiNT:−τ,0⟶RN and any measurable selection ϕi=ϕi1,ϕi2,…,ϕiNT:−τ,0⟶Rn, where ϕis∈co¯fiϕis for a.e s∈−τ,0. Absolute continuous function xit=xit,φ,ϕ associated with measurable function γit∈co¯fxit is said to be solution of IVP of system (1) on 0,T with initial value φis,ϕis if (5)x˙it=−Qxit+Aγit+Bγit−τ+c1∑j=1Ndijxjt+c2∑j=1Nhijxjt−τ+Jt+ξit+Δit,xit,for a.e.t∈0,Txis=φ˜is,∀s∈−τ,0,γis=ϕ˜is,for a.e.s∈−τ,0.Consider the following isolated neural network:(6)y˙t=−Qyt+Afyt+Bfyt−τ+Jt.Analogous to Definition2, the IVP associated with system (6) is obtained as follows:Definition 3.
(IVP [42]). For any ψ=ψ1,ψ2,…,ψNT:−τ,0⟶Rn and any measurable selection ω=ω1,ω2,…,ωNT:−τ,0⟶Rn, where ωis∈co¯fiψis for a.e. s∈−τ,0. Absolute continuous function yt=yt,ψ,ω associated with measurable function γ˜t∈co¯fyt is said to be solution of IVP of system (6) on 0,T with initial value ψs,ωs if(7)y˙t=−Qyt+Aγ˜t+Bγ˜t−τ+Jt,for a.e.t∈0,T,ys=ψ˜s,∀s∈−τ,0,γ˜s=ωs,for a.e.s∈−τ,0.In this paper, our objective is to design new feedback controllers to realize the robust fixed-time synchronization between CNNs (1) and isolated network (6).Set synchronization erroreit=xit−yt. Then,(8)e˙it=−Qeit+Af˜eit+Bf˜eit−τ+c1∑j=1Ndijejt+c2∑j=1Nhijejt−τ+uit+Δit,xit,where fi˜t=fxit−fyt and fi˜t−τ=fxit−τ−fyt−τ.According to Definitions2 and 3, IVP of error system can be written as(9)ei.t=−Qeit+Aγ^it+Bγ^it−τ+ξit+c1∑j=1Ndijejt+c2∑j=1Nhijejt−τ+Δit,xit,for a.e.t∈0,T,eis=e˜is,∀s∈−τ,0,γ^is=η˜is,for a.e.s∈−τ,0,where γ^it=γit−γ˜t∈co¯fxit−co¯fyt∈f˜eit, e˜is=φ˜is−ψ˜s, and η˜is=ϕ˜is−ωs.In order to derive the robust synchronization results of CNNs (1), for the terms Δit,x, we make the following assumption: (A3) The uncertain disturbancesΔit,x are bounded by(10)Δit,x≤Δmax,whereΔmax is a known nonnegative constant.Definition 4.
Under the designed controlleruit, if there exists time function Tei˜,ηi˜>0 such that limt⟶Tei˜,ηi˜eit=0 and eit≡0, t≥Tei˜,ηi˜, then system (1) is said to be globally robust finite-time synchronized with system (6). Moreover, if there exists scalar Tmax>0 such that Tei˜,ηi˜≤Tmax, then system (1) is said to be globally robust fixed-time synchronized with system (6). Tei˜,ηi˜ is called as the settling time function, and Tmax is the upper bound of the settling time function.Definition 5.
(see [43]). For function Vxi:Rn⟶R, if(i)
Vxi is regular in Rn,(ii)
forxi=0, V0=0, and for xi≠0, Vxi>0,(iii)
Vxi⟶+∞ as xi⟶+∞,then Vxi is called as C-regular.Lemma 1.
(chain rule [44]). If function Vxi is C-regular and xit is absolutely continuous on 0,+∞, then Vxit is differentiable for xitt∈0,+∞ and(11)ddtVxit=υTxit.,∀υ∈∂Vxi,where ∂Vxi is the Clarke generalized gradient of Vxi at x.Lemma 2.
(see [44]). Suppose that V:Rn⟶R is C-regular and xit:0,+∞⟶Rn is absolutely continuous. Let νi^t=Vxit. If there exist a continuous function ς:0,+∞⟶R with ς〉>0, 〉∈0,+∞, such that(12)νi^.t≤−ςνi^t,for a.e.,t>0,∫0νi^01ςϱdϱ=t1<+∞,then limt⟶t1xit=0 and xit=0, t≥t1. Especially, if for all ϱ∈0,+∞ and ςϱ=Mϱμ, where 0<μ<1 and M>0, then t1 can be calculated by(13)t1=ν^1−μ0M1−μ.Lemma 3.
(see [30]). Suppose that V:Rn⟶R is a C-regular function and eit:0,+∞⟶Rn is solution with initial value ei˜,ηi˜ of error system (9). If there exist constants a˜>0,b˜>0,δ˜>0,k˜>0, and δ˜k˜>1, such that(14)ddtVeit≤−a˜Vδ˜eit+b˜k˜,eit∈Rn\0,for a.e.t>0,then the upper bound settling time Tei˜,ηi˜ is estimated by(15)Tei˜,ηi˜≤Tmax≜1b˜k˜a˜b˜1/δ˜1+1δ˜k˜−1.Lemma 4.
(see [45, 46]). Let δ1,δ2,…,δn≥0, q>1, and p∈0,1. And then,(16)∑i=1nδiq≥n1−q∑i=1nδiq,∑i=1nδip≥∑i=1nδip.Lemma 5.
(see [47, 48]). Let ζ1,ζ2,…,ζn≥0, 0<p<q. Then,(17)∑i=1nζip1/p≥∑i=1nζiq1/q.


## 3. Main Results

Seta^=maxl,r∈1,nalr, b^=maxl,r∈1,nbrl, ν^=maxr∈1,nνr, and ω^=maxr∈1,nωr. Define L˜=l˜ijN×N and D˜=d˜ijN×N, i,j=1,2,…,N, where(18)d˜ij=2dij2/3,i≠j,0,i=j,l˜ij=−d˜ij,i≠j,∑j=1,j≠iNd˜ij,i=j,(19)Λ˜=diag2β˜12/3,2β˜22/3,…,2β˜N2/3.The controlleruit is designed as follows:(20)uit=−Γeit−τ12signeit+β∑j=1Ndijsig2ejt−eit−ββ˜sig2eit−asigneit−Keiteit2+K^∑l=1n∫t−τteil2sds3/2eiteit2,where Γ>0, β>0, signeit=diagsignei1t,…,signeint, a>0, K>0, K^>0, β˜=diagβ˜1,…,β˜N. There exists a path between network i and (6), if and only if, β˜i>0.Note that controller (20) is discontinuous, we have(21)co¯uit=−Γeit−τ12co¯signeit+β∑j=1Ndijco¯sig2ejt−eit−ββ˜co¯sig2eit−aco¯signeit−Keiteit2+K^∑l=1n∫t−τteil2sds3/2eiteit2,where eit>0, co¯signeit=1; eit=0, co¯signeit=−1,1; eit<0, co¯signeit=−1.Letξit∈co¯uit, then there exists Signeit∈co¯signeit, such that(22)ξit=−Γeit−τ12Signeit+β∑j=1Ndijejt−eit2Signejt−eit−ββ˜eit2Signeit+K^∑l=1n∫t−τteil2sds3/2eiteit2−aSigneit−Keiteit2.Theorem 1.
Suppose that (A1), (A2), and (A3) are satisfied, and the coupling interaction topology is undirected and connected. If the following conditions(23)Θ=Φ11Φ12Φ21Φ22<0,(24)a−Δmax−nω^b^−nω^a^>0,(25)Γ−g^b^>0,(26)χ2−2ng^a^>0,hold, where Φ11=−2IN⊗Q+2nν^a^+1InN−2c1L⊗In+c2H⊗InH⊗InT,Φ12=Φ21^=ν^b^⊗In,Φ21=ν^b^⊗In,Φ22=c2−1⊗In,χ2=λminL˜+Λ˜3/2βn˜−1/2,K^=χ2−2ng^a^,n˜=nN2, then CNNs (1) are globally robust fixed-time synchronized with system (6) under the designed controller (20). And the upper bound Tmax of settling time is estimated by(27)Tmax=32KK^2K2/3.Proof.
Construct the Lyapunov–Krasovskii functional(28)Vt=∑i=1N∑l=1n∫t−τteil2sds+∑i=1NeiTteit.
Calculating the derivative ofVt at time t along the trajectories of error system (9), it follows that(29)V.t=∑i=1N2eiTte˙it−∑i=1NeiTt−τeit−τ+∑i=1NeiTteit=∑i=1N2eiTt−Qeit+Af˜eit+Bf˜eit−τ+c1∑j=1Ndijejt+Δit,xit+c2∑j=1Nhijejt−τ+uit+∑i=1NeiTteit−∑i=1NeiTt−τeit−τ=2∑i=1NeiTtAγ^it−2∑i=1NeiTtQeit+2∑i=1NeiTtBγ^it−τ+2∑i=1NeiTtΔit,xit+2c2∑i=1N∑j=1NhijeiTtejt−τ−eit−τ+2c1∑i=1N∑j=1NdijeiTtejt+2∑i=1NeiTtuit−∑i=1NeiTt−τeit−τ+∑i=1NeiTteit.
Substituting (22) into (31), we can obtain(30)V.t=−2∑i=1NeiTtQeit+2∑i=1NeiTtAγ^it+∑i=1NeiTteit+2∑i=1NeiTtBγ^it−τ+2c1∑i=1N∑j=1NdijeiTtejt−eit+2c2∑i=1N∑j=1NhijeiTtejt−τ−eit−τ−2∑i=1NeiTtΓeit−τ12Signeit−∑i=1NeiTt−τeit−τ−2a∑i=1NeiTtSigneit+2∑i=1NeiTtβ∑j=1Ndijejt−eit2⋅Signejt−eit+2∑i=1NeiTtΔit,xit−2ββ˜∑i=1NeiTteit2Signeit−2K∑i=1NeiTteiteit2+2K^∑i=1NeiTt∑l=1n∫t−τteil2sds3/2eiteit2.
Due todij=dji, it follows that(31)∑i=1N∑j=1NdijeiTtejt−eit=12∑i=1N∑j=1NdijeiTtejt−eit+12∑i=1N∑j=1NdjiejTteit−ejt=12∑i=1N∑j=1NdijeiTtejt−eit−12∑i=1N∑j=1NdijejTtejt−eit=−12∑i=1N∑j=1Ndijejt−eitTejt−eit=−12∑i=1N∑j=1N∑l=1ndijejlt−eilt2=−eTtL⊗Inet.
It is easy to derive that(32)2c2∑i=1N∑j=1NhijeiTtejt−τ−eit−τ=2c2eTtH⊗Inet−τ≤c2eTtH⊗InH⊗InTet+c2eTt−τet−τ.
Similar to (31), one obtains(33)∑i=1N∑j=1NdijeiTtejt−eit2Signejt−eit=12∑i=1N∑j=1NdijeiTtejt−eit2Signejt−eit+12∑i=1N∑j=1NdjiejTtejt−eit2Signeit−ejt=12∑i=1N∑j=1NdjieiTtejt−eit2Signejt−eit−12∑i=1N∑j=1NdjiejTtejt−eit2Signejt−eit=−12∑i=1N∑j=1N∑l=1ndijejlt−eilt3.
By means of Assumptions (A1), (A2), and (A3), we get(34)∑i=1NeiTtAγ^it=∑i=1N∑r=1n∑l=1neiltalrγ˜irt≤∑i=1N∑r=1n∑l=1neiltalrγ˜irt≤∑i=1N∑r=1n∑l=1neiltalrgreirt2+νreirt+ωr≤nν^a^∑i=1NeiTteit+ng^a^∑i=1NeiTtei2t+nω^a^∑i=1N∑l=1neilt.
By Assumption (A2), one has(35)∑i=1NeiTtBγ^it−τ=∑i=1N∑r=1n∑l=1neiltblrγ˜irt−τ≤b^∑i=1N∑r=1n∑l=1neiltγ˜irt−τ≤b^∑i=1N∑r=1n∑l=1neiltgreirt−τ2+νreirt−τ+ωr≤g^b^∑i=1NeitTei2t−τ+ν^b^∑i=1NeitTeit−τ+nω^b^∑i=1N∑l=1neilt.
Under Assumption (A3), combining (30) with (35), it follows that(36)V.t≤−2eTtIN⊗Qet+2nν^a^∑i=1NeiTteit2ng^a^∑i=1NeiTtei2t+2nω^a^∑i=1N∑l=1neilt+2nω^b^∑i=1N∑l=1neilt+2ν^b^∑i=1N∑l=1neiTteit−τ−2c1eTtL⊗Inet+2g^b^∑i=1Neit1eit−τ12+c2eTtH⊗InH⊗InTet+c2eTt−τet−τ−2∑i=1N∑l=1neiTtχ2∫t−τteil2sds3/2eiteit2+2Δmax∑i=1N∑l=1neilt−2β∑i=1Nβ˜i∑l=1neilt3−2∑i=1NeiTtKeiteit−∑i=1NeiTt−τeit−τ+∑i=1NeiTteit−2Γ∑i=1NeiTt1eit−τ12−β∑i=1N∑j=1N∑l=1ndijejlt−eilt3−2a∑i=1N∑l=1neilt+2ng^a^∑i=1N∑l=1neiTt∫t−τteil2sds3/2.
The inequality above can be rewritten as(37)V.t≤ηΘηT−−2nω^a^−2nω^b^+2a−2Δmax∑i=1N∑l=1neilt+2ng^a^∑i=1N∑l=1neiTt∫t−τteil2sds3/2+2ng^a^∑i=1NeiTtei2t−−2g^b^+2Γ∑i=1Neiteit−τ2−2χ2∑i=1N∑l=1neiTt∫t−τteil2sds3/2−β∑i=1N∑j=1N∑l=1ndijejlt−eilt3−2β∑i=1Nβ˜i∑l=1neilt3−2K,where η=eit,eit−τT.
Together (24)–(26) and (37), one has(38)V.t≤2ng^a^∑i=1NeiTtei2t+2ng^a^∑i=1N∑l=1neiTt∫t−τteil2sds3/2−β∑i=1N∑j=1N∑l=1ndijejlt−eilt3−2β∑i=1Nβ˜i∑l=1neilt3−2K−2χ2∑i=1N∑l=1neiTt∫t−τteil2sds3/2.
Using Lemma4 yields(39)∑i=1N∑l=1neilt≥∑i=1N∑l=1neilt21/2=eTtet1/2.
By Lemma3 and (18), one has(40)−β∑i=1N∑j=1N∑l=1ndijejlt−eilt3−2β∑i=1Nβ˜i∑l=1neilt3≤−βn¯−1/2∑l=1n∑j=1N∑i=1Ndij2/3ejlt−eilt2+∑i=1N2β˜i2/3eilt23/2=−βn¯−1/2∑i=1N∑j=1N∑l=1n12d˜ijejlt−eilt2+∑l=1n∑i=1N2β˜i2/3eilt23/2=−βn¯−1/2eTtL˜⊗Iet+eTt∧˜⊗Iet3/2=−βn˜−1/2eTt∧˜+L˜⊗Iet3/2,where m is the real number nodes of β˜i, n¯=nNN−1+m, and mmax=N. Then, n˜=nNN−1+m=nN2≥n¯. Combining (39) with (40), we have(41)V.t≤−−2ng^a^+βn˜−1/2λmin∧˜+L˜⊗I3/2V3/2−2K.
Noting that∧˜+L˜ is positive definite, we can get(42)V.t≤−K^V3/2t+2K,where K^=−2ng^a^+βn˜−1/2λmin∧˜+L˜⊗I3/2>0.
By Lemma3, we can conclude that error system (9) is globally robust fixed-time stable. This shows that system (1) can achieve the global robust fixed-time synchronization with system (6) under the controller (20). The upper bound Tmax of settling time is estimated by Tmax=3/2KK^/2K2/3. The proof is completed.Remark 1.
In [30, 33–36], the global fixed-time synchronization issues were considered for delayed CNNs with discontinuous activations, where activation function f⋅ is subject to linear growth. However, in Theorem 1, discontinuous activations f⋅ are nonlinear growth and subject to a quadratic polynomial function. In addition, in [37], the Lyapunov function Vt=∑i=1NeiTteit is used to achieve the global fixed-time synchronization conditions. In this paper, the integral item ∑i=1N∑l=1n∫t−τteil2sds is introduced in Lyapunov functional (28). Obviously, compared with the above works in [30, 33–36], the result in this paper is more general.Remark 2.
It should be pointed out that the upper bound of the settling time,Tmax=3/2KK^/2K2/3, is independent on initial conditions. In addition, it is easy to see that, on the basis of the configuration for parameters K in the proposed control law, the upper bound Tmax can be determined in advance to any desired values.In the designed controller (20), the integral item ∑l=1n∫t−τteil2sds3/2 is used, which may bring difficulties in implement. We remove this integral item and design the following controller:(43)uit=−Γeit−τ12signeit+β∑j=1Ndijsig2ejt−eit−ββ˜sig2eit−asigneit−Keiteit2−Γ2eit−τ1signeit.Applying controller (43), we can obtain the following result.Corollary 1.
Suppose that Assumptions (A1), (A2), and (A3) are satisfied and couple interaction topology is connected and undirected. If the following conditions(44)Θ1=Ψ11Ψ12Ψ21Ψ22<0,a−Δmax−nω^b^−nω^a^>0,Γ−g^b^>0,hold, where Ψ11=−2IN⊗Q+2nν^a^InN−2c1L⊗In+c2H⊗InH⊗InT+IN⊗In,Ψ12=Φ21^=nν^b^InN,Ψ21=nν^b^⊗In,Ψ22=c2InN,χ2=λminL˜+Λ˜3/2βn˜−1/2,n˜=nN2, then CNNs (1) are globally robust fixed-time synchronized with system (6) under controller (43). Also, the settling time is estimated by(45)Tmax=32Γ2χ22Γ22/3.Proof.
ConsiderVt=∑i=1NeiTteit and follow the similar proof process of Theorem 1, and Corollary 1 can easily be proved, hence omitted here.
Next, we developed the global robust finite-time synchronization conditions for CNNs (1). Design the following controller:(46)uit=−Γeit−τ12signeit−Πeit12signeit−asigneit+K^∑l=1n∫t−τteil2sds1/2eiteit2,where Γ>0, Π>0, a>0, K^>0, and signeit=diagsignei1t,…,signeint.Theorem 2.
Suppose that Assumptions (A1), (A2), and (A3) are satisfied and couple interaction topology is connected and undirected. If the following conditions(47)Θ2=Φ11Φ12Φ21Φ22<0,(48)Π−ng^a^>0,(49)2Γ−2g^b^>0,hold, where Φ11=−2IN⊗Q+2nν^a^+1InN−2c1L⊗In+c2H⊗InH⊗InT,Φ12=Φ21^=nν^b^⊗In,Φ21=nν^b^⊗In,Φ22=c2−1InN,χ1=−nω^a^−nω^b^+a−Δmax,V0=∑i=1Nei01, then CNNs (1) are globally robust finite-time synchronized under the controller (46). And the setting time is estimated by(50)t1φ˜is,ϕ˜is=2V1/20χ1.Proof.
Consider Lyapunov–Krasovskii functional (28). Analogous as the proof of Theorem 1, we can get(51)V.t≤−−2nω^a^−2nω^b^+2a−2Δmax∑i=1N∑l=1neilt+ηΘ2ηT+2χ1∑i=1N∑l=1neiTt∫t−τteil2sds1/2−Π−2ng^a^∑i=1NeiTteit2−−2g^b^+2Γ∑i=1Neiteit−τ2,where η=eit,eit−τT.
By (47)–(49) and (51), it follows that(52)V.t≤2χ1∑i=1N∑l=1neiTt∫t−τteil2sds1/2−−2nω^a^−2nω^b^+2a−2Δmax∑i=1N∑l=1neilt.
Furthermore, we have(53)V.t≤χ1V1/2t.
Applying Lemma2, we can conclude that error system (9) is globally robust finite-time stable. This shows that system (1) can achieve the global robust finite-time synchronization with system (6) under the controller (46). The settling time is estimated by t1φ˜is,ϕ˜is=2V1/20/χ1. The proof is completed.


## 4. Numerical Examples

Example 1.
Consider an array CNNs (1) with five NNs, in which the dynamical equation of each network is described by(54)x˙ilt=−Qxilt+Afxilt+Bfxilt−τ+Δilt,xit+c1∑j=15dijxjlt−xilt+c2∑j=15hijxjlt−τ−xilt−τ+Jt+uit,l=1,2,i=1,…,5.
The corresponding isolated network (6) is given by(55)y˙t=−Qyt+Afyt+Bfyt−τ+Jt,where A=0.120.30.440.5,B=0.20.3−0.36−0.06,andQ=5.51006.91. Take D=1111111101111101011111011,L=4−1−1−1−1−13−10−1−1−13−10−10−13−1−1−10−13,andH=1−10−1−2−11−1−100−1101−1−101−1−20−1−11. The initial values are selected as φ˜1t=2t2+1,2cos2t+2T, φ˜2t=−2t2+1,2sin2t+3T, φ˜3t=0.2sin2t+1,0.1e3t+etT, φ˜4t=2sin2t−0.5,etT, and φ˜5t=2cos2t+1,0.1e3t+etT. The initial condition of the corresponding isolated network is ψ˜t=2sin2t,2sintT. Take activation function f as fxilt=0.001xil2t+0.05signxilt,i=1,2,…,5, where xilt∈D^,D^=−10,10×−10,10, then g^=0.001,ν^=0,ω^=0.45.(56)Δxit=0.7sinxi1t,0.6sinxi2tT,i=1,2,…,5,Δmax=0.85.
Setτ=1, c1=0.4, c2=1, a=4.15, β˜=diag3.14,5.19,0,0,0, K=0.25, β=1.412, and Γ=5.863. By simple calculation, a−Δmax−nω^b^−nω^a^=2.544>0, Γ−g^b^=5.862>0, χ2=λminL˜+Λ˜3/2βn˜−1/2=0.636, and K^=χ2−2ng^a^=0.6344>0.
Then, we calculateTmax=7.032s. Moreover, it is easy to verify that condition (23) in Theorem 1 is satisfied. As shown in Figures 1–3, the simulation results agree well with the theoretical analysis. Specifically, Figures 1 and 2 depict the state variables of system (54) and (55). Figure 3 shows the evolutions of eij, i,j=1,2,…,5. From Figure 3, it is easy to view that systems (54) and (55) can achieve the global robust synchronization in fixed time Tmax=1.2s under controller (20).Figure 1
State trajectories of variablesxi1t and y1t, i=1,…,5.Figure 2
State trajectories of variablesxi2t and y2t,i=1,…,5.Figure 3
Time evolutions ofeij,i,j=1,…,5 under the controller (20).Example 2.
Consider an array delayed CNNs (1) with three NNs, in which the dynamical equation of each network described by(57)x˙ilt=−Qxilt+Afxilt+Bfxilt−τ+Δilt,xit+c1∑j=13dijxjlt−xilt+c2∑j=13hijxjlt−τ−xilt−τ+Jt+uit,l=1,2,3,i=1,2,3.
The corresponding isolated network (6) is given by(58)y˙t=−Qyt+Afyt+Bfyt−τ+Jt,where A=0.60.11−0.10.40.65110.70.4,B=0.20.3−0.19−0.260.060.030.30.10.1,andQ=5.10002.20002.01. Take D=−1101−21011,L=1−10−12−10−11,andH=−1101−2101−1. The initial values are selected as φ˜1t=2t2+1,2.1cos2t+2,2.5t2+1T, φ˜2t=−2t2+1,2.5t+3,2sint+1T, and φ˜3t=0.2sin2t+1,0.1e3t+et,e3tT. The trivial point yt=0 is taken as the objective trajectory.
We take the activation functionsf as fxilt=0.01xil2t+0.3signxilt, i=1,2,3, where xilt∈D^,D^=−10,10×−10,10, then g^=0.01,ν^=0,ω^=0.5.
LetΔxit=0.2sinxi1t,0.3sinxi2t,0.4sinxi3tT,i=1,2,3, Δmax=0.29. Set τ=1, c1=0.04, c2=1.8, a=3.01, Π=0.48, K^=1.121, and Γ=0.42. By simple computation, Π−ng^a^=0.45>0, and Γ−g^b^=0.414>0.
From (50), χ1=−nω^a^−nω^b^+a−Δmax=0.522. Then, we calculate t1φ˜is,ϕ˜is=21.979s. Moreover, it is easy to verify that condition (50) in Theorem 2 also satisfied. As shown in Figure 4, the simulation results agree well with the theoretical analysis. Figure 4 shows the evolutions of eij, i,j=1,2,3, and it is obvious to view that systems (57) and (58) can achieve THE global robust finite-time synchronization before T=3.8s under controller (46).Figure 4
Time response of synchronization errorsei,i=1,2,3 under the controller (46).


## 5. Conclusion

In this paper, the global robust synchronization in fixed time and global synchronization in finite time have been investigated for a class of hybrid coupled delayed NNs with discontinuous activation functions. Under the designed discontinuous feedback controller, the global synchronization conditions have been presented in the forms of LMIs, and the settling time, which is independent on initial conditions, has been also evaluated. Compared with the existing works, where the neuron activations are supposed to be linear growth, the results proposed in this paper are more general.It is worth noting that the designed feedback controllers contain the sign function and the integral term, which may bring the chatter in actual implement, indicating that the designed control schemes have certain limitation.Future work will be focused on how to remove the chatter of the designed controller and to extend the results here obtained for stochastic sampled-date synchronization control for CNNs with delays and discontinuous activations.



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