# Novel Delay-Dependent Stability Criteria for Discrete-Time Neural Networks with Time-Varying Delay

Authors: Sreten Stojanovic; Milan Stojanovic; Milos Stevanovic
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The delay-dependent stability problem is investigated for discrete-time neural networks with time-varying delays. A new augmented Lyapunov-Krasovskii functional (LKF) with single and double summation terms and several augmented vectors is proposed by decomposing the time-delay interval into two nonequidistant subintervals to derive less conservative stability conditions. Then, by using Wirtinger-based inequality, reciprocally, and extended reciprocally convex combination lemmas, tight estimations for sum terms in the forward difference of the LKF are given. Several zero equalities are introduced to further relax the existing results. Less conservative stability criteria are proposed in terms of linear matrix inequalities (LMIs). Finally, numerical examples are proposed to show the effectiveness and less conservativeness of the proposed method.


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## Body

## 1. Introduction

During the past few decades, neural networks (NNs) have received great attention because of their wide applications in various fields such as image processing, signal processing, pattern recognition, associative memory, parallel computation, optimization, and error diagnosis [1, 2]. One of the most important questions in theoretical analysis of NNs is dynamical behaviors of the NNs, such as their stability, periodic oscillatory, and chaos. Among them, analysis of the stability has received much attention and various stability conditions have been obtained in [3, 4].It is well known that a time delay is inherent in various systems, including NNs, owing to the finite speed of signal transmission and conversion rate of the processors. Delays in a system may cause oscillation and divergence and further degrade the performance. It is of great importance to determine the admissible maximal delay bound such that the neural networks with a delay less than this upper bound remain stable. Hence, it is essential to investigate the stability of the neural networks with time delay [5–10].Since most systems use a digital processor to acquire information from computers at discrete instants of time, it is essential to formulate discrete-time neural networks (DNNs) that are an analogue of continuous ones. Therefore, it is significant to study the dynamics of the DNNs and many results for the DNNs with time delay has received increasing attention; see [11–16] and references therein.The stability criteria for the DNNs with time-varying delay can be classified into two categories: delay-independent stability criteria and delay-dependent ones [17, 18]. As is well known, delay-dependent stability criteria, which take advantage of the information on the size of time delays, are less conservative than delay-independent ones. The main aim of delay-dependent stability criteria is to get maximum delay bounds guaranteeing the addressed neural networks to be stable. In order to improve results regarding this problem, various methods have been applied to the delay-dependent category, such as augmented LKF method [13, 19–22], free-weighting matrix method [18, 23], summation inequality method [16, 24–27], delay-partitioning method [5, 28, 29], and reciprocally convex approach [20, 30, 31].It is known that the simple LKFs with the single and/or double sum terms lead to the criteria with high conservatism, while more augmented LKFs reduce conservatism but increase the number of decision variables. By constructing an augmented LKF with the sum terms of a state vector as well as the activation function, an improved stability condition for the asymptotic stability of DNNs was derived in [13] but with an enlarged number of decision variables. In recent years, the LKF including triple summation terms has also been applied to study the stability analysis of discrete-time DNNs for a further improvement of the results [19, 32]. For the purpose of reducing conservatism, the delay-partitioning LKFs, in which the delay interval was separated into several subintervals, were introduced [5, 28, 29].Various free-weighting matrix-based stability criteria [18, 23] were established to improve the results obtained by the Jensen-based summation inequality. However, introducing the free-weighting matrices may result in increased number of decision variables.After the computational complexity became one of the crucial aspects of a research in the area of the system stability, the direct bounding method based on summation inequalities once again becomes the most popular method [17, 19, 32, 33]. Very recently, various types of the Wirtinger-based summation inequalities, tighter than the Jensen-based summation inequality, have been proposed for discrete-time linear time-delay systems [25, 34, 35] and have also been used for the study of the discrete-time DNNs [36]. As one of useful methods to deal with the stability of delayed systems, the reciprocally convex approach was developed in [30] and has been extensively used to study the dynamical behaviours of time-delay systems since then (see [32, 37–39] and references therein).Not long ago, an extended reciprocally convex combination lemma (ERCCL) has been developed [26, 31] to replace the popular reciprocally convex combination lemma (RCCL). It has a potential to reduce the conservatism of the RCCL-based criteria without introducing any extra decision variable due to its advantage of reduced estimation gap using the same decision variables.Motivated by the aforementioned discussions, in this paper, the problem of delay-dependent stability for DNNs with interval time-varying delay is considered. The objective of the paper is to derive simple but efficient stability conditions for DNNs with interval time-varying delay. The major contributions of this paper are as follows. Delay-decomposition method is used with new augmented Lyapunov–Krasovskii functional, which contains several augmented vectors in single and double summation terms with a free parameterα that divides interval [k-h2,k-h1] into two asymmetric subintervals [k-h2,k-α-1] and [k-α,k-h1]. The value of parameter α is determined by solving linear matrix inequalities (LMIs) that are defined in the proposed stability criterion. In this way, a greater degree of freedom is enabled in estimating the stability of the DDN, which leads to a smaller conservative stability criterion. The convenient existing summations inequalities, as well as extended reciprocally convex combination lemma, have been used for calculation difference of KLF. In order to improve the stability criteria, the LKF is extended with an additional double summation term and zero equalities (ZEs) are introduced in calculation difference of LKF. As a result of applying the mentioned techniques, it is shown that the derived results are less conservative then the existing ones [24, 31, 40, 41].Notations. Throughout the paper, Z+ denotes the set of positive integers and Rn denotes the n-dimensional Euclidean space and Rn×m the set of all n×m real matrices. For the positive integers a and b(b>a), Za,b denotes the set of all positive integers z satisfying a≤z≤b. In and 0n×m denote n×n identity matrix and n×m zero matrix, respectively. XT denotes the matrix transpose of X and ∗ represent the elements below the main diagonal of a symmetric matrix. diag{a,b,…,z} denotes the block-diagonal matrix with elements a,b,…,z in the diagonal entries and SymX=X+XT. For any symmetric matrix X∈Rn×n the notation X>0(X≥0) means that X is positive definite (positive semidefinite) matrix and X<0(X≤0) means that X is negative definite (negative semidefinite) matrix. For the matrices Ai∈Rn×m, i=1,2,…,l, ColA1,A2,…,Al denotes the column block matrix A1TA2T⋯AlTT. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Preliminaries

Consider the discrete-time neural networks (DNNs) with interval time-varying delay of the form:(1)yk+1=Cyk+Agyk+Adgyk-hk+J,k≥0,yj=φj,j∈-h2,-h2+1,…,-1,0,where k∈Z+, y(k)∈Rn is the neural state vector, gy(k)=g1y1k,g2y2k,…,gn(yn(k))T denotes the neural activation functions, φ(j) is the initial condition, C=diag{c1,c2,…,cn}, ci>0 represents the state feedback coefficient matrix, i.e., the rate with which thei-th neuron will reset its potential to the resting state in isolation when disconnected from the networks and external inputs, A∈Rn×n and Ad∈Rn×n are the connection weight matrix and the delay connection weight matrix, respectively, and J=J1,J2,…,JnT is the external input vector. Integer h(k) is time-varying delay that satisfies(2)0<h1≤hk≤h2,where h1 and h2 are known positive integers representing the lower and upper bounds of h(k).In this paper, the following assumption on the system (1) is made.Assumption 1 (see [18]).
For anya,b∈R and a≠b, the continuous and bounded activation functions gi(·) satisfy(3)ki-≤gia-giba-b≤ki+,i=1,2,…,n,where ki-and ki+ are known constants.Since the activation functiongi(·) satisfies (3), the system (1) has an equilibrium point. Let us denote the equilibrium point by y∗=y1∗,⋯,yn∗T, and then y∗=Cy∗+Ag(y∗)+Adg(y∗)+J. In order to simplify our consideration, we shift the equilibrium point of the system (1) to the origin. Let xi(k)=yi(k)-yi∗ and fixi(k)=gixi(k)+yi∗-gi(yi∗). Then, system (1) can be transformed to the following form:(4)xk+1=Cxk+Afxk+Adfxk-hk,k≥0,xj=ϕj,j∈-h2,-h2+1,…,-1,0,where x(k)∈Rn is the state vector of the transformed system and ϕ(j) is the initial condition.After applying Assumption1, one can check that the function f(·) with f(0)=0 satisfies(5)ki-≤fia-fiba-b≤ki+,i=1,2,…,n,for all a≠b. If =0, we have(6)ki-≤fiaa≤ki+,i=1,2,…,n,for all a≠0.The following lemmas will be used in the sequel to establish the main results.Lemma 2 (Jensen’s inequality [42]).
For any positive definite symmetric matrixR∈Rn×n and two positive integers a and b>a, the sum term ∑j=ab-1xT(j)Rx(j) is estimated as(7)b-a∑j=ab-1xTjRxj≥∑j=ab-1xTjR∑j=ab-1xj.Lemma 3 (Wirtinger-based inequality [25]).
For a given positive definite symmetric matrixR, two positive integers a and b>a, and any sequence of discrete-time variable x:Za,b→Rn, the following inequality holds:(8)b-a∑i=ab-1ηTiRηi≥θ1θ2TR0∗3ρa,bRθ1θ2(9)≥θ1θ2TR0∗3Rθ1θ2,where (10)ηk=xk+1-xk,θ1=xb-xa,θ2=xb+xa-2b-a+1∑i=abxi,ρa,b=b-a+1b-a-1,b-a≠11,b-a=1.Remark 4.
In [27], stability criteria for discrete-time delay systems are considered by using a new summation inequality with three slack matrix variables. It has been shown that inequality (8) is obtained as a special case of the new summation inequality with convenient selected values of the slack matrix variables. Therefore, by using the mentioned summation inequality, the conservatism can be reduced, but the introduction of three slack matrix variables can increase the computation complexity of the obtained stability criterion.The reciprocally convex combination lemma (RCCL) plays an important role in the estimation of the forward difference of LKF.Lemma 5 (reciprocally convex combination lemma [30, 31, 43]).
For a real scalarδ∈(0,1), a symmetric matrix R>0, and any matrix S satisfying(11)RS∗R≥0,the following inequality holds:(12)1δR0∗11-δR≥RS∗R.An extended reciprocally convex combination lemma (ERCCL), which estimates the sum terms in the forward difference of the LKF tightly than RCCL, is presented as follows.Lemma 6 (extended reciprocally convex combination lemma [31, 40]).
For a real scalarδ∈(0,1), a symmetric matrix R>0, and any matrix S, the following inequality holds:(13)1δR0∗11-δR≥R+1-δT1S∗R+δT2,where T1=R-SR-1ST and T2=R-STR-1S.Remark 7.
Unlike RCCL, the condition (11) is not used in ERCCL. Thus, the matrix S in ERCCL can be chosen more freely then the one in RCCL such that the feasibility criterion based on (13) is better than one based on (11). Further, if (11) holds then Ti≥0, i=1,2, and (14)R+1-δT1S∗R+δT2≥RS∗R.Thus, the terms (1-δ)T and δT2, which appear in (13), reduce estimation gap between two sides of (13) and inequality (13) in ERCCL is tighter then inequality (12) in RCCL. Note that the RCCL and ERCCL require the same number of decision variables (R and S).Remark 8.
In [20], the specific reciprocally convex inequality is proposed. For comparison, we rewrite the result of Theorem 1 in [20] in a block form as(15)1δR10∗11-δR2≥R1+1-δT1δS1+1-δS2∗R2+δT2,where T1=R1-Y1R2-1Y1T and T2=R2-Y2TR1-1Y2. If taking R1=R2=R and S1=S2=S, inequality (15) immediately reduces to (13), which means that ERCCL (Lemma 6) is a special case of Theorem 1 in [20]. During the calculation of the difference of LKF in (31), the inequality (13) is used instead of (15), because the same matrix R=Z~2 appears twice on the block diagonal in (31), so that the condition R1=R2=R is automatically satisfied.Further, by applying inequality (13) to (32), only one matrix S was used instead of two (S1 and S2), as in the case of application of inequality (15). Thereby, a computation complexity of the proposed stability criterion (13) is decreased, but there is a possibility of increasing conservatism (13) in regard to (15).Lemma 9 (see [40]).
For a positive definite symmetric matrixR and matrices Υ and Ξ, the following statements are equivalent:
( i )   Ξ - Υ T R Υ < 0
( i i ) There exists a matrix Ψ with appropriate dimension such that(16)Ξ+SymΥTΨΨT∗-R<0.


## 3. Main Results

In this section, we will establish a new criterion on asymptotical stability for the discrete-time neural networks with interval time-varying delay (1).We introduce the following notations for later use:(17)ξk=xTkxTk-h1xTk-hkxTk-h2xTk-αv1Tkv2Tkv3Tkv4TkfTxkfTxk-hkT,ηk=xk+1-xk,ζ1k=xTk∑j=k-h1k-1xTj∑j=k-h2k-h1-1xTj∑j=k-αk-h1-1xTjT,ζ2k=xTkfTxkT,ξ3k=xTkηTkT,v1k=∑j=k-h1kxjh1+1,v2k=∑j=k-hkk-h1xjhk-h1+1,v3k=∑j=k-h2k-hkxjh2-hk+1,v4k=∑j=k-αk-h1xjα-h1+1,ω1=e1-e2ξk,ω2=e1+e2-2e6ξk,ω3=e2-e5ξk,ω4=e2+e5-2e9ξk,ei=0n×i-1n,In,0n×11-in,i=1,2,…,11,es=Ce1+Ae10+Ade11,ρa,b=b-a+1b-a-1,b-a>11,b-a=1,δh=hk-h1h12,h12=h2-h1,Σhk,α=Σ11+Σ12hk,α+Σ2+Σ3+Σ41-Σ42-Σ43hk+Σ51α-Σ52α-ΣS,Σ11=Π1TPΠ1-Π2TPΠ2,Π1=esh1+1e6-e2-e3-e4-e5,Π2=e1h1+1e6-e1-e2-e3-e2,Σ12hk,α=SymΠ1-Π2TPΠhk,α,Πhk,α=00hk-h1+1e7+h2-hk+1e8α-h1+1e9,Σ2=e1TQ1e1+e2TQ3-Q1e2+e5TQ2-Q3e5-e4TQ2e4,Σ3=e1e10Th12+1Re1e10-e3e11TRe3e11,Σ4hk=Σ41-Σ42-Σ43hk,Σ41=es-e1Th12Z1+h122Z2es-e1,Σ42=E1TZ10∗3ρ0,h1Z1E1,E1=e1-e2e1+e2-2e6,Σ43hk=E2E3TZ~2S∗Z~2+1-δhT10∗δhT2E2E3,T1=Z~2-SZ~2-1ST,T2=Z~2-STZ~2-1S,Σ~43h1=E2E3TZ~2S∗Z~2E2E3+E2TZ~2E2,Σ~43h2=E2E3TZ~2S∗Z~2E2E3+E3TZ~2E3,Z~2=Z20∗3Z2,E2=e2-e3e2+e3-2e7,E3=e3-e4e3+e4-2e8,E4=e2-e5e2+e5-2e9,E5hk=hk-h1+1e7-e2e2-e3,E6hk=h2-hk+1e8-e3e3-e4,E56hk=E5hkE6hk,Σ51α=es-e1Tα-h12Z3es-e1,Σ52α=E4TZ30∗3ρh1,αZ3E4,ΣS=Syme10-K1e1TM1e10-K2e1+Syme11-K1e3TM2e11-K2e3+Syme10-e11-K1e1-e3TM3××e10-e11-K2e1-e3,Σ6=Σ61-Σ62hk,Σ61=h122e1es-e1TGe1es-e1+h12e2TH1e2-h12e4TH2e4-h12e3TH1-H2e3,Σ62hk=E56ThkG12E56hk,Gi=G+0Hi∗Hi,i=1,2,G12=G1X∗G2,K1=diagk1-,k2-,…,kn-,K2=diagk1+,k2+,…,kn+.Theorem 10.
For given positive integersh1 and h2, system (4) with interval time-varying delay satisfying condition (2) is asymptotically stable, if there exist positive-definite matrices P∈R4n×4n, Qi∈Rn×n, i=1,2,3, R∈R2n×2n, Zi∈Rn×n, i=1,2,3, positive definite diagonal matrices Mi∈Rn×n, i=1,2,3, and any matrix S∈R2n×2n such that the following LMIs hold: (18)Φh1,αE2TS∗-Z~2<0,(19)Φh2,αE3TST∗-Z~2<0,where(20)Φhi,α=Σ11+Σ12hi,α+Σ2+Σ3+Σ41-Σ42-Σ~43hi+Σ51α-Σ52α-Σs,i=1,2.Proof.
Construct a LKF for the DNN (4) as follows:(21)Vk=∑i=15Vik,where(22)V1k=ζ1TkPζ1k,V2k=∑j=k-h1k-1xTjQ1xj+∑j=k-h2k-α-1xTjQ2xj+∑j=k-αk-h1-1xTjQ3xj,V3k=∑j=k-hkk-1ζ2TjRζ2j+∑i=-h2+1-h1∑j=k+ik-1ζ2TjRζ2j,V4k=h1∑i=-h1-1∑j=k+ik-1ηTjZ1ηj+h12∑i=-h2-h1-1∑j=k+ik-1ηTjZ2ηj,V5k=α-h1∑i=-α-h1-1∑j=k+ik-1ηTjZ3ηj.The forward difference of LKF (21), ΔV(k)=V(k+1)-V(k), along the trajectories of the DNN (4) gives(23)ΔV1k=ζ1Tk+1Pζ1k+1-ζ1TkPζ1k.Since(24)ζ1k=Cole1,h1+1e6-e1,hk-h1+1e7+h2-hk+1e8-e2-e3,α-h1+1e9-e2ξk=Π2+Πhk,αξk,(25)ζ1k+1=Coles,h1+1e6-e2,hk-h1+1e7+h2-hk+1e8-e3-e4,α-h1+1e9-e5ξk=Π1+Πhk,αξk,it follows that(26)ΔV1k=ξTkΠ1+Πhk,αTP×Π1+Πhk,αξk-ξTkΠ2+Πhk,αTP×Π2+Πhk,αξk=ξTkΠ1TPΠ1-Π2TPΠ2+2Π1-Π2TPΠhk,αξk=ξTkΣ11+Σ12hk,αξk.The forward differences of V2(k) and V3(k) can be obtained as (27)ΔV2k=∑j=k+1-h1kxTjQ1xj-∑j=k-h1k-1xTjQ1xj+∑j=k+1-h2k-αxTjQ2xj-∑j=k-h2k-α-1xTjQ2xj+∑j=k+1-αk-h1xTjQ3xj-∑j=k-αk-h1-1xTjQ3xj=xTkQ1xk-xTk-h1Q1xk-h1+xTk-αQ2xk-α-xTk-h2Q2xk-h2+xTk-h1Q3xk-h1-xTk-αQ3xk-α=xTkQ1xk+xTk-h1Q3-Q1xk-h1+xTk-αQ2-Q3xk-α-xTk-h2Q2xk-h2=ξTke1TQ1e1+e2TQ3-Q1e2+e5TQ2-Q3e5-e4TQ2e4ξk=ξTkΣ2ξk,(28)ΔV3k=∑j=k+1-hk+1kζ2TjRζ2j-∑j=k-hkk-1ζ2TjRζ2j+∑i=-h2+1-h1∑j=k+1+ikζ2TjRζ2j-∑i=-h2+1-h1∑j=k+ik-1ζ2TjRζ2j=∑j=k+1-hk+1k-h1ζ2TjRζ2j+∑j=k-h1+1k-1ζ2TjRζ2j+ζ2TkRζ2k-∑j=k-hkk-1ζ2TjRζ2j+∑i=-h2+1-h1∑j=k+ik-1ζ2TjRζ2j+ζ2TkRζ2k-ζ2Tk+iRζ2k+i-∑j=k+ik-1ζ2TjRζ2j≤∑j=k+1-h2k-h1ζ2TjRζ2j+∑j=k-hk+1k-1ζ2TjRζ2j+ζ2TkRζ2k-∑j=k-hk+1k-1ζ2TjRζ2j-ζ2Tk-hkRζ2k-hk+∑i=-h2+1-h1ζ2TkRζ2k-∑i=-h2+1-h1ζ2Tk+iRζ2k+i=∑j=k+1-h2k-h1ζ2TjRζ2j+ζ2TkRζ2k-ζ2Tk-hkRζ2k-hk+h12ζ2TkRζ2k-∑j=k+1-h2k-h1ζ2TjRζ2j=h12+1ζ2TkRζ2k-ζ2Tk-hkRζ2k-hk=ξTke1e10Th12+1Re1e10ξk-ξTke3e11TRe3e11ξk=ξTkΣ3ξk,(29)ΔV4k=h1∑i=-h1-1∑j=k+1+ikηTjZ1ηj-∑j=k+ik-1ηTjZ1ηj+h12∑i=-h2-h1-1∑j=k+1+ikηTjZ2ηj-∑j=k+ik-1ηTjZ2ηj=h1∑i=-h1-1ηTkZ1ηk-ηTk+iZ1ηk+i+h12∑i=-h2-h1-1ηTkZ2ηk-ηTk+iZ2ηk+i=ηTkh12Z1+h122Z2ηk-h1∑i=-h1-1ηTk+iZ1ηk+i-h12∑i=-h2-h1-1ηTk+iZ2ηk+i=ξTkes-e1Th12Z1+h122Z2es-e1ξk-h1∑j=k-h1k-1ηTjZ1ηj-h12∑j=k-h2k-h1-1ηTjZ2ηj.Using inequality (8) from Lemma 3 to estimate Z1-dependent summation term yields (30)h1∑j=k-h1k-1ηTjZ1ηj≥ω1ω2TZ10∗3ρ0,h1Z1ω1ω2=ξTkΣ42ξk.Similarly, by using (9) we estimate Z2-dependent summation term(31)h12∑j=k-h2k-h1-1ηTjZ2ηj=h12∑j=k-h2k-hk-1ηTjZ2ηj+h12∑j=k-hkk-h1-1ηTjZ2ηj≥1δhξTkE2TZ2003Z2E2ξk+11-δhξTkE3TZ2003Z2E3ξk=ξTkE2E3T1δhZ~20011-δhZ~2E2E3ξk.Based on Lemma 6, for any matrix S∈R2n×2n we have(32)h12∑j=k-h2k-h1-1ηTjZ2ηj≥ξTkE2E3TZ~2+1-δhT1S∗Z~2+δhT2E2E3ξk=ξTkΣ43hkξTk.Then, we can get the upper bound of ΔV4(k) as(33)ΔV4k≤ξTkΣ41-Σ42-Σ43hkξk=ξTkΣ4hkξk.Calculating ΔV5(k) gives(34)ΔV5k=α-h1∑i=-α-h1-1∑j=k+ik-1ηTjZ3ηj+ηTkZ3ηk-ηTk+iZ3ηk+i-∑j=k+ik-1ηTjZ3ηj=α-h12ηTkZ3ηk-α-h1∑i=-α-h1-1ηTk+iZ3ηk+i=ξTkes-e1Tα-h12Z3es-e1ξk-α-h1∑j=k-αk-h1-1ηTjZ3ηj.By inequality (8), the second term of ΔV5(k) can be written as(35)α-h1∑j=k-αk-h1-1ηTjZ3ηj≥ω3ω4TZ30∗3ρh1,αZ3ω3ω4.Then, we can get the upper bound of ΔV5(k) as(36)ΔV5k≤ξTkΣ51α-E4TZ30∗3ρh1,αZ3E4ξk=ξTkΣ51α-Σ52αξk.
Under the assumption on the activation functions (5) and (6), for any positive definite diagonal matrices Mj=diagmj1,mj2,…,mjn, j=1,2,3, the following inequality holds:(37)0≥2∑i=1nm1ifixik-ki-xik×fixik-ki+xik+2∑i=1nm2ifixik-hk-ki-xik-hk×fixik-hk-ki+xik-hk+2∑i=1nm3ifixik-fixik-hk-ki-xik-xik-hk×fixik-fixik-hk-ki+xik-xik-hk.Then(38)0≥2fxk-K1xkM1fxk-K2xk+2fxk-hk-K1xk-hk×M2fxk-hk-K2xk-hk+2fxk-fxk-hk-K1xk-xk-hkM3×fxk-fxk-hk-K2xk-xk-hk,(39)0≥2ξTke10-K1e1TM1e10-K2e1ξTk+2ξTke11-K1e3TM2e11-K2e3ξTk+2ξTke10-e11-K1e1-e3TM3×e10-e11-K2e1-e3ξTk.So, we have (40)ξTkΣSξk≤0.Therefore, by combining (26)-(28), (33), (36), and (40), the forward difference of V(k) is obtained as(41)ΔVk≤ξTkΣhk,αξk.Since the matrix Σ(h(k),α) is affine with respect to the delay h(k), the condition Σ(h(k),α)<0 is satisfied if and only if(42)Σhi,α<0,i=1,2.By calculating, we have(43)Σ12h1,α=SymΠ1-Π2TPΠh1,α,Σ12h2,α=SymΠ1-Π2TPΠh2,α,(44)Σ43h1=E2E3TZ~2S∗Z~2+T10∗0E2E3=E2E3TZ~2S∗Z~2E2E3+E2TZ~2E2-E2TSZ~2-1STE2=Σ~43h1-E2TSZ~2-1STE2,(45)Σ43h2=E2E3TZ~2S∗Z~2+00∗T2E2E3=E2E3TZ~2S∗Z~2E2E3+E3TZ~2E3-E3TSTZ~2-1SE3=Σ~43h2-E3TSTZ~2-1SE3,(46)Σh1,α=Φh1,α+E2TSZ~2-1STE2<0,(47)Σh2,α=Φh2,α+E3TSTZ~2-1SE3<0.Then, by taking Schur complement, it can be seen that (46) and (47) are equivalent to (18) and (19), respectively. Therefore, when (18) and (19) hold, ΔV(k)<0, which shows that system (4) is asymptotically stable.Remark 11.
In Theorem10, the Wirtinger-based summation inequality (8) is applied to summation with the constant lower and upper bound (h1∑j=k-h1k-1ηT(j)Z1η(j)). However, in the case of the summation with the time-varying lower or upper bound(48)h12∑j=k-h2k-hk-1ηTjZ2ηj+h12∑j=k-hkk-h1-1ηTjZ2ηjcombination of the Wirtinger-based inequality (9) and the reciprocally convex approach (13) is applied.By introducing an augmented LKF and zero equations, a further improved stability condition of system (4) can be obtained as follows.Theorem 12.
For given positive integersh1 and h2, system (4) with interval time-varying delay satisfying condition (2) is asymptotically stable, if there exist positive-definite matrices P∈R4n×4n, Qi∈Rn×n, i=1,2,3, R∈R2n×2n, Zi∈Rn×n, i=1,2,3, G∈R2n×2n, positive definite diagonal matrices Mi∈Rn×n, i=1,2,3, symmetric matrices Hi∈Rn×n, i=1,2, and any matrices S∈R2n×2n and X∈R2n×2n such that the following LMIs hold:(49)G1X∗G2>0,(50)Φh1,α+Σ61-Σ62h1E2TS∗-Z~2<0,(51)Φh2,α+Σ61-Σ62h2E3TST∗-Z~2<0,where(52)Φhi,α=Σ11+Σ12hi,α+Σ2+Σ3+Σ41-Σ42-Σ~43hi+Σ51α-Σ52α-Σs,i=1,2.Proof.
Construct the following LKF candidate:(53)V~k=Vk+V6k,where(54)V6k=h12∑i=-h2-h1-1∑j=k+ik-1ζ3TjGξ3j.The forward difference of (54) along the trajectories of the DNN (4) gives(55)ΔV6k=h122ζ3TkGξ3k-h12∑j=k-h2k-hk-1ζ3TjGξ3j-h12∑j=k-hkk-h1-1ζ3TjGξ3j.For any symmetric matrices Hi, i=1,2, we have(56)∑j=k-l2k-l1-1xTj+1Hixj+1-xTjHixj=xTk-l1Hixk-l1-xTk-l2Hixk-l2.Since(57)xTj+1Hixj+1-xTjHixj=xj+ηkTHixj+ηk-xTjHixj=ηTjHiηj+2xTjHiηj=xjηjT0Hi∗Hixjηj=ζ3Tk0Hi∗Hiξ3j,then the following zero equation holds:(58)EQl1,l2,Hi=xTk-l1Hixk-l1-xTk-l2Hixk-l2-∑j=k-l2k-l1-1ζ3Tk0Hi∗Hiξ3j=0.Now, from (58) we get the following two-zero equations:(59)EQh1,hk,H1=xTk-h1H1xk-h1-xTk-hkH1xk-hk-∑j=k-hkk-h1-1ζ3Tk0H1∗H1ξ3j=0,(60)EQhk,h2,H2=xTk-hkH2xk-hk-xTk-h2H2xk-h2-∑j=k-h2k-hk-1ζ3Tk0H2∗H2ξ3j=0.Adding the inequalities (59) and (60) to (55) implies(61)ΔV6k=h122ζ3TkGξ3k+h12xTk-h1H1xk-h1-h12xTk-hkH1-H2xk-hk-h12xTk-h2H2xk-h2-h12∑j=k-hkk-h1-1ζ3TjG1ξ3j-h12∑j=k-h2k-hk-1ζ3TjG2ξ3j.Applying Jensen inequality (7), ΔV6(k) is estimated as(62)
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.Using Lemma 5, for any matrix X∈R2n×2n we have (63)ΔV6k≤ξTkΣ61-E56ThkG1X∗G2E56hkξk=ξTkΣ61-E56ThkG12E56hkξk=ξTkΣ61-Σ62hkξk=ξTkΣ6hkξk.Finally(64)ΔV~k≤ξTkΣhk,α+Σ61-Σ62hkξk.Based on Lemma 9, the equivalent condition of Σ(h(k),α)+Σ61-E56T(h(k))G12E56(h(k))<0 is that matrix Ψ exists such that (65)
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61
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.Since the matrix Θ(h(k),α) is affine with respect to the delay h(k), the condition Θ(h(k),α)<0 is satisfied if and only if(66)Θhi,α<0,i=1,2.Based on Lemma 9, (66) is equivalent to(67)Σhi,α+Σ61-E56ThiG12E56hi=Σhi,α+Σ61-Σ62hi<0,i=1,2,i.e.,(68)Φh1,α+E2TSZ~2-1STE2+Σ61-Σ62h1<0,(69)Φh2,α+E3TSTZ~2-1SE3+Σ61-Σ62h2<0.Then, by taking Schur complement, it can be seen that (68) and (69) are equivalent to (50) and (51), respectively. Therefore, when (50) and (51) hold, ΔV~(k)<0, which shows that system (4) is asymptotically stable.Remark 13.
In order to improve the stability criterion proposed in Theorem10, a new double summation term V6(k) with vector ξ3(k)=xT(k)ηT(k)T and free matrix G is introduced in LKF. Further, in the proof of Theorem 12, two zero equalities with symmetric matrices H1 and H2 are introduced in ΔV6(k). In this way, the conservatism of the criterion is reduced.Remark 14.
In order to reduce the number of decision variables, Lemma9 is used twice. First, by using the lemma, nonaffine term (70)Σhk,α+Σ61-E56ThkG12E56hk<0with respect to h(k) is transformed into the affine term Θ(h(k),α). On that occasion, additional matrix Ψ∈R11n×2n is introduced in (65) and the number of decision variables has increased significantly (for 22n2). Second, if we apply Lemma 9 on (66), then the matrix Ψ∈R11n×2n can be eliminated from Θ(hi,α). In this way, the significant reduction of decision variables was performed.Remark 15.
The proposed stability criterion depends on the lower and upper bounds of time-delay,h1 and h2. In order to compare our results with existing ones, a maximum allowable upper bound (MAUB) of the time-delay, h2max, is adopted, such that the concerned system is asymptotically stable for any delay size less than the MAUB. Note that a criterion that gives a lower value of MAUB is less conservative with respect to other criteria.


## 4. Demonstrative Examples

In the section bellow, we provide three numerical examples to illustrate the effectiveness of the stability criteria proposed in this paper.Example 1.
Consider the DNN (4) with the following parameters:(71)C=0.8000.9,A=0.001000.005,Ad=-0.10.01-0.2-0.1,which was used to check the feasible region of stability criteria in [24, 31, 40, 41]. The activation function is in the form of fx=tanh(x) and satisfies Assumption 1 with k1-=k2-=0 and k1+=k2+=1. By using Theorem 10, the MAUB h2max is computed with different lover bounds h1 and the obtained results are given in Table 1. The number of decision variables (NDVs) is also given to show computation complexity. From this table, we can see that our results are less conservative and/or have less decision variables than [24, 31, 40, 41].Table 1
MAUBh2max for given h1 in Example 1.
Method h 1 NDVs 2 4 6 8 10 15 20 Th. 2 [40] 19 19 20 21 22 24 27 17.5 n 2 + 8.5 n Th. 1 [40] 20 20 21 21 22 24 27 53.5 n 2 + 8.5 n Th. 1 [31] - 20 20 21 22 24 - 13.5 n 2 + 11.5 n Th. 3.1 [24] 19 19 20 20 21 24 27 11 n 2 + 6 n Cor. 2 [41] 19 20 20 21 21 24 27 154 n 2 + 6 n  Theorem10 19 20 20 21 22 24 27 17 n 2 + 9 n  Theorem12 20 20 21 21 22 24 27 23 n 2 + 11 nThe time-varying delay and the state trajectories of the DNN (4) with parameters (71) for initial value ϕ(j)=31T, j∈-22,-21,…,-1,0 are shown in Figures 1 and 2, which shows that the DNN (4) is asymptotically stable for 10≤h(k)≤22.Figure 1
The time-varying delay.Figure 2
State trajectories of the DNN.Example 2.
Consider the DNN (4) with the following parameters [31]:(72)C=0.1000.3,A=0.02000.004,Ad=-0.010.01-0.02-0.01,and activation functions that satisfy Assumption 1 with k1-=k2-=0 and k1+=k2+=1. In this example, the MAUB h2max is computed by using Theorems 10 and 12 for different lover bounds h1 and results are shown in Table 2. From this table, it can be seen that the proposed stability criteria, Theorems 10 and 12, provide significantly larger MAUB than the previous result [31], while the numbers of decision variables are slightly higher.Table 2
MAUBh2max for given h1 in Example 2.
Method h 1 NDVs 2 4 6 8 10 20 Th. 1 [31] 99 101 103 105 107 117 13.5 n 2 + 11.5 n  Theorem10 2028 2030 2032 2034 2036 2046 17 n 2 + 9 n  Theorem12 3140 3142 3144 3146 3148 3158 23 n 2 + 11 nExample 3.
Consider the DNN (4) with the following [40]:(73)C=0.40000.30000.3,A=0.2-0.20.10-0.30.3-0.2-0.1-0.2,Ad=-0.20.10-0.20.30.10.1-0.20.3,k1-=0,k2-=-0.4,k3-=-0.2,k1+=0.6,k2+=0,k3+=0.By applying Theorems 10 and 12 to the DNN (4) with parameters (73), the MAUB h2max is computed for different lover bounds h1 and results are shown in Table 3. From this table it can be seen that stability criterion proposed in Theorem 12 gives approximately the same results but with significantly smaller number of decision variables than [40].Table 3
MAUBh2max for given h1 in Example 3.
Method h 1 NDVs 2 4 6 10 15 20 Th. 2 [40] 13 15 17 21 26 31 17.5 n 2 + 8.5 n  Th. 1 [40] 18 20 22 26 31 36 53.5 n 2 + 8.5 n  Theorem10 13 15 16 19 24 29 17 n 2 + 9 n  Theorem12 19 20 22 26 31 36 23 n 2 + 11 n


## 5. Conclusion

In this study, a novel Lyapunov-Krasovskii functional with single and double summation terms and several augmented vectors is constructed by decomposing the time-delay interval into two nonequidistant subintervals. Using Wirtinger-based inequality, extended reciprocally convex approach, and several zero equalities, new stability conditions are developed in the form of linear matrix inequalities. It is shown that the proposed results are less conservative than the existing ones. Several numerical examples are presented to show the effectiveness and less conservativeness of the proposed method.



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