# Global Asymptotical Stability of Neutral-Type Neural Networks withD-Operator and Mixed Delays

Authors: Qing Yang; Bo Du; Xiwang Cheng
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this manuscript, we investigate the stability problems of neutral-type neural networks withD-operator and mixed delays. Some sufficient conditions are obtained for guaranteeing the existence, uniqueness, and global asymptotical stability of periodic solutions to the considered neural networks. Finally, a numerical example is performed to illustrate the theoretical results.


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## Body

## 1. Introduction

In this article, we consider the following neutral neural networks withD-operator and mixed delays:(1)Ax′t=−Bαxt+Kfxt+Ugxt−τt+W∫t−μtthsds+It,t>0,xs=ϕs,s∈−ρ,0,where xt=x1t,…,xnt⊤ corresponds to the state of the ith unit at time t and Axt=A1x1t,…,Anxnt⊤, which is defined by(2)Axt=xt−Cxt−σ,t∈ℝ,where σ>0 is a constant delay, C=diagc1,…,cn with ci<1; αxt=Axt denotes the behaved function; the functions fx⋅=f1x1⋅,…,fnxn⋅⊤,gx⋅=g1x1⋅,…,gnxn⋅⊤, and hx⋅=h1x1⋅,…,hnxn⋅⊤ denote the neuron activations; τt and μt are the discrete time-varying delay and the distributed time-varying delay, respectively, which satisfy 0≤τt≤τ,0≤μt≤μ, where τ and μ are positive constants; and ρ=maxσ,τ,μ, for ϕ∈C−ρ,0,ℝn, and the norm is defined by ϕρ=sup−ρ≤s≤0ϕs. The matrix B=bijn×n>0 is a constant matrix, and K=kijn×n>0,U=uijn×n,andW=wijn×n are some unknown constant matrices. It denotes the exogenous inputs at time t.Remark 1.
The neutral-type operator A in (2) represents the D-operator form which was put forward by Hale [1]. Then, we give the following properties for the difference operator A. Define the operator A on CT:(3)A:CT⟶CT,Axt=xt−cxt−τ,∀t∈ℝ,where CT=x:x∈Cℝ,ℝ,xt+T≡xtandc is a constant.Lemma 1 (see [2, 3]).
Whenc≠1, then A has a unique continuous bounded inverse A−1 satisfying(4)A−1ft=∑j≥0cjft−jτ,ifc<1,∀f∈CT,−∑j≥1c−jft+jτ,ifc>1,∀f∈CT,A−1≤11−c.
When the neutral-type operatorA is defined on continuous function space Cℝ,ℝ, we have the following lemma:Lemma 2 (see [3]).
Ifc<1, then the inverse of difference operator A denoted by A−1 exists, and(5)A−1≤11−c.Proof.
The proof of Lemma2 is similar to the proof in [3]. For the convenience of the reader, we provide a detailed proof. Let Bxt=cxt−τ, then B=c<1. Thus, A−1=I−B−1 exists, and(6)A−1=I−B−1≤11−c.Remark 2.
Since the matricesK,U, and W in (1) are uncertain matrices, system (1) is interval neural networks. In the real world, the dynamic systems are often destroyed by various inevitable factors, such as parameter fluctuation, external disturbance, and random disturbance. Hence, the research for interval neural networks has important practical application value. There are lots of results for interval neural networks, e.g., [4, 5].
Let us recall the research for the stability problems of neural networks. Zhang [6] studied the stability problem of periodic solutions for a time-varying recurrent cellular neural network with time delays and impulses. Then, Zhang and Wang [7] further investigated dynamic properties of periodic solutions for high-order Hopfield neural networks with time delays and impulses. The models in [6, 7] are nonneutral-type models which are different from the model in the present paper. In [8], periodic oscillation problems of discrete-time bidirectional associative memory neural networks have been studied by employing the theory of coincidence degree and Halanay-type inequality technique. In the present paper, we obtain the existence, uniqueness, and global asymptotical stability of periodic solutions by using LMI approach, Lyapunov function, and a blend of matrix theory which are different from the methods in [8]. Zhu and Cao [9, 10] studied two different types of stochastic neural networks which are not neutral-type neural networks and are different from the model in the present paper. In [11, 12], the authors considered the following two kinds of neutral-type neural system:(7)xi′t=−aitxit+∑j=1nbijtfjt,xjt+dijtgjt,xj′t−τijt+Iit,xit=ϕit,t∈−τ,0,i=1,2,…,n,y′t=−Ayt+Bgyt+Cgyt−τt+Dy′t−ht,t≠tk,Δyt=Ikyt,t=tk,yt0++s=ϕs,s∈t0−ρ,t0,k∈ℕ.
Liu et al. [13] studied a class of Markovian jumping neutral-type neural networks with mode-dependent mixed time delays: (8)x˙t=Ertx˙t−τ1,rt−Artxt−Brtfxt+crtgxt−τ2,rt+Drt∫t−τ3,rtt−τ4,rthxsds.
By the above work, we note that the neutral character in neural networks shows by the nonlinear termgt,x′t−τt or differential operator x′t−τt. In this paper, we will study the neutral-type neural networks when the neutral term has the D-operator form which is shown by Axt. For more references about neutral-type neural networks with mixed delays, see [14, 15].
To best knowledge of the authors, few authors have studied the global asymptotical stability problems of periodic solutions to neutral-type neural networks with mixed delays. The main areas of challenge are as follows: (1) since system (1) contains neutral-type operator A, constructing a viable Lyapunov–Krasovskii functional seems to be very difficult; (2) as the mixed delays exist in system (1), the corresponding LMI approach becomes more complicated since the LMI approach is required to reflect mixed delay’s influence; and (3) it is extraordinary to establish a unified framework to dispose of the uncertain matrices, the neutral-type terms, and mixed delays’ influence. Therefore, the main purpose of this article is to try for the first time to address the challenges listed.
In this paper, we will study global asymptotical stability of interval neural networks for a neutral-type neural network with mixed delays. Note that system (1) contains uncertain matrices, the neutral-type terms, and mixed delays that are all dependent on the properties of the neutral operator.Remark 3.
The main purpose of this paper is to obtain some sufficient conditions for guaranteeing asymptotic stability of system (1). We develop LMI approach to answer the addressed challenges. A simulation example is given to show the usefulness of the main results in the present paper. There are three aspects to the contribution of this paper: (1) For using the LMI method, we have to take into account the properties of the neutral-type operator. (2) Unlike the most existing results, we develop a new unified framework to deal with global asymptotical stability of interval neural networks by LMI approach, Lyapunov function, and a blend of matrix theory which may be of independent interest. It is worth pointing out that our main methods are also important for the case of nonneutral system with constant delays. (3) This article uses some new inequality techniques. In particular, using the properties of the neutral-type operator, we construct an appropriate Lyapunov–Krasovskii functional to handle the considered system.
The following sections are organized as follows: Section2 gives some preliminaries including some useful lemmas and definitions. In Section 3, we obtain some sufficient conditions for existence, uniqueness, and global asymptotic stability of periodic solution to system (1). In Section 4, a numerical example verifies the accuracy of the results in the present paper.


## 2. Some Preliminaries

Throughout the paper,Λ=1,2,…,n,andℝn and ℝn×m denote the n-dimensional Euclidean space and the set of all n×m real matrices, respectively. The superscript “T” represents the matrix transposition. A>0 (or A<0) denotes that A is a symmetric and positive definite (or negative definite) matrix. z denotes the Euclidean norm of a vector z and A denotes the induced norm of the matrix A, that is, A=λmaxA⊤A, where λmax⋅ means the largest eigenvalue of A. If their dimensions are not explicitly stated, they are assumed to be compatible for algebraic operations.Lemma 3 (see [16]).
For any vectors withx,y∈ℝn, the inequality(9)±2x⊤y≤x⊤ℳx+y⊤ℳy,holds, where ℳ is any n×n matrix with ℳ>0.Lemma 4 (see [17]).
Letp,q,γ,τ, and σ be the positive constants, and the function f∈PCℝ,ℝ+ satisfies the scalar impulsive differential inequality:(10)D+ft≤−pft+qsupt−τ≤s≤tfs+∫0σrsft−sds,t≠tk,t≥0,ftk≤ftk−,k∈ℤ+,where r⋅∈PC0,σ,ℝ+. Assume that p>q+γ∫0σrsds, then(11)ft≤f¯0e−λt,t≥0,where f¯0=sup−ρ≤s≤0fs,ρ=maxσ,τ, and λ>0 satisfies λ<p−qeλτ−γ∫0σrseλsds.Remark 4.
Whenftk=ftk−, then ft is a continuous function on 0,∞, and the results of Lemma 4 also hold.
Throughout this paper, the following assumptions are needed:(H1) For i∈Λ, the neuron activation functions in (1) satisfy(12)li−≤fis1−fis2s1−s2≤li+,σi−≤gis1−gis2s1−s2≤σi+,υi−≤his1−his2s1−s2≤υi+,whereli−,li+,σi−,σi+,υi−,andυi+ are the real constants and they may be positive, zero, or negative.(H2) τt,μt, and It are all continuously T−periodic functions on 0,∞.


## 3. Main Results

Assume that the unknown constant matricesK,U,andW satisfy(13)K∈K,U∈U,W∈W,where(14)K=K¯,K¯≕kijn×nk¯ij≤kij≤k¯ij,i,j∈Λ,U=U¯,U¯≕uijn×nu¯ij≤uij≤u¯ij,i,j∈Λ,W=W¯,B¯≕wijn×nw¯ij≤wij≤w¯ij,i,j∈Λ,with(15)K¯=k¯ijn×n,K¯=k¯ijn×n,U¯=u¯ijn×n,U¯=u¯ijn×n,W¯=w¯ijn×n,W¯=w¯ijn×n.Let(16)Δ≕diagδ11,…,δ1n,…,δn1,…,δnn∈ℝn2×n2,δij≤1,i,j∈Λ,K0=K¯+K¯2,U0=U¯+U¯2,W0=W¯+W¯2,ηijn×n=K¯−K¯2,ιijn×n=U¯−U¯2,κijn×n=W¯−W¯2,Γ1=η11e1,…,η1ne1,…,ηn1en,…,ηnnenn×n2,Γ2=ι11e1,…,ι1ne1,…,ιn1en,…,ιnnenn×n2,Γ3=κ11e1,…,κ1ne1,…,κn1en,…,κnnenn×n2,ϒ1=η11e1,…,η1ne1,…,ηn1en,…,ηnnenn2×n⊤,ϒ2=ι11e1,…,ι1ne1,…,ιn1en,…,ιnnenn2×n⊤,ϒ3=κ11e1,…,κ1ne1,…,κn1en,…,κnnenn2×n⊤,where ei∈ℝnandi∈Λ denotes the column vector with ith element to be 1 and others to be 0. Then, system (1) can be rewritten as(17)Ax′t=−Bαxt+K0+Γ1Δ1ϒ1fxt+U0+Γ2Δ2ϒ2gxt−τt+W0+Γ3Δ3ϒ3∫t−μtthsds+It,t>0,xs=ϕs,s∈−ρ,0,i.e.,(18)Ax′t=−Bαxt+K0fxt+U0gxt−τt+W0∫t−μtthsds+ΘΨt+It,t>0,xs=ϕs,s∈−ρ,0,where Θ=Γ1,Γ2,Γ3n×3n2,(19)Ψt=Δ1ϒ1000Δ2ϒ2000Δ3ϒ33n2×n×fxtgxt−τt∫t−μtthsds,Δi∈Δ,i=1,2,3.Now, we present the main results of the present paper.Theorem 1.
Assume that the assumptions (H1) and (H2) hold. Then, system (1) has a global asymptotically stable periodic solution if there exist three constants εi>0,i=1,2,3, n×n diagonal matrices P>0, and Qi>0,i=1,2,3, and three n2×n2 diagonal matrices Qi>0,i=4,5,6, such that the following inequalities hold:(20)Ξ≥0,ε2P−∑i=1n11−ci2LgQ2Lg−∑i=1n11−ci2Lgϒ1⊤Q5ϒ1Lg≥0,ε3P−∑i=1n11−ci2LhQ3Lh−∑i=1n11−ci2Lhϒ3⊤Q6ϒ3Lh≥0,ε1−ε2−ε3μ>0,where(21)Ξ=Ξ1PK0PU0PW0PΘ∗Q1000∗∗Q200∗∗∗Q30∗∗∗∗Ξ2,Ξ1=2PB−ε1P−∑i=1n11−ci2LfQ1Lf−∑i=1n11−ci2Lfϒ1⊤Q4ϒ1Lf,Ξ2=Q400∗Q50∗∗Q6,Lf=diagl1,…,ln,lj=maxlj−,lj+,j∈Λ,Lg=diagσ1,…,σn,σj=maxσj−,σj+,j∈Λ,Lh=diagυ1,…,υn,υj=maxυj−,υj+,j∈Λ.Proof.
Letxt,ϕ be an arbitrary solution of system (1) with initial value 0,ϕ. Define yt,ψ=xt+T,ϕ, where T>0 is a constant and ψ=xs+T,ϕ,−ρ≤s≤0. Then yt,ψ is a solution of system (1) with initial value 0,ψ. Let zt=yt,ψ−xt,ϕ. In view of system (18), we have(22)Ax′t=−Bαyt−xt+K0fyt−fxt+U0gyt−τt−gxt−τt+W0∫0μthyt−s−hxt−sds+ΘΨzt,t>0,xs=ψs−ϕs,s∈−ρ,0,where(23)Ψzt=Δ1ϒ1000Δ2ϒ2000Δ3ϒ33n2×n×fyt−fxtgyt−τt−gxt−τt∫0μthyt−s−hxt−sds.
Construct the following Lyapunov function:(24)Vt=Azt⊤PAzt,t≥0.
Derivation of the above Lyapunov function along the solution of (22) gives(25)D+Vt=2Azt⊤P−Bαyt−xt+K0fyt−fxt+U0gyt−τt−gxt−τt+W0∫0μthyt−s−hxt−sds+ΘΨzt=−2Azt⊤PBAzt+2Azt⊤PK0fyt−fxt+2Azt⊤PU0gyt−τt−gxt−τt+2Azt⊤W0∫0μthyt−s−hxt−sds+2Azt⊤ΘΨzt.
We calculate every term ofD+Vt. It follows from the assumption (H1), Lemma 2, and Lemma 3 that(26)2Azt⊤PK0fyt−fxt≤Azt⊤PK0Q1−1K0⊤PAzt+gyt−τt−gxt−τt⊤Q2gyt−τt−gxt−τt≤Azt⊤PK0Q1−1K0⊤PAzt+∑i=1n11−ci2Azt⊤LfQ1LfAzt,(27)2Azt⊤PU0gyt−τt−gxt−τt≤Azt⊤PU0Q2−1U0⊤PAzt+gyt−τt−gxt−τt⊤Q2gyt−τt−gxt−τt≤Azt⊤PU0Q2−1U0⊤PAzt+∑i=1n11−ci2Azt−τt⊤LgQ2LgAzt−τt,(28)2Azt⊤W0∫0μthyt−s−hxt−sds≤Azt⊤PW0Q3−1W0⊤PAzt+∫0μthyt−s−hxt−sds⊤Q3∫0μthyt−s−hxt−sds≤Azt⊤PW0Q3−1W0⊤PAzt+∫0μt∑i=1n11−ci2Azt−v⊤LhQ3LhAzt−vdt.
Using the definition ofΨzt, assumption (H1), and Lemma 2, we have(29)Ψzt⊤Q4000Q5000Q6Ψzt=fxt−fytgxt−τt−gyt−τt∫0μthxt−s−hyt−sds⊤×ϒ1⊤Δ1Q4Δ1ϒ1000ϒ2⊤Δ2Q5Δ2ϒ2000ϒ3⊤Δ3Q6Δ3ϒ3×fxt−fytgxt−τt−gyt−τt∫0μthxt−s−hyt−sds≤∑i=1n11−ci2Azt⊤Lfϒ1⊤Q4ϒ1LfAzt+∑i=1n11−ci2Azt−τt⊤Lgϒ1⊤Q5ϒ1LgAzt−τt+∫0μt∑i=1n11−ci2Azt−v⊤Lhϒ3⊤Q6ϒ3LhAzt−vdt.
By (29), we have(30)2Azt⊤ΘΨzt≤Azt⊤PΘQ4−1000Q5−1000Q6−1Θ⊤PAzt+Ψzt⊤Q4000Q5000Q6Ψzt≤Azt⊤PΘQ4−1000Q5−1000Q6−1Θ⊤P+∑i=1n11−ci2Lfϒ1⊤Q4ϒ1LfAzt+∑i=1n11−ci2Azt−τt⊤Lgϒ1⊤Q5ϒ1LgAzt−τt+∫0μt∑i=1n11−ci2Azt−v⊤Lhϒ3⊤Q6ϒ3LhAzt−vdt.
Adding the terms on the right-hand side of (26)–(28) and (30) to (25), and using condition (20), we get(31)D+Vt≤Azt⊤−2PB+PK0Q1−1K0⊤P+∑i=1n11−ci2LfQ1Lf+PU0Q2−1U0⊤P+PW0Q3−1W0⊤P+PΘQ4−1000Q5−1000Q6−1Θ⊤P+∑i=1n11−ci2Lfϒ1⊤Q4ϒ1LfAzt+∑i=1n11−ci2Azt−τt⊤LgQ2Lg+Lgϒ1⊤Q5ϒ1LgAzt−τt+∫0μt∑i=1n11−ci2Azt−v⊤LhQ3Lh+Lhϒ3⊤Q6ϒ3LhAzt−vdt≤−ε1Azt⊤PAzt+ε2Azt−τt⊤PAzt−τt+ε3∫0μtAzt−v⊤PAzt−vdv≤−ε1Vt+ε2Vt−τt+ε3∫0μtVt−vdv,which together with Remark 4 yields(32)Vt≤V^ρe−λt,t≥0,where V^ρ=sups∈−ρ,0Vs and λ>0 satisfies λ<ε1−ε2eλτ−ε3eλμ/λ. Thus,(33)λminPAzt2≤Vt≤λmaxPmax−ρ≤s≤0Aϕs−ψs2e−λt,t≥0,Azt≤λmaxPλminPmax−ρ≤s≤0Aϕs−ψse−0.5λt,t≥0.
By (33) and Lemma 2, we have(34)zt≤∑i=1n11−ci2Azt≤∑i=1n11−ci2λmaxPλminPmax−ρ≤s≤0Aϕs−ψse−0.5λt,t≥0,i.e.,(35)yt,ψ−xt,ϕ≤∑i=1nλmaxP1−ci2λminPmax−ρ≤s≤0Aϕs−ψse−0.5λt,t≥0.
Let(36)Xmt=xt,ϕ+∑k=1mxt+kT,ϕ−xt+k−1T,ϕ,t≥0,m∈Z+.
We show thatXmt is uniformly convergent. For s∈Z+, by (35), we have(37)Xm+st−Xmt=∑k=m+1m+sxt+kT,ϕ−xt+k−1T,ϕ≤∑k=m+1m+sxt+kT,ϕ−xt+k−1T,ϕ=∑k=m+1m+syt+k−1T,ψ−xt+k−1T,ϕ≤∑i=1nλmaxP1−ci2λminPmax−ρ≤s≤0Aϕs−ψse−0.5λt∑k=m+1m+se−0.5λk−1T≤∑i=1nλmaxP1−ci2λminPmax−ρ≤s≤0Aϕs−ψse−0.5λTm1−e−0.5λT.
It is obvious that the function sequenceXmtm=1∞ is uniformly convergent by the Cauchy convergence criterion. In addition,(38)xt+mT,ϕ=xt,ϕ+∑k=1mxt+kT,ϕ−xt+k−1T,ϕ=Xmt.
Hence,xt+mT,ϕ is also uniformly convergent, which implies that(39)limm⟶∞xt+mT,ϕ=x0t,(40)x0′t=limΔ⟶0x0t+Δ−x0tΔ=limΔ⟶0limm⟶∞xt+Δ+mT,ϕ−xt+mT,ϕΔ=limm⟶∞x0′t+mT,ϕ.
Hence, we can deduce thatx0t is a T−periodic solution of system (1) since xt+mT,ϕ is a solution of system (1). Now, we prove that x0t is a unique T−periodic solution of system (1). Let x1t=x1t,ϕ1 and x2t=x2t,ϕ2 be different T−periodic solutions of system (1), where ϕ1,ϕ2∈C−ρ,0,ℝn. It follows by (35) that(41)x1t,ϕ1−x2t,ϕ2=x1t+kT,ϕ1−x2t+kT,ϕ2≤∑i=1nλmaxP1−ci2λminPmax−ρ≤s≤0Aϕ1s−ϕ2se−0.5kt⟶0ask⟶∞.
Thus,x1t,ϕ1=x2t,ϕ2 for t≥0.Remark 5.
In the proof of Theorem1, since zt is not a continuous T−periodic solution, Lemma 1 cannot be used for the proof of Theorem 1. We generalize the results of Lemma 1 to the continuous function space which can be used for the proof of Theorem 1. However, when the parameter c≥1 in (2), we cannot prove the result of Theorem 1. We hope that subsequent researchers can solve the above problems.Remark 6.
Obviously, the periodic solution of system (1) is global asymptotical stability by Theorem 1. On the contrary, using Lemma 2, we have(42)ϕ−ψρ≤∑i=1n11−ci2max−ρ≤s≤0Aϕs−ψs.
If there exists suitablec˜i,i∈Λ, such that(43)ϕ−ψρ=∑i=1n11−c˜i2max−ρ≤s≤0Aϕs−ψs.
Hence, it follows by (35) and (43) that(44)yt,ψ−xt,ϕ≤λmaxPλminPϕ−ψρe−0.5λt,t≥0,and the periodic solution of system (1) is global exponential stability.Corollary 1.
Assume thatB,U, and K are some known constant matrices. Assume that the assumptions (H1) and (H2) hold. Then, system (1) exists a global asymptotically stable periodic solution if there exist three constants εi>0,i=1,2,3, n×n diagonal matrices P>0, and Qi>0,i=1,2,3, such that the following inequalities hold:(45)Ξ≥0,ε2P−∑i=1n11−ci2LgQ2Lg≥0,ε3P−∑i=1n11−ci2LhQ3Lh≥0,ε1−ε2−ε3μ>0,where(46)Ξ=Ξ1PKPUPW∗Q100∗∗Q20∗∗∗Q3,Ξ1=2PB−ε1P−∑i=1n11−ci2LfQ1Lf,Lf=diagl1,…,ln,lj=maxlj−,lj+,j∈Λ,Lg=diagσ1,…,σn,σj=maxσj−,σj+,j∈Λ,Lh=diagυ1,…,υn,υj=maxυj−,υj+,j∈Λ.


## 4. Numerical Example

In this section, a simulation example is presented for illustrating the usefulness of our main results. Consider a 2-neuron neural network (1) with the following parameters:(47)C=diag12,12,σ=π2,τt=μt=0.1,It=6sin2πTt,5cos2πTt⊤,B=0.91.21.50.6,K∈0.2,1.4−0.8,0.2−0.2,0.40,1.2,U∈0.3,1.5−0.1,0.9−1.2,−0.60.6,1.8,W∈−1.3,−0.10.1,1.1−0.9,−0.30.3,1.5,fx=tanhx,gx=0.1sinx,hx=3cosx.By simple calculation,Lf=I2×2,Lg=0.1I2×2,Lh=3I2×2,(48)Γi=0.60.500000.30.6,ϒi=0.60.500000.30.6⊤,i=1,2,3.Letε1=6,ε2=2,andε3=4.2. We can obtain the following feasible solutions to LMIs in Theorem 1:(49)P=2.8042I2×2,Q1=2.7821I2×2,Q2=4.4652I2×2,Q3=1.6052I2×2,Q4=2.9372I4×4,Q5=12.1125I4×4,Q6=1.5632I4×4.Hence, all the conditions in Theorem1 hold, and the 2-neuron neural networks (47) have a stationary oscillation with T period.Particularly, let the concrete parameter matrices in (47) be(50)K0=0.2−0.6−0.2,10.8,U0=0.5−0.1−0.51.7,W0=−0.20.7−0.31.2.The corresponding numerical simulations for differentT are presented in Figures 1–4. Figure 1 shows the state trajectories of system (47) with the period T=10. Figure 2 shows the phase plots of system (47) with the period T=10. Figure 3 shows the state trajectories of system (47) with the period T=0.2. Figure 4 shows the phase plots of system (47) with the period T=0.2.Figure 1
State trajectories of system (47) with T=10.Figure 2
Phase plots of system (47) with T=10.Figure 3
State trajectories of system (47) with T=0.2.Figure 4
Phase plots of system (47) with T=0.2.Remark 7.
In [4, 5, 8, 9, 11], the authors studied the stability problems of some nonneutral interval neural networks with delays. However, there are few results for neutral-type interval neural networks with impulsive delays which is the problem we will solve in the future.


## 5. Conclusions

In this paper, we had investigated the stability problems of neutral-type neural networks withD-operator and mixed delays. It is interesting and challenging to extend our results to the stochastic delay system [18], impulsive stochastic delay differential systems [19], semi-Markov switched stochastic systems [20], and stochastic systems with Lévy noise [21].



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