# Exponential Stabilization of Neutral-Type Neural Networks with Interval Nondifferentiable and Distributed Time-Varying Delays

Authors: W. Weera; P. Niamsup
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The problem of exponential stabilization of neutral-type neural networks
with various activation functions and interval nondifferentiable and distributed time-varying
delays is considered. The interval time-varying delay function is not required
to be differentiable. By employing new and improved Lyapunov-Krasovskii functional
combined with Leibniz-Newton’s formula, the stabilizability criteria are formulated in
terms of a linear matrix inequalities. Numerical examples are given to illustrate and show the
effectiveness of the obtained results.


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## Body

## 1. Introduction

In recent years, there have been great attentions on the stability analysis of neural networks due to its real world application to various systems such as signal processing, pattern recognition, content-addressable memory, and optimization [1–5]. In performing a periodicity or stability analysis of a neural network, the conditions to be imposed on the neural network are determined by the characteristics of various activation functions and network parameters. When neural networks are designed for problem solving, it is desirable that their activation functions are not too restrictive, [3, 6, 7]. It is known that time delays cannot be avoided in the hardware implementation of neural networks due to the finite switching speed of amplifies in electronic neural networks or the finite signal propagation time in biological networks. The existence of time delays may result in instability or oscillation of a neural network. Therefore, many researchers have focused on the study of stability analysis of delayed neural networks with various activation functions with more general conditions during the last decades [2, 8–11].The stability criteria for system with time delays can be classified into two categories: delay independent and delay dependent. Delay-independent criteria do not employ any information on the size of the delay; while delay-dependent criteria make use of such information at different levels. Delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the delay is small. In many situations, time delays are time-varying continuous functions which vary from 0 to a given upper bound. In addition, the range of time delays may vary in a range for which the lower bound is not restricted to be 0; in which case time delays are called interval time-varying delay. A typical example with interval time delay is the networked control system, which has been widely studied in the recent literature (see, e.g., [2, 11–14]). Therefore, it is of great significance to investigate the stability of system with interval time-varying delay. Another important type of time delay is distributed delay where stability analysis of neural networks with distributed delayed has been studied extensively recently, see [2, 5, 8–11, 15–17].It is know that exponential stability is more important than asymptotic stability since it provides information on convergence rate of solutions of systems to equilibrium points. It is particularly important for neural networks where the exponential convergence rate is used to determine the speed of neural computations. Therefore, it is important to determine the exponential stability and to estimate the exponential convergence rate for delayed neural networks. Consequently, many researchers have considered the exponential stability analysis problem for delayed neural networks and several results on this topic that have been reported in the literatures [3, 9, 13, 14, 17].In practical control designs, due to systems uncertainty, failure modes, or systems with various modes of operation, the simultaneous stabilization problem has often to be taken into account. The problem is concerned with designing a single controller which can simultaneously stabilize a set of systems. Recently, the exponential stability and stabilization problems for time-delay systems have been studied by many researchers, see [8, 12, 18, 19]. Among the usual approach, there are many results on the stabilization problem of neural networks being reported in the literature (see [2, 9, 15, 16, 20, 21]). In [21], robust stabilization criterion are provided via designing a memoryless state feedback controller for the time-delay dynamical neural networks with nonlinear perturbation. However, time-delay is required to be constant. In [16], global robust stabilizing control are presented for neural network with time-varying delay with the lower bound restricted to be 0. For neural network with interval time-varying delay, global stability analysis is considered with control input in [2, 9, 12]. Nonetheless, in most studies, the time-varying delays are required to be differentiable [2, 9, 15, 16]. Therefore, their methods have a conservatism which can be improved upon.It is noted that these stability conditions are either with testing difficulty or with conservatism to some extent. It is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions. Such systems are called neutral-type systems in which the systems contain both state delay as well as state derivative delay, so called neutral delay. The phenomena on neutral delay often appears in the study of heat exchanges, distributed networks containing lossless transmission lines, partial element equivalent circuits and population ecology are examples of neutral systems see [1, 10, 11, 22, 23] and references cited therein.Based on the above discussions, we consider the problem of exponential stabilization of neutral-type neural networks with interval and distributed time-varying delays. There are various activation functions which are considered in the system and the restriction on differentiability of interval time-varying delays is removed, which means that a fast interval time-varying delay is allowed. Based on the construction of improved Lyapunov-Krasovskii functionals combined with Liebniz-Newton's formula and some appropriate estimation of integral terms, new delay-dependent sufficient conditions for the exponential stabilization of the system are derived in terms of LMIs without introducing any free-weighting matrices. The new stability conditions are much less conservative and more general than some existing results. Numerical examples are given to illustrate the effectiveness and less conservativeness of our theoretical results. To the best of our knowledge, our results are among the first on investigation of exponential stabilization for neutral-type neural networks with discrete, neutral, and distributed delays.The rest of this paper is organized as follows. In Section2, we give notations, definitions, propositions, and lemmas which will be used in the proof of the main results. Delay-dependent sufficient conditions for the exponential stabilization of neutral-type neural networks with various activation functions, interval and distributed time-varying delays, and designs of memoryless feedback controls are presented in Section 3. Numerical examples illustrated the obtained results are given in Section 4. The paper ends with conclusions in Section 5 and cited references.


## 2. Preliminaries

The following notation will be used in this paper:ℝ+ denotes the set of all real nonnegative numbers; ℝn denotes the n-dimensional space and the vector norm ∥·∥; Mn×r denotes the space of all matrices of (n×r)-dimensions.AT denotes the transpose of matrix A; A is symmetric if A=AT; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λmax(A)=max{Reλ;λ∈λ(A)}.xt:={x(t+s):s∈[-h,0]},∥xt∥=sups∈[-h,0]∥x(t+s)∥;C([0,t],ℝn) denotes the set of all ℝn-valued continuous functions on [0,t]; L2([0,t],ℝm) denotes the set of all the ℝm-valued square integrable functions on [0,t].MatrixA is called semi-positive definite (A≥0) if 〈Ax,x〉≥0, for all x∈ℝn;A is positive definite (A>0) if 〈Ax,x〉>0 for all x≠0;A>B means A-B>0. The symmetric term in a matrix is denoted by *.Consider the following neural networks with mixed time-varying delays and control input(2.1)ẋ(t)=-(A+ΔA(t))x(t)+(W0+ΔW0)f(x(t))+(W1+ΔW1)g(x(t-h(t)))+(W2+ΔW2)∫t-k(t)th(x(s))ds+B0ẋ(t-η(t))+Bu(t),x(t)=ϕ(t),t∈[-d,0],d=max⁡{h2,k,η},
where x(t)∈ℝn is the state of the neural networks, u(·)∈L2([0,t],ℝm) is the control, n is the number of neurals, and (2.2)f(x(t))=[f1(x1(t)),f2(x2(t)),…,fn(xn(t))]T,g(x(t))=[g1(x1(t)),g2(x2(t)),…,gn(xn(t))]T,h(x(t))=[h1(x1(t)),h2(x2(t)),…hn(xn(t))]T
are the activation functions, A=diag(a¯1,a¯2,…,a¯n),a¯i>0 represents the self-feedback term and W0,W1,W2 denote the connection weights, the discretely delayed connection weights, and the distributively delayed connection weight, respectively. In this paper, we consider various activation functions and assume that the activation functions f(·),g(·),h(·) are Lipschitzian with the Lipschitz constants ai,bi,ci>0(2.3)|fi(ξ1)-fi(ξ2)|≤ai|ξ1-ξ2|,i=1,2,…,n,∀ξ1,ξ2∈R,|gi(ξ1)-gi(ξ2)|≤bi|ξ1-ξ2|,i=1,2,…,n,∀ξ1,ξ2∈R,|hi(ξ1)-hi(ξ2)|≤ci|ξ1-ξ2|,i=1,2,…,n,∀ξ1,ξ2∈R.
The time-varying delay functions h(t),k(t),η(t) satisfy the condition(2.4)0≤h1≤h(t)≤h2,0≤k(t)≤k,0≤η(t)≤η,η̇(t)≤δ<1.
It is worth noting that the time delay is assumed to be a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is bounded but not restricted to being zero. The initial functions ϕ(t)∈C1([-d,0],ℝn), with the norm (2.5)∥ϕ∥=sup⁡t∈[-d,0]∥ϕ(t)∥2+∥ϕ̇(t)∥2.
The uncertainties satisfy the following condition:(2.6)ΔA(t)=EaFa(t)Ha,ΔW0(t)=E0F0(t)H0,ΔW1(t)=E1F1(t)H1,ΔW2(t)=E2F2(t)H2,
where Ei,Hi,i=a,0,1,2 are given constant matrices with appropriate dimensions, Fi(t),i=a,0,1,2 are unknown, real matrices with Lebesgue measurable elements satisfying(2.7)FiT(t)Fi(t)≤I,i=a,0,1,2∀t≥0.Definition 2.1.
The zero solution of system (2.8) is exponentially stabilizable if there exists a feedback control u(t)=Kx(t),K∈ℝm×n such that the resulting closed-loop system
(2.8)ẋ(t)=-[A-BK]x(t)+W0f(x(t))+W1g(x(t-h(t)))+W2∫t-k(t)th(x(s))ds+B0ẋ(t-η(t))
is α-stable.Definition 2.2.
Givenα>0. The zero solution of system (2.1) with u(t)=0 is α-stable if there exists a positive number N>0 such that every solution x(t,ϕ) satisfies the following condition:
(2.9)‖x(t,ϕ)‖≤Ne-αt‖ϕ‖,∀t≥0.We introduce the following technical well-known propositions and lemma, which will be used in the proof of our results.Proposition 2.3 (Cauchy inequality).
For any symmetric positive definite matrixN∈Mn×n and x,y∈ℝn one has
(2.10)±2xTy≤xTNx+yTN-1y.Proposition 2.4 (see [24]).
For any symmetric positive definite matrixM>0, scalar γ>0 and vector function ω:[0,γ]→ℝn such that the integrations concerned are well defined, the following inequality holds:
(2.11)(∫0γω(s)ds)TM(∫0γω(s)ds)≤γ(∫0γωT(s)Mω(s)ds).Proposition 2.5 ([24, Schur complement lemma]).
Given constant symmetric matricesX,Y,Z with appropriate dimensions satisfying X=XT,Y=YT>0. Then X+ZTY-1Z<0 if and only if
(2.12)(XZTZ-Y)<0or(-YZZTX)<0.Lemma 2.6 (see [25]).
Given matricesQ=QT,H,E, and R=RT>0 with appropriate dimensions. Then
(2.13)Q+HFE+ETFTHT<0,
for all F satisfying FTF≤R, if and only if there exists an ϵ>0 such that
(2.14)Q+ϵHHT+ϵ-1ETRE<0.


## 3. Main Results

### 3.1. Exponential Stabilization for Nominal Interval Time-Varying Delay Systems

The nominal system is given by(3.1)ẋ(t)=-Ax(t)+W0f(x(t))+W1g(x(t-h(t)))+W2∫t-k(t)th(x(s))ds+B0ẋ(t-η(t))+Bu(t)x(t)=ϕ(t),t∈[-d,0].
First, we present a delay-dependent exponential stabilizability analysis conditions for the given nominal interval time-varying delay system (3.1) with ΔA(t)=ΔW0(t)=ΔW1(t)=ΔW2(t)=0. Let us set (3.2)λ1=λmin⁡(P-1),λ2=λmax⁡(P-1)+2h2λmax⁡(P-1QP-1)+2h22λmax⁡(P-1RP-1)+ηλmax⁡(P-1Q1P-1)+2λmax⁡(HD2-1H)+(h2-h1)2λmax⁡(P-1UP-1).Assumption 3.1.
All the eigenvalues of matrixB0 are inside the unit circle.Theorem 3.2.
Givenα>0. The system (3.1) is α-exponentially stabilizable if there exist symmetric positive definite matrices P,Q,R,U,Q1, three diagonal matrices Di,i=0,1,2 such that the following LMI holds:
(3.3)M1=M-[000-I0I]T×e-2αh2U[000-I0I]<0,(3.4)M2=M-[00I00-I]T×e-2αh2U[00I00-I]<0,(3.5)M3=[-0.1BBT-0.1(e-2αh1+e-2αh2)R2kPH2PF*-2kD20**-2D0]<0,(3.6)M4=[-0.1e-2αh2U2PG*-2D1]<0,(3.7)M=[M11M12M13M14M150*M2200M250**M3300M36***M440M46****M550*****M66],
where
(3.8)M11=[-A+αI]P+P[-A+αI]T-0.9BBT+2Q+W0D0W0T+W1D1W1T+ke2αkW2D2W2T-0.9e-2αh1R-0.9e-2αh2R]M12=B0P,M13=e-2αh1R,M14=e-2αh2R,M15=-PAT-0.5BBT,M22=-(1-δ)e-2αηQ1,M25=PB0T,M33=-e-2αh1Q-e-2αh1R-e-2αh2U,M36=e-2αh2U,M44=-e-2αh2Q-e-2αh2R-e-2αh2U,M46=e-2αh2U,M55=h12R+h22R+(h2-h1)2U+Q1-2P+W0D0W0T+W1D1W1T+ke2αkW2D2W2T]M66=-1.9e-2αh2U.
Moreover, the memoryless feedback control is
(3.9)u(t)=-0.5BTP-1x(t),t≥0,
and the solution x(t,ϕ) of the system satisfies
(3.10)‖x(t,ϕ)‖≤λ2λ1e-αt‖ϕ‖,∀t≥0.Proof.
LetY=P-1,y(t)=Yx(t).  Using the feedback control (3.9) we consider the following Lyapunov-Krasovskii functional:
(3.11)V(t,xt)=∑i=18Vi,
where
(3.12)V1=xT(t)Yx(t),V2=∫t-h1te2α(s-t)xT(s)YQYx(s)ds,V3=∫t-h2te2α(s-t)xT(s)YQYx(s)ds,V4=h1∫-h10∫t+ste2α(τ-t)ẋT(τ)YRYẋ(τ)dτds,V5=h2∫-h20∫t+ste2α(τ-t)ẋT(τ)YRYẋ(τ)dτds,V6=(h2-h1)∫-h2-h1∫t+ste2α(τ-t)ẋT(τ)YUYẋ(τ)dτds,V7=∫t-η(t)te2α(s-t)ẋT(s)YQ1Yẋ(s)ds,V8=2∫-k0∫t+ste2α(τ-t)hT(x(τ))D2-1h(x(τ))dτds.
It easy to check that
(3.13)λ1∥x(t)∥2≤V(t,xt)≤λ2∥xt∥2,∀t≥0.
Taking the derivative of V(xt) along the solution of system (3.1) we have
(3.14)V̇1=2xT(t)Yẋ(t),=2yT(t)[-Ax(t)+W0f(x(t))+W1g(x(t-h(t)))+W2∫t-k(t)th(x(s))ds+B0ẋ(t-η(t))-0.5BBTP-1x(t)∫t-k(t)t]=2yT(t)[-APy(t)+W0f(x(t))+W1g(x(t-h(t)))+W2∫t-k(t)th(x(s))ds+B0Pẏ(t-η(t))-0.5BBTy(t)∫t-k(t)t](3.15)=yT(t)[-AP-PAT]y(t)+2yT(t)W0f(x(t))+2yT(t)W1g(x(t-h(t)))+2yT(t)W2∫t-k(t)th(x(s))ds+2yT(t)B0Pẏ(t-η(t))-yT(t)BBTy(t)+2αyT(t)Py(t)-2αV1,V̇2=yT(t)Qy(t)-e-2αh1yT(t-h1)Qy(t-h1)-2αV2,V̇3=yT(t)Qy(t)-e-2αh2yT(t-h2)Qy(t-h2)-2αV3,V̇4≤h12ẏT(t)Rẏ(t)-h1e-2αh1∫t-h1tẏT(s)Rẏ(s)ds-2αV4,V̇5≤h22ẏT(t)Rẏ(t)-h2e-2αh2∫t-h2tẏT(s)Rẏ(s)ds-2αV5,V̇6≤(h2-h1)2ẏT(t)Uẏ(t)-(h2-h1)e-2αh2∫t-h2t-h1ẏT(s)Uẏ(s)ds-2αV6,V̇7≤ẏT(t)Q1ẏ(t)-(1-δ)e-2αηẏT(t-η(t))Q1ẏ(t-η(t))-2αV7,V̇8≤2khT(x(t))D2-1h(x(t))-2e-2αk∫t-kthT(x(s))D2-1h(x(s))ds-2αV8.
Using the condition (2.3) and since the matrices Di-1>0,i=0,1,2 are diagonal, we have
(3.16)khT(x(t))D2-1h(x(t))≤kxT(t)HD2-1Hx(t)=kyT(t)PHD2-1HPy(t),fT(x(t))D0-1f(x(t))≤xT(t)FD0-1Fx(t)=yT(t)PFD0-1FPy(t),gT(x(t-h(t)))D1-1g(x(t-h(t)))≤xT(t-h(t))GD1-1Gx(t-h(t)),=yT(t-h(t))PGD1-1GPy(t-h(t)),
and using (2.3) and the Proposition (2.3) for the following estimations:
(3.17)2yT(t)W0f(x(t))≤yT(t)W0D0W0Ty(t)+fT(x(t))D0-1f(x(t))≤yT(t)W0D0W0Ty(t)+xT(t)FD0-1Fx(t)≤yT(t)W0D0W0Ty(t)+yT(t)PFD0-1FPy(t),2yT(t)W1g(x(t-h(t)))≤yT(t)W1D1W1Ty(t)+gT(x(t-h(t)))D1-1g(x(t-h(t)))≤yT(t)W1D1W1Ty(t)+xT(t-h(t))GD1-1Gx(t-h(t))≤yT(t)W1D1W1Ty(t)+yT(t-h(t))PGD1-1GPy(t-h(t)),2yT(t)W2∫t-k(t)th(x(s))ds≤ke2αkyT(t)W2D2W2Ty(t)+k-1e-2αk(∫t-k(t)th(x(s)ds))TD2-1(∫t-k(t)th(x(s))ds)≤ke2αkyT(t)W2D2W2Ty(t)+e-2αk∫t-kthT(x(s))D2-1h(x(s))ds.
Applying Proposition 2.4 and the Leibniz-Newton formula, we have
(3.18)-h1∫t-h1tẏT(s)Rẏ(s)ds≤-[∫t-h1tẏ(s)]TR[∫t-h1tẏ(s)]≤-[y(t)-y(t-h1)]TR[y(t)-y(t-h1)]=-yT(t)Ry(t)+2yT(t)Ry(t-h1)-yT(t-h1)Ry(t-h1),-h2∫t-h2tẏT(s)Rẏ(s)ds≤-[∫t-h2tẏ(s)]TR[∫t-h2tẏ(s)]≤-[y(t)-y(t-h2)]TR[y(t)-y(t-h2)]=-yT(t)Ry(t)+2yT(t)Ry(t-h2)-yT(t-h2)Ry(t-h2).
Note that
(3.19)-(h2-h1)∫t-h2t-h1ẏT(s)Uẏ(s)ds=-(h2-h1)∫t-h2t-h(t)ẏT(s)Uẏ(s)ds-(h2-h1)∫t-h(t)t-h1ẏT(s)Uẏ(s)ds=-(h2-h(t))∫t-h2t-h(t)ẏT(s)Uẏ(s)ds-(h(t)-h1)∫t-h2t-h(t)ẏT(s)Uẏ(s)ds-(h(t)-h1)∫t-h(t)t-h1ẏT(s)Uẏ(s)ds-(h2-h(t))∫t-h(t)t-h1ẏT(s)Uẏ(s)ds.
Using Proposition 2.4 gives
(3.20)-(h2-h(t))∫t-h2t-h(t)ẏT(s)Uẏ(s)ds≤-[∫t-h2t-h(t)ẏ(s)ds]TU[∫t-h2t-h(t)ẏ(s)ds]≤-[y(t-h(t))-y(t-h2)]TU[y(t-h(t))-y(t-h2)],(3.21)-(h(t)-h1)∫t-h(t)t-h1ẏT(s)Uẏ(s)ds≤-[∫t-h(t)t-h1ẏ(s)ds]TU[∫t-h(t)t-h1ẏ(s)ds]≤-[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))].
Let β=(h2-h(t))/(h2-h1)≤1. Then
(3.22)-(h2-h(t))∫t-h(t)t-h1ẏT(s)Uẏ(s)ds=-β∫t-h(t)t-h1(h2-h1)ẏT(s)Uẏ(s)ds≤-β∫t-h(t)t-h1(h(t)-h1)ẏT(s)Uẏ(s)ds≤-β[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))],(3.23)-(h(t)-h1)∫t-h2t-h(t)ẏT(s)Uẏ(s)ds=-(1-β)∫t-h2t-h(t)(h2-h1)ẏT(s)Uẏ(s)ds≤-(1-β)∫t-h2t-h(t)(h2-h(t))ẏT(s)Uẏ(s)ds≤-(1-β)[y(t-h(t))-y(t-h2)]T×U[y(t-h(t))-y(t-h2)].
Therefore from (3.20)–(3.23), we obtain
(3.24)-(h2-h1)∫t-h2t-h1ẏT(s)Uẏ(s)ds≤-[y(t-h(t))-y(t-h2)]TU[y(t-h(t))-y(t-h2)]-[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))]-β[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))]-(1-β)[y(t-h(t))-y(t-h2)]T×U[y(t-h(t))-y(t-h2)].
By using the following identity relation:
(3.25)-Pẏ(t)-APy(t)+W0f(x(t))+W1g(x(t-h(t)))+W0∫t-k(t)th(x(s))ds+B0Pẏ(t-η(t))-0.5BBTy(t)=0,
we have
(3.26)-2ẏT(t)Pẏ(t)-2ẏT(t)APy(t)+2ẏT(t)W0f(x(t))+2ẏT(t)W1g(x(t-h(t)))+2ẏT(t)W0∫t-k(t)th(x(s))ds+2ẏT(t)B0Pẏ(t-η(t))-ẏT(t)BBTy(t)=0.
By using Propositions 2.3 and 2.4, we have
(3.27)2ẏT(t)W0f(x(t))≤ẏT(t)W0D0W0Tẏ(t)+fT(x(t))D0-1f(x(t))≤ẏT(t)W0D0W0Tẏ(t)+yT(t)PFD0-1FPy(t),(3.28)2ẏT(t)W1g(x(t-h(t)))≤ẏT(t)W1D1W1Tẏ(t)+gT(x(t-h(t)))D1-1g(x(t-h(t)))≤ẏT(t)W1D1W1Tẏ(t)+yT(t-h(t))PGD1-1GPy(t-h(t)),(3.29)2ẏT(t)W2∫t-k(t)th(x(s))ds≤ke2αkẏT(t)W2D2W2Tẏ(t)+e-2αk∫t-kthT(x(s))D2-1h(x(s))ds.
From (3.15)–(3.29), we obtain
(3.30)V̇(t,xt)+2αV(t,xt)≤yT(t)[-AP-PAT+2αP-BBT+2Q+2kPHD2-1HP+W0D0W0T+W1D1W1T+ke2αkW2D2W2T-e-2αh1R-e-2αh2R+PFD1-1FP]y(t)+2yT(t)B0Pẏ(t-η(t))+yT(t-h1)[-e-2αh1Q-e-2αh1R-e-2αh2U]y(t-h1),+(t-h2)[-e-2αh2Q-e-2αh2R-e-2αh2U]y(t-h2),+ẏT(t)[h12R+h22R+(h2-h1)2U+Q1-2P+W0D0W0T+W1D1W1T+ke2αkW2D2W2T]ẏ(t)-(1-δ)e-2αηẏT(t-η(t))Q1ẏ(t-η(t))+yT(t-h(t))[2PGD1-1GP-2e-2αh2U]y(t-h(t))+2e-2αh1yT(t)Ry(t-h1)+2e-2αh2yT(t)Ry(t-h2)+2e-2αh2yT(t-h(t))Uy(t-h2)+2e-2αh2yT(t-h(t))Uy(t-h1)-2ẏT(t)APy(t)+2ẏT(t)B0Pẏ(t-η(t))-ẏT(t)BBTy(t)-e-2αh2β[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))]-e-2αh2(1-β)[y(t-h(t))-y(t-h2)]TU[y(t-h(t))-y(t-h2)]=ξT(t)Ωξ(t)-e-2αh2β[y(t-h1)-y(t-h(t))]TU×[y(t-h1)-y(t-h(t))]-e-2αh2(1-β)[y(t-h(t))-y(t-h2)]T×U[y(t-h(t))-y(t-h2)]=ξT(t)[(1-β)M1+βM2]ξ(t)+yT(t)M3y(t)+yT(t-h(t))M4y(t-h(t)),
where
(3.31)M3=-0.1BBT-0.1(e-2αh1+e-2αh2)R+2kPHD2-1HP+2PFD0-1FP,M4=-0.1e-2αh2U+2PGD1-1GP,ζ(t)=[y(t),ẏ(t-η(t)),y(t-h1),y(t-h2),ẏ(t),y(t-h(t))].
Since 0≤β≤1, (1-β)ℳ1+βℳ2 is a convex combination of ℳ1 and ℳ2. Therefore, (1-β)ℳ1+βℳ2<0 is equivalent to ℳ1<0 and ℳ2<0. Applying Schur complement lemma Proposition 2.5, the inequalities M3<0 and M4<0 are equivalent to ℳ3<0 and ℳ4<0, respectively. Thus, it follows from (3.3)–(3.6) and (3.30), we obtain
(3.32)V̇(t,xt)≤-2αV(t,xt),∀t≥0.
Integrating both sides of (3.32) from 0 to t, we obtain
(3.33)V(t,xt)≤V(ϕ)e-2αt,∀t≥0.
Furthermore, taking condition (3.13) into account, we have
(3.34)λ1‖x(t,ϕ)‖2≤V(xt)≤V(ϕ)e-2αt≤λ2e-2αt‖ϕ‖2,
then
(3.35)‖x(t,ϕ)‖≤λ2λ1e-αt‖ϕ‖,t≥0.
Therefore, nominal system (3.1) is α-exponentially stabilizable. The proof is completed.


### 3.2. Exponential Stabilization for Interval Time-Varying Delay Systems

Based on Theorem3.2, we derive robustly α-exponential stabilizability conditions of uncertain linear control systems with interval time-varying delay (2.1) in terms of LMIs.Theorem 3.3.
Givenα>0. The system (2.1) is α-exponentially stabilizable if there exist symmetric positive definite matrices P,Q,R,U,Q1, three diagonal matrices Di,i=0,1,2 and ϵi>0,i=1,2,…,6 such that the following LMI holds:
(3.36)W1=W-[000-I0I]T×e-2αh2U[000-I0I]<0,(3.37)W2=W-[00I00-I]T×e-2αh2U[00I00-I]<0,(3.38)W3=[Δ4kPH2PFPHaTPHaTPFH0TPFH0T*-4kD200000**-2D00000***-ϵ1I000****-ϵ4I00*****-ϵ2I0******-ϵ5I]<0,(3.39)W4=[-0.1e-2αh2U2PGPGH1TPGH1T*-2D100**-ϵ3I0***-ϵ6I]<0,W=[W11W12W13W14W150*W2200W250**W3300W36***W440W46****W550*****W66],
where
(3.40)Δ=-0.1BBT-0.1(e-2αh1+e-2αh2)R,W11=[-A+αI]P+P[-A+αI]T-0.9BBT+2Q+W0D0W0T+W1D1W1T+ke2αkW2D2W2T-0.9e-2αh1R-0.9e-2αh2R+ϵ1EaTEa+ϵ3E1TE1+ϵ2E0TE0+ke2αkE2TH2TD2H2E2,W12=B0P,W13=e-2αh1R,W14=e-2αh2R,W15=-PAT-0.5BBT,W22=-(1-δ)e-2αηQ1,W25=PB0T,W33=-e-2αh1Q-e-2αh1R-e-2αh2U,W36=e-2αh2U,W44=-e-2αh2Q-e-2αh2R-e-2αh2U,W46=e-2αh2U,W55=h12R+h22R+(h2-h1)2U+Q1-2P+W0D0W0T+W1D1W1T+ke2αkW2D2W2T+ϵ4EaTEa+ϵ5E0TE0+ϵ6E1TE1+ke2αkE2TH2TD2H2E2,W66=-1.9e-2αh2U.Proof.
Choose Lyapunov-Krasovskii functional as in (3.11) but change V8 to V8=4∫-k0∫t+ste2α(τ-t)hT(x(τ))D2-1h(x(τ))dτds, we may proof the Theorem by using a similar argument as in the proof of Theorem 3.2. By replacing A,W0,W1, and W2 with A+EaFa(t)Ha,W0+E0F0(t)H0,W1+E1F1(t)H1, and W2+E2F2(t)H2, respectively. We have the following:
(3.41)V̇(t,xt)+2αV(t,xt)≤yT(t)[(-A+EaFa(t)Ha)P+P(-A+EaFa(t)Ha)T-BBT+2αP+2Q-e-2αh1R-e-2αh2RyT]y(t)+2yT(t)(W0+E0F0(t)H0)f(x(t))+2yT(t)(W1+E1F1(t)H1)g(x(t-h(t)))+2yT(t)(W2+E2F2(t)H2)×∫t-k(t)th(x(s))ds+2yT(t)B0Pẏ(t-η(t))-e-2αh1yT(t-h1)Qy(t-h1)-e-2αh2yT(t-h2)Qy(t-h2)+h12ẏT(t)Rẏ(t)+h22ẏT(t)Rẏ(t)+(h2-h1)2ẏT(t)Uẏ(t)+e-2αh12yT(t)Ry(t-h1)-e-2αh1yT(t-h1)Ry(t-h1)+e-2αh22yT(t)Ry(t-h2)-e-2αh2yT(t-h2)Ry(t-h2)-e-2αh2[y(t-h(t))-y(t-h2)]TU[y(t-h(t)-y(t-h2))]T-e-2αh2[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))]T-βe-2αh2[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))]T-e-2αh2(1-β)[y(t-h(t))-y(t-h2)]TU[y(t-h(t))-y(t-h2)]T+ẏT(t)Q1ẏ(t)-(1-δ)e-2αηẏT(t-η(t))Q1ẏ(t-η(t))+4kyT(t)PHD2-1HPy(t)-4e-2αk∫t-kthT(x(s))D2-1h(x(s))ds-2ẏT(t)Pẏ(t)-2ẏT(t)(A+EaFa(t)Ha)Py(t)+2ẏT(t)(W0+E0F0(t)H0)f(x(t))+2ẏT(t)(W1+E1F1(t)H1)g(x(t-h(t)))+2ẏT(t)(W2+E2F2(t)H2)∫t-k(t)th(x(s))ds+2ẏT(t)B0Pẏ(t-η(t))-ẏT(t)BBTy(t)=ξT(t)[(1-β)M1+βM2]ξ(t)+yT(t)M3y(t)+yT(t-h(t))M4y(t-h(t)).
Applying Proposition 2.3 and Lemma 2.6, the following estimations hold:
(3.42)yT(t)[(-A+EaFa(t)H)a)P+P(-AT+HaTFaT(t)EaT)]y(t)≤yT(t)[-PAT-AP]y(t)+ϵ1yT(t)EaTEay(t)+ϵ1-1yT(t)PHaTHaPy(t),2yT(t)[W0+E0F0(t)H0]f(x(t))=2yT(t)W0f(x(t))+2yT(t)E0F0(t)H0f(x(t))≤yT(t)W0D0W0Ty(t)+yT(t)PFD0-1FPy(t)+ϵ2yT(t)E0TE0y(t)+ϵ2-1yT(t)PFH0TH0FPy(t),2yT(t)(W1+E1F1(t)H1)g(x(t-h(t)))=2yT(t)W1g(x(t-h(t)))+2yT(t)E1F1(t)H1g(x(t-h(t)))≤yT(t)W1D1W1Ty(t)+yT(t-h(t))PGD1-1GPy(t-h(t))+ϵ3yT(t)E1TE1y(t)+ϵ3-1yT(t-h(t))PGH1TH1GPyT(t-h(t)),2yT(t)(W2+E2F2(t)H2)∫t-k(t)th(x(s))ds=2yT(t)W2∫t-k(t)th(x(s))ds+2yT(t)E2F2(t)H2∫t-k(t)th(x(s))ds≤ke2αkyT(t)W2D2W2Ty(t)+e-2αk∫t-kthT(x(s))D2-1h(x(s))ds+ke2αkyT(t)E2TH2D2H2TE2y(t)+k-1e-2αk[∫t-k(t)th(x(s))ds]TH2TH2-TD2-1H2-1H2×[∫t-k(t)th(x(s))ds]≤ke2αkyT(t)W2D2W2Ty(t)+e-2αk∫t-kthT(x(s))D2-1h(x(s))ds+ke2αkyT(t)E2TH2D2H2TE2y(t)+e-2αk∫t-kthT(x(s))D2-1h(x(s))ds-2ẏT(t)(A+EaFa(t)Ha)Py(t)≤-2ẏT(t)APy(t)-2ẏT(t)EaFa(t)HaPy(t)≤-2ẏT(t)APy(t)+ϵ4ẏT(t)EaTEaẏ(t)+ϵ4-1yT(t)PHaTHaPy(t),2ẏT(t)(W0+E0F0(t)H0)f(x(t))=2ẏT(t)[W0+E0F0(t)H0]f(x(t))≤ẏT(t)W0D0W0Tẏ(t)+yT(t)PFD0-1FPy(t)+ϵ5ẏT(t)E0TE0ẏ(t)+ϵ5-1yT(t)PFH0TH0FPy(t),2ẏT(t)(W1+E1F1(t)H1)g(x(t-h(t)))≤ẏT(t)W1D1W1Tẏ(t)+yT(t-h(t))PGD1-1G×Py(t-h(t))+ϵ6ẏT(t)E1TE1ẏ(t)+ϵ6-1yT(t-h(t))PGH1TH1GPyT(t-h(t)),2ẏT(t)(W2+E2F2(t)H2)≤ke2αkẏT(t)W2D2W2Tẏ(t)+2e-2αk∫t-kthT(x(s))D2-1h(x(s))ds+ke2αkẏT(t)E2TH2TD2H2E2ẏ(t).Remark 3.4.
In [10, 13, 14], exponential stability of neutral-type neural networks with time-varying delays were investigated. However, the distributed delays have not been considered. Stability conditions in [13, 26–28] are not applicable to our work, since we consider more activation functions than them. Therefore, our stability conditions are less conservative than some other existing results.Remark 3.5.
In this paper, the restriction that the state delay is differentiable is not required which allows the state delay to be fast time-varying. Meanwhile, this restriction is required in some existing result, see [13, 14, 26–28].


## 3.1. Exponential Stabilization for Nominal Interval Time-Varying Delay Systems

The nominal system is given by(3.1)ẋ(t)=-Ax(t)+W0f(x(t))+W1g(x(t-h(t)))+W2∫t-k(t)th(x(s))ds+B0ẋ(t-η(t))+Bu(t)x(t)=ϕ(t),t∈[-d,0].
First, we present a delay-dependent exponential stabilizability analysis conditions for the given nominal interval time-varying delay system (3.1) with ΔA(t)=ΔW0(t)=ΔW1(t)=ΔW2(t)=0. Let us set (3.2)λ1=λmin⁡(P-1),λ2=λmax⁡(P-1)+2h2λmax⁡(P-1QP-1)+2h22λmax⁡(P-1RP-1)+ηλmax⁡(P-1Q1P-1)+2λmax⁡(HD2-1H)+(h2-h1)2λmax⁡(P-1UP-1).Assumption 3.1.
All the eigenvalues of matrixB0 are inside the unit circle.Theorem 3.2.
Givenα>0. The system (3.1) is α-exponentially stabilizable if there exist symmetric positive definite matrices P,Q,R,U,Q1, three diagonal matrices Di,i=0,1,2 such that the following LMI holds:
(3.3)M1=M-[000-I0I]T×e-2αh2U[000-I0I]<0,(3.4)M2=M-[00I00-I]T×e-2αh2U[00I00-I]<0,(3.5)M3=[-0.1BBT-0.1(e-2αh1+e-2αh2)R2kPH2PF*-2kD20**-2D0]<0,(3.6)M4=[-0.1e-2αh2U2PG*-2D1]<0,(3.7)M=[M11M12M13M14M150*M2200M250**M3300M36***M440M46****M550*****M66],
where
(3.8)M11=[-A+αI]P+P[-A+αI]T-0.9BBT+2Q+W0D0W0T+W1D1W1T+ke2αkW2D2W2T-0.9e-2αh1R-0.9e-2αh2R]M12=B0P,M13=e-2αh1R,M14=e-2αh2R,M15=-PAT-0.5BBT,M22=-(1-δ)e-2αηQ1,M25=PB0T,M33=-e-2αh1Q-e-2αh1R-e-2αh2U,M36=e-2αh2U,M44=-e-2αh2Q-e-2αh2R-e-2αh2U,M46=e-2αh2U,M55=h12R+h22R+(h2-h1)2U+Q1-2P+W0D0W0T+W1D1W1T+ke2αkW2D2W2T]M66=-1.9e-2αh2U.
Moreover, the memoryless feedback control is
(3.9)u(t)=-0.5BTP-1x(t),t≥0,
and the solution x(t,ϕ) of the system satisfies
(3.10)‖x(t,ϕ)‖≤λ2λ1e-αt‖ϕ‖,∀t≥0.Proof.
LetY=P-1,y(t)=Yx(t).  Using the feedback control (3.9) we consider the following Lyapunov-Krasovskii functional:
(3.11)V(t,xt)=∑i=18Vi,
where
(3.12)V1=xT(t)Yx(t),V2=∫t-h1te2α(s-t)xT(s)YQYx(s)ds,V3=∫t-h2te2α(s-t)xT(s)YQYx(s)ds,V4=h1∫-h10∫t+ste2α(τ-t)ẋT(τ)YRYẋ(τ)dτds,V5=h2∫-h20∫t+ste2α(τ-t)ẋT(τ)YRYẋ(τ)dτds,V6=(h2-h1)∫-h2-h1∫t+ste2α(τ-t)ẋT(τ)YUYẋ(τ)dτds,V7=∫t-η(t)te2α(s-t)ẋT(s)YQ1Yẋ(s)ds,V8=2∫-k0∫t+ste2α(τ-t)hT(x(τ))D2-1h(x(τ))dτds.
It easy to check that
(3.13)λ1∥x(t)∥2≤V(t,xt)≤λ2∥xt∥2,∀t≥0.
Taking the derivative of V(xt) along the solution of system (3.1) we have
(3.14)V̇1=2xT(t)Yẋ(t),=2yT(t)[-Ax(t)+W0f(x(t))+W1g(x(t-h(t)))+W2∫t-k(t)th(x(s))ds+B0ẋ(t-η(t))-0.5BBTP-1x(t)∫t-k(t)t]=2yT(t)[-APy(t)+W0f(x(t))+W1g(x(t-h(t)))+W2∫t-k(t)th(x(s))ds+B0Pẏ(t-η(t))-0.5BBTy(t)∫t-k(t)t](3.15)=yT(t)[-AP-PAT]y(t)+2yT(t)W0f(x(t))+2yT(t)W1g(x(t-h(t)))+2yT(t)W2∫t-k(t)th(x(s))ds+2yT(t)B0Pẏ(t-η(t))-yT(t)BBTy(t)+2αyT(t)Py(t)-2αV1,V̇2=yT(t)Qy(t)-e-2αh1yT(t-h1)Qy(t-h1)-2αV2,V̇3=yT(t)Qy(t)-e-2αh2yT(t-h2)Qy(t-h2)-2αV3,V̇4≤h12ẏT(t)Rẏ(t)-h1e-2αh1∫t-h1tẏT(s)Rẏ(s)ds-2αV4,V̇5≤h22ẏT(t)Rẏ(t)-h2e-2αh2∫t-h2tẏT(s)Rẏ(s)ds-2αV5,V̇6≤(h2-h1)2ẏT(t)Uẏ(t)-(h2-h1)e-2αh2∫t-h2t-h1ẏT(s)Uẏ(s)ds-2αV6,V̇7≤ẏT(t)Q1ẏ(t)-(1-δ)e-2αηẏT(t-η(t))Q1ẏ(t-η(t))-2αV7,V̇8≤2khT(x(t))D2-1h(x(t))-2e-2αk∫t-kthT(x(s))D2-1h(x(s))ds-2αV8.
Using the condition (2.3) and since the matrices Di-1>0,i=0,1,2 are diagonal, we have
(3.16)khT(x(t))D2-1h(x(t))≤kxT(t)HD2-1Hx(t)=kyT(t)PHD2-1HPy(t),fT(x(t))D0-1f(x(t))≤xT(t)FD0-1Fx(t)=yT(t)PFD0-1FPy(t),gT(x(t-h(t)))D1-1g(x(t-h(t)))≤xT(t-h(t))GD1-1Gx(t-h(t)),=yT(t-h(t))PGD1-1GPy(t-h(t)),
and using (2.3) and the Proposition (2.3) for the following estimations:
(3.17)2yT(t)W0f(x(t))≤yT(t)W0D0W0Ty(t)+fT(x(t))D0-1f(x(t))≤yT(t)W0D0W0Ty(t)+xT(t)FD0-1Fx(t)≤yT(t)W0D0W0Ty(t)+yT(t)PFD0-1FPy(t),2yT(t)W1g(x(t-h(t)))≤yT(t)W1D1W1Ty(t)+gT(x(t-h(t)))D1-1g(x(t-h(t)))≤yT(t)W1D1W1Ty(t)+xT(t-h(t))GD1-1Gx(t-h(t))≤yT(t)W1D1W1Ty(t)+yT(t-h(t))PGD1-1GPy(t-h(t)),2yT(t)W2∫t-k(t)th(x(s))ds≤ke2αkyT(t)W2D2W2Ty(t)+k-1e-2αk(∫t-k(t)th(x(s)ds))TD2-1(∫t-k(t)th(x(s))ds)≤ke2αkyT(t)W2D2W2Ty(t)+e-2αk∫t-kthT(x(s))D2-1h(x(s))ds.
Applying Proposition 2.4 and the Leibniz-Newton formula, we have
(3.18)-h1∫t-h1tẏT(s)Rẏ(s)ds≤-[∫t-h1tẏ(s)]TR[∫t-h1tẏ(s)]≤-[y(t)-y(t-h1)]TR[y(t)-y(t-h1)]=-yT(t)Ry(t)+2yT(t)Ry(t-h1)-yT(t-h1)Ry(t-h1),-h2∫t-h2tẏT(s)Rẏ(s)ds≤-[∫t-h2tẏ(s)]TR[∫t-h2tẏ(s)]≤-[y(t)-y(t-h2)]TR[y(t)-y(t-h2)]=-yT(t)Ry(t)+2yT(t)Ry(t-h2)-yT(t-h2)Ry(t-h2).
Note that
(3.19)-(h2-h1)∫t-h2t-h1ẏT(s)Uẏ(s)ds=-(h2-h1)∫t-h2t-h(t)ẏT(s)Uẏ(s)ds-(h2-h1)∫t-h(t)t-h1ẏT(s)Uẏ(s)ds=-(h2-h(t))∫t-h2t-h(t)ẏT(s)Uẏ(s)ds-(h(t)-h1)∫t-h2t-h(t)ẏT(s)Uẏ(s)ds-(h(t)-h1)∫t-h(t)t-h1ẏT(s)Uẏ(s)ds-(h2-h(t))∫t-h(t)t-h1ẏT(s)Uẏ(s)ds.
Using Proposition 2.4 gives
(3.20)-(h2-h(t))∫t-h2t-h(t)ẏT(s)Uẏ(s)ds≤-[∫t-h2t-h(t)ẏ(s)ds]TU[∫t-h2t-h(t)ẏ(s)ds]≤-[y(t-h(t))-y(t-h2)]TU[y(t-h(t))-y(t-h2)],(3.21)-(h(t)-h1)∫t-h(t)t-h1ẏT(s)Uẏ(s)ds≤-[∫t-h(t)t-h1ẏ(s)ds]TU[∫t-h(t)t-h1ẏ(s)ds]≤-[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))].
Let β=(h2-h(t))/(h2-h1)≤1. Then
(3.22)-(h2-h(t))∫t-h(t)t-h1ẏT(s)Uẏ(s)ds=-β∫t-h(t)t-h1(h2-h1)ẏT(s)Uẏ(s)ds≤-β∫t-h(t)t-h1(h(t)-h1)ẏT(s)Uẏ(s)ds≤-β[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))],(3.23)-(h(t)-h1)∫t-h2t-h(t)ẏT(s)Uẏ(s)ds=-(1-β)∫t-h2t-h(t)(h2-h1)ẏT(s)Uẏ(s)ds≤-(1-β)∫t-h2t-h(t)(h2-h(t))ẏT(s)Uẏ(s)ds≤-(1-β)[y(t-h(t))-y(t-h2)]T×U[y(t-h(t))-y(t-h2)].
Therefore from (3.20)–(3.23), we obtain
(3.24)-(h2-h1)∫t-h2t-h1ẏT(s)Uẏ(s)ds≤-[y(t-h(t))-y(t-h2)]TU[y(t-h(t))-y(t-h2)]-[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))]-β[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))]-(1-β)[y(t-h(t))-y(t-h2)]T×U[y(t-h(t))-y(t-h2)].
By using the following identity relation:
(3.25)-Pẏ(t)-APy(t)+W0f(x(t))+W1g(x(t-h(t)))+W0∫t-k(t)th(x(s))ds+B0Pẏ(t-η(t))-0.5BBTy(t)=0,
we have
(3.26)-2ẏT(t)Pẏ(t)-2ẏT(t)APy(t)+2ẏT(t)W0f(x(t))+2ẏT(t)W1g(x(t-h(t)))+2ẏT(t)W0∫t-k(t)th(x(s))ds+2ẏT(t)B0Pẏ(t-η(t))-ẏT(t)BBTy(t)=0.
By using Propositions 2.3 and 2.4, we have
(3.27)2ẏT(t)W0f(x(t))≤ẏT(t)W0D0W0Tẏ(t)+fT(x(t))D0-1f(x(t))≤ẏT(t)W0D0W0Tẏ(t)+yT(t)PFD0-1FPy(t),(3.28)2ẏT(t)W1g(x(t-h(t)))≤ẏT(t)W1D1W1Tẏ(t)+gT(x(t-h(t)))D1-1g(x(t-h(t)))≤ẏT(t)W1D1W1Tẏ(t)+yT(t-h(t))PGD1-1GPy(t-h(t)),(3.29)2ẏT(t)W2∫t-k(t)th(x(s))ds≤ke2αkẏT(t)W2D2W2Tẏ(t)+e-2αk∫t-kthT(x(s))D2-1h(x(s))ds.
From (3.15)–(3.29), we obtain
(3.30)V̇(t,xt)+2αV(t,xt)≤yT(t)[-AP-PAT+2αP-BBT+2Q+2kPHD2-1HP+W0D0W0T+W1D1W1T+ke2αkW2D2W2T-e-2αh1R-e-2αh2R+PFD1-1FP]y(t)+2yT(t)B0Pẏ(t-η(t))+yT(t-h1)[-e-2αh1Q-e-2αh1R-e-2αh2U]y(t-h1),+(t-h2)[-e-2αh2Q-e-2αh2R-e-2αh2U]y(t-h2),+ẏT(t)[h12R+h22R+(h2-h1)2U+Q1-2P+W0D0W0T+W1D1W1T+ke2αkW2D2W2T]ẏ(t)-(1-δ)e-2αηẏT(t-η(t))Q1ẏ(t-η(t))+yT(t-h(t))[2PGD1-1GP-2e-2αh2U]y(t-h(t))+2e-2αh1yT(t)Ry(t-h1)+2e-2αh2yT(t)Ry(t-h2)+2e-2αh2yT(t-h(t))Uy(t-h2)+2e-2αh2yT(t-h(t))Uy(t-h1)-2ẏT(t)APy(t)+2ẏT(t)B0Pẏ(t-η(t))-ẏT(t)BBTy(t)-e-2αh2β[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))]-e-2αh2(1-β)[y(t-h(t))-y(t-h2)]TU[y(t-h(t))-y(t-h2)]=ξT(t)Ωξ(t)-e-2αh2β[y(t-h1)-y(t-h(t))]TU×[y(t-h1)-y(t-h(t))]-e-2αh2(1-β)[y(t-h(t))-y(t-h2)]T×U[y(t-h(t))-y(t-h2)]=ξT(t)[(1-β)M1+βM2]ξ(t)+yT(t)M3y(t)+yT(t-h(t))M4y(t-h(t)),
where
(3.31)M3=-0.1BBT-0.1(e-2αh1+e-2αh2)R+2kPHD2-1HP+2PFD0-1FP,M4=-0.1e-2αh2U+2PGD1-1GP,ζ(t)=[y(t),ẏ(t-η(t)),y(t-h1),y(t-h2),ẏ(t),y(t-h(t))].
Since 0≤β≤1, (1-β)ℳ1+βℳ2 is a convex combination of ℳ1 and ℳ2. Therefore, (1-β)ℳ1+βℳ2<0 is equivalent to ℳ1<0 and ℳ2<0. Applying Schur complement lemma Proposition 2.5, the inequalities M3<0 and M4<0 are equivalent to ℳ3<0 and ℳ4<0, respectively. Thus, it follows from (3.3)–(3.6) and (3.30), we obtain
(3.32)V̇(t,xt)≤-2αV(t,xt),∀t≥0.
Integrating both sides of (3.32) from 0 to t, we obtain
(3.33)V(t,xt)≤V(ϕ)e-2αt,∀t≥0.
Furthermore, taking condition (3.13) into account, we have
(3.34)λ1‖x(t,ϕ)‖2≤V(xt)≤V(ϕ)e-2αt≤λ2e-2αt‖ϕ‖2,
then
(3.35)‖x(t,ϕ)‖≤λ2λ1e-αt‖ϕ‖,t≥0.
Therefore, nominal system (3.1) is α-exponentially stabilizable. The proof is completed.


## 3.2. Exponential Stabilization for Interval Time-Varying Delay Systems

Based on Theorem3.2, we derive robustly α-exponential stabilizability conditions of uncertain linear control systems with interval time-varying delay (2.1) in terms of LMIs.Theorem 3.3.
Givenα>0. The system (2.1) is α-exponentially stabilizable if there exist symmetric positive definite matrices P,Q,R,U,Q1, three diagonal matrices Di,i=0,1,2 and ϵi>0,i=1,2,…,6 such that the following LMI holds:
(3.36)W1=W-[000-I0I]T×e-2αh2U[000-I0I]<0,(3.37)W2=W-[00I00-I]T×e-2αh2U[00I00-I]<0,(3.38)W3=[Δ4kPH2PFPHaTPHaTPFH0TPFH0T*-4kD200000**-2D00000***-ϵ1I000****-ϵ4I00*****-ϵ2I0******-ϵ5I]<0,(3.39)W4=[-0.1e-2αh2U2PGPGH1TPGH1T*-2D100**-ϵ3I0***-ϵ6I]<0,W=[W11W12W13W14W150*W2200W250**W3300W36***W440W46****W550*****W66],
where
(3.40)Δ=-0.1BBT-0.1(e-2αh1+e-2αh2)R,W11=[-A+αI]P+P[-A+αI]T-0.9BBT+2Q+W0D0W0T+W1D1W1T+ke2αkW2D2W2T-0.9e-2αh1R-0.9e-2αh2R+ϵ1EaTEa+ϵ3E1TE1+ϵ2E0TE0+ke2αkE2TH2TD2H2E2,W12=B0P,W13=e-2αh1R,W14=e-2αh2R,W15=-PAT-0.5BBT,W22=-(1-δ)e-2αηQ1,W25=PB0T,W33=-e-2αh1Q-e-2αh1R-e-2αh2U,W36=e-2αh2U,W44=-e-2αh2Q-e-2αh2R-e-2αh2U,W46=e-2αh2U,W55=h12R+h22R+(h2-h1)2U+Q1-2P+W0D0W0T+W1D1W1T+ke2αkW2D2W2T+ϵ4EaTEa+ϵ5E0TE0+ϵ6E1TE1+ke2αkE2TH2TD2H2E2,W66=-1.9e-2αh2U.Proof.
Choose Lyapunov-Krasovskii functional as in (3.11) but change V8 to V8=4∫-k0∫t+ste2α(τ-t)hT(x(τ))D2-1h(x(τ))dτds, we may proof the Theorem by using a similar argument as in the proof of Theorem 3.2. By replacing A,W0,W1, and W2 with A+EaFa(t)Ha,W0+E0F0(t)H0,W1+E1F1(t)H1, and W2+E2F2(t)H2, respectively. We have the following:
(3.41)V̇(t,xt)+2αV(t,xt)≤yT(t)[(-A+EaFa(t)Ha)P+P(-A+EaFa(t)Ha)T-BBT+2αP+2Q-e-2αh1R-e-2αh2RyT]y(t)+2yT(t)(W0+E0F0(t)H0)f(x(t))+2yT(t)(W1+E1F1(t)H1)g(x(t-h(t)))+2yT(t)(W2+E2F2(t)H2)×∫t-k(t)th(x(s))ds+2yT(t)B0Pẏ(t-η(t))-e-2αh1yT(t-h1)Qy(t-h1)-e-2αh2yT(t-h2)Qy(t-h2)+h12ẏT(t)Rẏ(t)+h22ẏT(t)Rẏ(t)+(h2-h1)2ẏT(t)Uẏ(t)+e-2αh12yT(t)Ry(t-h1)-e-2αh1yT(t-h1)Ry(t-h1)+e-2αh22yT(t)Ry(t-h2)-e-2αh2yT(t-h2)Ry(t-h2)-e-2αh2[y(t-h(t))-y(t-h2)]TU[y(t-h(t)-y(t-h2))]T-e-2αh2[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))]T-βe-2αh2[y(t-h1)-y(t-h(t))]TU[y(t-h1)-y(t-h(t))]T-e-2αh2(1-β)[y(t-h(t))-y(t-h2)]TU[y(t-h(t))-y(t-h2)]T+ẏT(t)Q1ẏ(t)-(1-δ)e-2αηẏT(t-η(t))Q1ẏ(t-η(t))+4kyT(t)PHD2-1HPy(t)-4e-2αk∫t-kthT(x(s))D2-1h(x(s))ds-2ẏT(t)Pẏ(t)-2ẏT(t)(A+EaFa(t)Ha)Py(t)+2ẏT(t)(W0+E0F0(t)H0)f(x(t))+2ẏT(t)(W1+E1F1(t)H1)g(x(t-h(t)))+2ẏT(t)(W2+E2F2(t)H2)∫t-k(t)th(x(s))ds+2ẏT(t)B0Pẏ(t-η(t))-ẏT(t)BBTy(t)=ξT(t)[(1-β)M1+βM2]ξ(t)+yT(t)M3y(t)+yT(t-h(t))M4y(t-h(t)).
Applying Proposition 2.3 and Lemma 2.6, the following estimations hold:
(3.42)yT(t)[(-A+EaFa(t)H)a)P+P(-AT+HaTFaT(t)EaT)]y(t)≤yT(t)[-PAT-AP]y(t)+ϵ1yT(t)EaTEay(t)+ϵ1-1yT(t)PHaTHaPy(t),2yT(t)[W0+E0F0(t)H0]f(x(t))=2yT(t)W0f(x(t))+2yT(t)E0F0(t)H0f(x(t))≤yT(t)W0D0W0Ty(t)+yT(t)PFD0-1FPy(t)+ϵ2yT(t)E0TE0y(t)+ϵ2-1yT(t)PFH0TH0FPy(t),2yT(t)(W1+E1F1(t)H1)g(x(t-h(t)))=2yT(t)W1g(x(t-h(t)))+2yT(t)E1F1(t)H1g(x(t-h(t)))≤yT(t)W1D1W1Ty(t)+yT(t-h(t))PGD1-1GPy(t-h(t))+ϵ3yT(t)E1TE1y(t)+ϵ3-1yT(t-h(t))PGH1TH1GPyT(t-h(t)),2yT(t)(W2+E2F2(t)H2)∫t-k(t)th(x(s))ds=2yT(t)W2∫t-k(t)th(x(s))ds+2yT(t)E2F2(t)H2∫t-k(t)th(x(s))ds≤ke2αkyT(t)W2D2W2Ty(t)+e-2αk∫t-kthT(x(s))D2-1h(x(s))ds+ke2αkyT(t)E2TH2D2H2TE2y(t)+k-1e-2αk[∫t-k(t)th(x(s))ds]TH2TH2-TD2-1H2-1H2×[∫t-k(t)th(x(s))ds]≤ke2αkyT(t)W2D2W2Ty(t)+e-2αk∫t-kthT(x(s))D2-1h(x(s))ds+ke2αkyT(t)E2TH2D2H2TE2y(t)+e-2αk∫t-kthT(x(s))D2-1h(x(s))ds-2ẏT(t)(A+EaFa(t)Ha)Py(t)≤-2ẏT(t)APy(t)-2ẏT(t)EaFa(t)HaPy(t)≤-2ẏT(t)APy(t)+ϵ4ẏT(t)EaTEaẏ(t)+ϵ4-1yT(t)PHaTHaPy(t),2ẏT(t)(W0+E0F0(t)H0)f(x(t))=2ẏT(t)[W0+E0F0(t)H0]f(x(t))≤ẏT(t)W0D0W0Tẏ(t)+yT(t)PFD0-1FPy(t)+ϵ5ẏT(t)E0TE0ẏ(t)+ϵ5-1yT(t)PFH0TH0FPy(t),2ẏT(t)(W1+E1F1(t)H1)g(x(t-h(t)))≤ẏT(t)W1D1W1Tẏ(t)+yT(t-h(t))PGD1-1G×Py(t-h(t))+ϵ6ẏT(t)E1TE1ẏ(t)+ϵ6-1yT(t-h(t))PGH1TH1GPyT(t-h(t)),2ẏT(t)(W2+E2F2(t)H2)≤ke2αkẏT(t)W2D2W2Tẏ(t)+2e-2αk∫t-kthT(x(s))D2-1h(x(s))ds+ke2αkẏT(t)E2TH2TD2H2E2ẏ(t).Remark 3.4.
In [10, 13, 14], exponential stability of neutral-type neural networks with time-varying delays were investigated. However, the distributed delays have not been considered. Stability conditions in [13, 26–28] are not applicable to our work, since we consider more activation functions than them. Therefore, our stability conditions are less conservative than some other existing results.Remark 3.5.
In this paper, the restriction that the state delay is differentiable is not required which allows the state delay to be fast time-varying. Meanwhile, this restriction is required in some existing result, see [13, 14, 26–28].


## 4. Numerical Examples

In this section, we now provide an example to show the effectiveness of the result in Theorem3.2.Example 4.1.
Consider the neural networks with interval time-varying delay and control input with the following parameters:(4.1)ẋ(t)=-Ax(t)+W0f(x(t))+W1g(x(t-h(t)))+Bu(t),
where
(4.2)A=[-0.2012],W0=[0.40.10.1-0.2],W1=[0.30.10.50.2],F=[0.3000.5],G=[0.1000.4],B=[0.30.1].
It is worth noting that, the delay functions h(t)=0.1+0.1|sint|. Therefore, the methods used in [2, 9] are not applicable to this system. We have h1=0.1,h2=0.2. Given α=0.2 and any initial function ϕ(t)=C1([-0.2,0],ℝ2). Using the Matlab LMI toolbox, we obtain
(4.3)P=[0.03700.00100.00100.2938],Q=[0.00080.00290.00290.0250],U=[0.01530.00800.0080.6201],R=[0.03770.00550.00550.8173],D0=[0.0353000.2833],D1=[0.0215000.5025].
Thus, the system (4.1) is 0.2-exponentially stabilizable and the value λ2/λ1=1.6469, so the solution of the closed-loop system satisfies
(4.4)‖x(t,ϕ)‖≤1.6469e-0.2t‖ϕ‖,∀t∈R+.Example 4.2.
Consider the neural networks with mixed interval time-varying delays and control input with the following parameters:(4.5)ẋ(t)=-(A+ΔA(t))x(t)+(W0+ΔW0)f(x(t))+(W1+ΔW1)g(x(t-h(t)))+(W2+ΔW2)∫t-k(t)th(x(s))ds+B0ẋ(t-η(t))+Bu(t),
where
(4.6)A=[0.15001],W0=[0.50.120.1-0.3],W1=[0.20.10.10.2],W2=[0.10.20.50.1],B0=[0.15000.15],F=[0.4000.5],G=[0.1000.2],H=[0.5000.3],B=[0.10],Ha=H0=H1=H2=Ea=E0=E1=E2=[0.1000.1].
It is worth noting that, the delay functions h(t)=0.2+0.2|sint|,k(t)=|cost| are nondifferentiable and η(t)=0.2sin2(t). Therefore, the methods used in [13, 14] are not applicable to this system. We have h1=0.2,h2=0.4,k=0.1,δ=0.1,η=0.2. Given α=0.1 and any initial function ϕ(t)=C1([-0.4,0],ℝ2). Using the Matlab LMI toolbox, we obtain e1=0.0173,e2=0.0128,e3=0.0111,e4=0.0263,e5=0.0209,e6=0.0192,
(4.7)P=[0.00610.00020.00020.0228],Q=[0.00030.00050.00010.0031],Q1=[0.00050.00010.00010.0024],U=[0.00280.00040.00040.00382],R=[0.00520.00080.00080.0543],D0=[0.0068000.0304],D1=[0.0038000.0145],D2=[0.0433000.0275].
Thus, the system (4.1) is 0.1-exponentially stabilizable and the value λ2/λ1=2.2939, so the solution of the closed-loop system satisfies
(4.8)‖x(t,ϕ)‖≤2.2939e-0.1t‖ϕ‖,∀t∈R+.


## 5. Conclusions

In this paper, we have investigated the exponential stabilization of neutral-type neural networks with various activation functions and interval nondifferentiable and distributed time-varying delays. The interval time-varying delay function is not necessary to be differentiable which allows time-delay function to be a fast time-varying function. By constructing a set of improved Lyapunov-Krasovskii functional combined with Leibniz-Newton's formula, the proposed stability criteria have been formulated in the form of a linear matrix inequalities. Numerical examples illustrate the effectiveness of the results.



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*Source: 101426-2012-03-12.xml*