# New Robust Exponential Stability Criterion for Uncertain Neutral Systems with Discrete and Distributed Time-Varying Delays and Nonlinear Perturbations

Authors: Sirada Pinjai; Kanit Mukdasai
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We investigate the problem of robust exponential stability for uncertain neutral systems with discrete and distributed time-varying delays and nonlinear perturbations. Based on the combination of descriptor model transformation, decomposition technique of coefficient matrix, and utilization of zero equation and new Lyapunov functional, sufficient conditions for robust exponential stability are obtained and formulated in terms of linear matrix inequalities (LMIs). The new stability conditions are less conservative and more general than some existing results.


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## Body

## 1. Introduction

In the past decades, the problem of stability for neutral differential systems, which have delays in both its state and the derivatives of its states, has been widely investigated by many researchers. Such systems are often encountered in engineering, biology, and economics. The existence of time delay is frequently a source of instability or poor performances in the systems. Recently, some stability criteria for neutral system with time delay have been given [1–13]. The problem of delay-dependent stability for uncertain neutral systems with time-varying delays was considered [6]. Reference [13] investigated the problem of delay-dependent robust stability for delay neutral type control system with time-varying structured uncertainties and time-varying delays. The problem of a new delay-dependent robust stability criterion for neutral systems was investigated. The considered system has time-varying structured uncertainties and interval time-varying delays in [7]. However, many researchers have studied the problem of stability for systems with discrete and distributed delays such as [12] which presented some stability conditions for uncertain neutral systems with discrete and distributed delays. The robust stability of uncertain linear neutral systems with discrete and distributed delays has been studied in [3]. In [14, 15], they studied the problem of stability for linear switching system with discrete and distributed delays. Moreover, a descriptor model transformation and a corresponding Lyapunov-Krasovskii functional have been introduced for stability analysis of systems with delays in [10, 16].In this paper, we will study the problems of robust exponential stability for uncertain neutral systems with discrete and distributed time-varying delays and nonlinear perturbations. Based on a combination of descriptor model transformation, decomposition technique of coefficient matrix, and utilization of zero equation and new Lyapunov functional which improved delay-dependent stability criteria for the considered systems, sufficient conditions for the robust exponential stability are obtained and formulated in terms of linear matrix inequalities (LMIs). The new stability conditions will be less conservative and more general than some existing results.


## 2. Problem Formulation and Preliminaries

We introduce some notations and definitions that will be used throughout the paper.ℝ+ denotes the set of all real nonnegative numbers; ℝn denotes the n-dimensional space with the vector norm ∥·∥; ∥x∥ denotes the Euclidean vector norm of x∈ℝn; ℝn×r denotes the set of n×r real matrices; AT denotes the transpose of the 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)}; λmin(A)=min{Reλ:λ∈λ(A)}; matrix A is called semipositive definite (A≥0) if xTAx≥0, for all x∈ℝn;A is positive definite (A>0) if xTAx>0 for all x≠0; matrix B is called seminegative definite (B≤0) if xTBx≤0, for all x∈ℝn;B is negative definite (B<0) if xTBx<0 for all x≠0; A>B means A-B>0; A≥B means A-B≥0; C([-h,0],ℝn) denotes the space of all continuous vector functions mapping [-h,0] into ℝn; * represents the elements below the main diagonal of a symmetric matrix.Consider the uncertain neutral system with discrete and distributed time-varying delays and nonlinear perturbations of the form(2.1)ẋ(t)-Cẋ(t-r(t))=A(t)x(t)+B(t)x(t-h(t))+f1(t,x(t))+f2(t,x(t-h(t)))+D(t)∫t-g(t)tx(s)ds,t≥0,x(t0+θ)=ϕ(θ),ẋ(t0+θ)=φ(θ),∀θ∈[-max⁡{r,h,g},0],A(t)=[A+ΔA(t)],B(t)=[B+ΔB(t)],D(t)=[D+ΔD(t)],
where x(t)∈ℝn is the state, A,B,C,D∈ℝn×n are known real constant matrices. r(t),h(t), and g(t) are neutral, discrete, and distributed time-varying delays, respectively, (2.2)0≤r(t)≤r,ṙ(t)≤δ<1,0≤h(t)≤h,ḣ(t)≤μ<+∞,0≤g(t)≤g,
where h,r,g,μ,andδ are given positive constants. The initial condition functions ϕ(t),φ(t)∈C([-max{r,h,g},0],ℝn) denote continuous vector-valued initial function of t∈[-max{r,h,g},0] with the norm ∥ϕ∥=sups∈[-max{r,h,g},0]∥ϕ(s)∥, ∥φ∥=sups∈[-max{r,h,g},0]∥φ(s)∥. The uncertain matrices ΔA(t), ΔB(t), and ΔD(t) are norm bounded and can be described as (2.3)ΔA(t)=EF(t)N1,ΔB(t)=EF(t)N2,ΔD(t)=EF(t)N3,
where E,Ni,i=1,2,3, are known constant matrices with appropriate dimensions. The uncertain matrix F(t) satisfies (2.4)F(t)TF(t)≤I.
The uncertainties f1(·) and f2(·) represent the nonlinear parameter perturbations with respect to the current state x(t) and the delayed state x(t-h(t)), respectively, and are bounded in magnitude: (2.5)f1T(t,x(t))f1(t,x(t))≤β1xT(t)x(t),f2T(t,x(t-h(t)))f2(t,x(t-h(t)))≤β2xT(t-h(t))x(t-h(t)),
where η,ρ are given positive constants. First, we consider nominal system (2.1) which is defined to be (2.6)ẋ(t)-Cẋ(t-r(t))=Ax(t)+Bx(t-h(t))+f1(t,x(t))+f2(t,x(t-h(t)))+D∫t-g(t)tx(s)ds,t≥0,x(t0+θ)=ϕ(θ),ẋ(t0+θ)=φ(θ),∀θ∈[-max⁡{r,h,g},0].
Rewrite the system (2.6) in the following descriptor system: (2.7)ẋ(t)=y(t),y(t)=Ax(t)+Bx(t-h(t))+f1(t,x(t))+f2(t,x(t-h(t)))+Cy(t-r(t))+D∫t-g(t)tx(s)ds.
In order to improve the bound of the discrete delay h(t) in (2.6), let us decompose the constant matrix B as (2.8)B=B1+B2,
where B1,B2∈ℝn×n are constant matrices. Utilize the following zero equation: (2.9)0=Gx(t)-Gx(t-h(t))-G∫t-h(t)tẋ(s)ds,
where G∈ℝn×n will be chosen to guarantee the exponential stability of the system (2.6). By (2.8) and (2.9), the system (2.7) can be represented in the form of a descriptor system with discrete and distributed delays and nonlinear perturbations: (2.10)ẋ(t)=y(t)+Gx(t)-Gx(t-h(t))-G∫t-h(t)ty(s)ds,y(t)=A1x(t)+B2x(t-h(t))+f1(t,x(t))+f2(t,x(t-h(t)))+Cy(t-r(t))-B1∫t-h(t)ty(s)ds+D∫t-g(t)tx(s)ds,
where A1=A+B1.Definition 2.1.
The system (2.1) is robustly exponentially stable if there exist real scalars α>0, M>0 such that, for each ϕ(t)∈C([-max{r,h,g},0],ℝn), the solution x(t,ϕ,φ) of the system (2.1) satisfies
(2.11)‖x(t,ϕ,φ)‖≤Mmax⁡[‖ϕ‖,‖φ‖]e-αt,∀t∈R+.Lemma 2.2 (Jensen's inequality [17]).
For any constant symmetric matrixR>0, scalar  h>0, and vector function  ẋ(t):[-h,0]→ℝn such that the following integral is well defined, one has
(2.12)-h∫t-htẋT(s)Rẋ(s)ds≤[x(t)x(t-h)]T[-RRR-R][x(t)x(t-h)].


## 3. Exponential Stability Conditions

In this section, we first study the exponential stability criteria for the nominal system (2.1) by using the combination of linear matrix inequality (LMI) technique and Lyapunov method.Assumption 1.
All the eigenvalues of matrixC are inside the unit circle.Theorem 3.1.
Under Assumption1 and given positive constants  h,r,g,μ,δ,α,β1, and  β2, the nominal system  (2.1) is exponentially stable if there exist positive definite matrices  Pi,  i=1,2,…,6, any appropriate dimensional matrices  W1,Q1,Q2,Mi,  i=1,2,…,8, and positive constants  ϵ1,ϵ2 satisfying the following LMI:
(3.1)Ω=[Ω11Ω12Ω13Ω14Ω15Ω16Ω17Ω18*Ω22Ω23Ω24Q2TQ2TQ2TCQ2TD**Ω33Ω34-M5-M6-M7-M8***Ω44-M5-M6-M7-M8****-ϵ1I000*****-ϵ2I00******Ω770*******Ω88]<0,
where Ω11=W1+W1T+Q1TA1+A1TQ1+P2-e-2αhP4+g2P6+M1T+M1+ϵ1β12I+2αP1,Ω22=-Q2-Q2T+h2P3+h2P4+P5,Ω23=Q2TB2-M2T,Ω24=Q2TB1-M2T,Ω33=-e-2αhP2+μP2-e-2αhP4-M3T-M3+ϵ2β22I, Ω34=-M3T-M4,Ω44=-e-2αhP3-M4-M4T,Ω77=-(1-δ)e-2αrP5, Ω88=-e-2αgP6,Ω12=P1-Q1T+A1TQ2+M2,Ω13=-W1+Q1TB2+e-2αhP4-M1T+M3,Ω14=-W1-Q1TB1-M1T+M4,Ω15=Q1T+M5,Ω16=Q1T+M6,Ω17=Q1TC+M7, Ω18=Q1TD+M8.Moreover, the solution x(t,ϕ,φ) satisfies the inequality
(3.2)‖x(t,ϕ,φ)‖≤Nλmin⁡(P1)max⁡[‖ϕ‖,‖φ‖]e-αt,t∈R+,
where
(3.3)N=λmax⁡(P1)+hλmax⁡(P2)+h3λmax⁡(P3)+h3λmax⁡(P4)+rλmax⁡(P5)+g3λmax⁡(P6).Proof.
Construct a Lyapunov functional candidate for the nominal system (2.1) of the form
(3.4)V(t)=∑i=16Vi(t),
where
(3.5)V1(t)=xT(t)P1x(t)=[x(t)y(t)]T[I000][P10Q1Q2][x(t)y(t)],V2(t)=∫t-h(t)te2α(s-t)xT(s)P2x(s)ds,V3(t)=h∫-h0∫t+ste2α(s-t)yT(θ)P3y(θ)dθds,V4(t)=h∫-h0∫t+ste2α(s-t)yT(θ)P4y(θ)dθds,V5(t)=∫t-r(t)te2α(s-t)yT(s)P5y(s)ds,V6(t)=g∫-g0∫t+ste2α(s-t)xT(θ)P6x(θ)dθds.
The derivative of V(·) along the trajectories of nominal system (2.1) is given by
(3.6)V̇(t)=∑i=16V̇i(t),
where
(3.7)V̇1(t)=2[x(t)y(t)]T[P1Q1T0Q2T][ẋ(t)0]=2xT(t)P1[y(t)+Gx(t)-Gx(t-h(t))-G∫t-h(t)ty(s)ds]+2xT(t)Q1T[D∫t-g(t)tx(s)ds-y(t)+A1x(t)+B2x(t-h(t))+f1(t,x(t))+f2(t,x(t-h(t)))+Cy(t-r(t))-B1∫t-h(t)ty(s)ds+D∫t-g(t)tx(s)ds]+2yT(t)Q2T[D∫t-g(t)tx(s)ds-y(t)+A1x(t)+B2x(t-h(t))+f1(t,x(t))+f2(t,x(t-h(t)))+Cy(t-r(t))-B1∫t-h(t)ty(s)ds+D∫t-g(t)tx(s)ds]+2αxT(t)P1x(t)-2αV1(t),V̇2(t)=xT(t)P2x(t)-(1-ḣ(t))e-2αh(t)xT(t-h(t))P2x(t-h(t))-2αV2(t)≤xT(t)P2x(t)-e-2αhxT(t-h(t))P2x(t-h(t))+μxT(t-h(t))P2x(t-h(t))-2αV2(t).
Obviously, for any scalar s∈[t-h,t], we get e-2αh≤e-2α(s-t)≤1. Together with Lemma 2.2 (Jensen's inequality), we obtain
(3.8)V̇3(t)=h2yT(t)P3y(t)-h∫-h0e2αsyT(t+s)P3y(t+s)ds-2αV3(t)≤h2yT(t)P3y(t)-h∫t-hte2α(s-t)yT(s)P3y(s)ds-2αV3(t)≤h2yT(t)P3y(t)-he-2αh∫t-htyT(s)P3y(s)ds-2αV3(t)≤h2yT(t)P3y(t)-e-2αh∫t-htyT(s)dsP3∫t-hty(s)ds-2αV3(t)≤h2yT(t)P3y(t)-e-2αh∫t-h(t)tyT(s)dsP3∫t-h(t)ty(s)ds-2αV3(t),V̇4(t)≤h2yT(t)P4y(t)-e-2αh∫t-h(t)tyT(s)dsP4∫t-h(t)ty(s)ds-2αV4(t)=h2yT(t)P4y(t)-e-2αh[xT(t)-xT(t-h(t))]P4[x(t)-x(t-h(t))]-2αV4(t),V̇5(t)=xT(t)P5x(t)-(1-ṙ(t))e-2αr(t)xT(t-r(t))P2x(t-r(t))-2αV5(t)≤xT(t)P5x(t)-(1-δ)e-2αrxT(t-r(t))P5x(t-r(t))-2αV5(t),V̇6(t)≤g2xT(t)P6x(t)-e-2αg∫t-g(t)txT(s)dsP6∫t-g(t)tx(s)ds-2αV6(t).
From the Leibniz-Newton formula, the following equation is true for any real matrices Mi,i=1,2,…,8, with appropriate dimensions
(3.9)2[xT(t)M1T+yT(t)M2T+xT(t-h(t))M3T+∫t-h(t)tyT(s)dsM4+f1T(t,x(t))M5+f2T(t,x(t-h(t)))M6T+yT(t-r(t))M7T+∫t-g(t)txT(s)dsM8]×[x(t)-x(t-h(t))-∫t-h(t)ty(s)ds]=0.
From (2.5), we obtain for any scalars ϵ1,ϵ2>0(3.10)0≤ϵ1β1xT(t)x(t)-ϵ1f1T(t,x(t))f1(t,x(t)),0≤ϵ2β2xT(t-h(t))x(t-h(t))-ϵ2f2T(t,x(t-h(t)))f2(t,x(t-h(t))).
Let us define W1=P1G. From (3.6), (3.9), and (3.10), we obtain
(3.11)V̇(t)+2αV(t)≤ξT(t)Ωξ(t),
where ξT(t)=[xT(t),yT(t),xT(t-h(t)),∫t-h(t)tyT(s)ds,fT(t,x(t)),gT(t,x(t-h(t))),yT(t-r(t)),∫t-g(t)txT(s)ds] and
(3.12)Ω=[Ω11Ω12Ω13Ω14Ω15Ω16Ω17Ω18*Ω22Ω23Ω24Q2TQ2TQ2TCQ2TD**Ω33Ω34-M5-M6-M7-M8***Ω44-M5-M6-M7-M8****-ϵ1I000*****-ϵ2I00******Ω770*******Ω88].
It is a fact that, if Ω<0, then
(3.13)V̇(t)+2αV(t)≤0,∀t∈R+,
which gives
(3.14)V(t)≤V(0)e-2αt,∀t∈R+.
From (3.14), it is easy to see that
(3.15)λmin⁡(P1)‖x(t)‖2≤V(t)≤V(0)e-2αt,V(0)=∑i=16Vi(0),
where
(3.16)V1(0)=xT(0)P1x(0),V2(0)=∫-h(0)0e2αsxT(s)P2x(s)ds,V3(0)=h∫-h0∫s0e2αsyT(θ)P3y(θ)dθds,V4(0)=h∫-h0∫s0e2αsyT(θ)P4y(θ)dθds,V5(0)=∫-r(0)0e2αsyT(s)P5y(s)ds,V6(0)=g∫-g0∫s0e2αsxT(θ)P6x(θ)dθds.
Therefore, we get
(3.17)λmin⁡(P1)‖x(t)‖2≤V(0)e-2αt≤Nmax⁡[‖ϕ‖,‖φ‖]2e-2αt,
where N=λmax(P1)+hλmax(P2)+h3λmax(P3)+h3λmax(P4)+rλmax(P5)+g3λmax(P6). From (3.17), we get
(3.18)‖x(t,ϕ,φ)‖≤Nλmin⁡(P1)max⁡[‖ϕ‖,‖φ‖]e-αt,∀t∈R+.
This means that the system (2.1) is exponentially stable. The proof of the theorem is complete.IfC=ΔA(t)=ΔB(t)=f1(t,x(t))=f2(t,x(t-h(t)))=D(t)=0, then system (2.1) reduces to the following system: (3.19)ẋ(t)=Ax(t)+Bx(t-h(t)),t≥0,x(t0+θ)=ϕ(θ),ẋ(t0+θ)=φ(θ),∀θ∈[-h,0].
Take the Lyapunov functional as (3.20)V̇(t)=∑i=14V̇i(t),
where V1(t),V2(t),V3(t), and V4(t) are defined in Theorem 3.1. According to Theorem 3.1, we have the following corollary for the delay-dependent exponential stability criterion of system (3.19).Corollary 3.2.
For given positive constantsh,μ,andα, the system  (3.19) is exponentially stable if there exist positive definite matrices  Pi,  i=1,2,…,4, and any appropriate dimensional matrices  W1,Q1,Q2,Mi,  i=1,2,…,4, satisfying the following LMI:
(3.21)Φ=[Φ11Φ12Φ13Φ14*Φ22Φ23Φ24**Φ33Φ34***Φ44]<0,
where Φ11=W1+W1T+Q1TA1+A1TQ1+P2-e-2αhP4+M1T+M1+2αP1,Φ22=-Q2-Q2T+h2P3+h2P4,Φ23=Q2TB2-M2T,Φ24=Q2TB1-M2T,Φ33=-e-2αhP2+μP2-e-2αhP4-M3T-M3,Ω34=-M3T-M4,Φ44=-e-2αhP3-M4-M4T,Φ12=P1-Q1T+A1TQ2+M2, Φ13=-W1+Q1TB2+e-2αhP4-M1T+M3, Φ14=-W1-Q1TB1-M1T+M4. Moreover, the solution x(t,ϕ,φ) satisfies the inequality
(3.22)‖x(t,ϕ,φ)‖≤Kλmin⁡(P1)max⁡[‖ϕ‖,‖φ‖]e-αt,t∈R+,
where
(3.23)K=λmax⁡(P1)+hλmax⁡(P2)+h3λmax⁡(P3)+h3λmax⁡(P4).IfΔA(t)=ΔB(t)=D(t)=0, then system (2.1) reduces to the following system: (3.24)ẋ(t)-Cẋ(t-r(t))=Ax(t)+Bx(t-h(t))+f1(t,x(t))+f2(t,x(t-h(t))),t≥0,x(t0+θ)=ϕ(θ),ẋ(t0+θ)=φ(θ),∀θ∈[-max⁡{r,h},0].
Take the Lyapunov functional as (3.25)V̇(t)=∑i=15V̇i(t),
where V1(t),V2(t),V3(t),V4(t), and V5(t) are defined in Theorem 3.1. According to Theorem 3.1, we have the following corollary for the delay-dependent exponential stability criterion of system (3.24).Corollary 3.3.
For given positive constantsh,r,μ,δ,α,β1, and  β2, the system (3.24) is exponentially stable if there exist positive definite matrices  Pi,  i=1,2,…,5, any appropriate dimensional matrices  W1,Q1,Q2,Mi,  i=1,2,…,7, and positive constants  ϵ1,ϵ2 satisfying the following LMI:
(3.26)Υ=[Υ11Υ12Υ13Υ14Υ15Υ16Υ17*Υ22Υ23Υ24Q2TQ2TQ2TC**Υ33Υ34-M5-M6-M7***Υ44-M5-M6-M7****-ϵ1I00*****-ϵ2I0******Υ77]<0,
where Υ11=W1+W1T+Q1TA1+A1TQ1+P2-e-2αhP4+M1T+M1+ϵ1β12I+2αP1,Υ22=-Q2-Q2T+h2P3+h2P4+P5, Υ23=Q2TB2-M2T, Υ24=Q2TB1-M2T, Υ33=-e-2αhP2+μP2-e-2αhP4-M3T-M3+ϵ2β22I,Υ34=-M3T-M4,Υ44=-e-2αhP3-M4-M4T, Υ12=P1-Q1T+A1TQ2+M2,Υ13=-W1+Q1TB2+e-2αhP4-M1T+M3,Υ14=-W1-Q1TB1-M1T+M4,Υ15=Q1T+M5,Υ16=Q1T+M6,Υ17=Q1TC+M7,Υ77=-(1-δ)e-2αrP5. Moreover, the solution x(t,ϕ,φ) satisfies the inequality
(3.27)‖x(t,ϕ,φ)‖≤Kλmin⁡(P1)max⁡[‖ϕ‖,‖φ‖]e-αt,t∈R+,
where
(3.28)K=λmax⁡(P1)+hλmax⁡(P2)+h3λmax⁡(P3)+h3λmax⁡(P4).


## 4. Robust Exponential Stability Conditions

Theorem 4.1.
Under Assumption1 and given positive constants  h,r,g,μ,δ,α,β1, and  β2, the system (2.1) is exponentially stable if there exist positive definite matrices  Pi,  i=1,2,…,6, any appropriate dimensional matrices  W1,Q1,Q2,Mi,  i=1,2,…,8, and positive constants  ϵ1,ϵ2,andε satisfying the following LMI:
(4.1)Ω̃=[Ω11Ω12Ω13Ω14Ω15Ω16Ω17Ω18Q1TEεN1T*Ω22Ω23Ω24Q2TQ2TQ2TCQ2TDQ1TE0**Ω33Ω34-M5-M6-M7-M80εN2T***Ω44-M5-M6-M7-M800****-ϵ1I00000*****-ϵ2I0000******Ω77000*******Ω880εN3T********-εI0*********-εI]<0,
where Ω11=W1+W1T+Q1TA1+A1TQ1+P2-e-2αhP4+g2P6+M1T+M1+ϵ1β12I+2αP1,Ω22=-Q2-Q2T+h2P3+h2P4+P5,Ω23=Q2TB2-M2T,Ω24=Q2TB1-M2T, Ω33=-e-2αhP2+μP2-e-2αhP4-M3T-M3+ϵ2β22I,Ω34=-M3T-M4,Ω44=-e-2αhP3-M4-M4T,Ω77=-(1-δ)e-2αrP5,Ω88=-e-2αgP6,Ω12=P1-Q1T+A1TQ2+M2,Ω13=-W1+Q1TB2+e-2αhP4-M1T+M3, Ω14=-W1-Q1TB1-M1T+M4,Ω15=Q1T+M5,Ω16=Q1T+M6,Ω17=Q1TC+M7,Ω18=Q1TD+M8. Moreover, the solution x(t,ϕ,φ) satisfies the inequality
(4.2)‖x(t,ϕ,φ)‖≤Nλmin⁡(P1)max⁡[‖ϕ‖,‖φ‖]e-αt,t∈R+,
where
(4.3)N=λmax⁡(P1)+hλmax⁡(P2)+h3λmax⁡(P3)+h3λmax⁡(P4)+rλmax⁡(P5)+g3λmax⁡(P6).Proof.
ReplacingA1,B2, and D in (3.26) with A1(t)=A+B1+EF(t)N1, B2(t)=B2+EF(t)N2, and D(t)=D+EF(t)N3, respectively, we find that condition (3.26) for system (2.1) is equivalent to the following condition:
(4.4)Ω+ΓF(t)Π+ΠTF(t)TΓT<0,
where ΓT=[ETQ1,ETQ2,0,0,0,0,0,0] and Π=[N1,0,N2,0,0,0,0,N3]. Using Lemma  1 in [9], we have that there exists a positive number ε>0. We can find that (4.4) is equivalent to the LMI as follows:
(4.5)Ω+ε-1ΓΓT+εΠTΠ<0.
Applying Lemma  1 (Schur complement lemma) in [12], we obtain
(4.6)[ΩΓεΠTΓT-εI0εΠ0-εI]<0.
Hence, we conclude that (4.6) is equivalent to (4.1). The proof of the theorem is complete.Iff1(t,x(t))=f2(t,x(t-h(t)))=D(t)=0, then system (2.1) reduces to the following system: (4.7)ẋ(t)-Cẋ(t-r(r))=[A+EF(t)N1]x(t)+[B+EF(t)N2]x(t-h(t)),t≥0,x(t0+θ)=ϕ(θ),ẋ(t0+θ)=φ(θ),∀θ∈[-max⁡{r,h},0].
Take the Lyapunov functional as (4.8)V̇(t)=∑i=15V̇i(t),
where V1(t),V2(t),V3(t),V4(t), and V5(t) are defined in Theorem 3.1. According to Theorems 3.1 and 4.1, we have the following corollary for the delay-dependent robust exponential stability of system (4.7).Corollary 4.2.
Under Assumption1 and given positive constants  h,r,μ,δ, and  α, the system (4.7) is robustly exponentially stable if there exist positive definite matrices  Pi,  i=1,2,…,5, any appropriate dimensional matrices  W1,Q1,Q2,M7,Mi,  i=1,2,…,4, and positive constant  ε satisfying the following LMI:
(4.9)Ω̃=[Ω11Ω12Ω13Ω14Ω17Q1TEεN1T*Ω22Ω23Ω24Q2TCQ1TE0**Ω33Ω34-M70εN2T***Ω44-M700****Ω7700*****-εI0******-εI]<0,
where Ω11=W1+W1T+Q1TA1+A1TQ1+P2-e-2αhP4+M1T+M1+2αP1,Ω22=-Q2-Q2T+h2P3+h2P4+P5,Ω23=Q2TB2-M2T,Ω24=Q2TB1-M2T,Ω33=-e-2αhP2+μP2-e-2αhP4-M3T-M3,Ω34=-M3T-M4,Ω44=-e-2αhP3-M4-M4T,Ω77=-(1-δ)e-2αrP5,Ω12=P1-Q1T+A1TQ2+M2,Ω13=-W1+Q1TB2+e-2αhP4-M1T+M3, Ω14=-W1-Q1TB1-M1T+M4,Ω17=Q1TC+M7. Moreover, the solution x(t,ϕ,φ) satisfies the inequality
(4.10)‖x(t,ϕ,φ)‖≤Zλmin⁡(P1)max⁡[‖ϕ‖,‖φ‖]e-αt,t∈R+,
where
(4.11)Z=λmax⁡(P1)+hλmax⁡(P2)+h3λmax⁡(P3)+h3λmax⁡(P4)+rλmax⁡(P5).


## 5. Numerical Examples

In order to show the effectiveness of the approaches presented in Sections3 and 4, three numerical examples are provided.Example 5.1.
Consider the asymptotic stability(α=0) of the linear system (3.19) with time-varying delays when
(5.1)A=[-200-0.9],B=[-10-1-1].
Decompose matrix B as follows: B=B1+B2, where
(5.2)B1=[-0.830-1-0.83],B2=[-0.1700-0.17].
Table1 lists the comparison of the upper bounds delays for asymptotic stability (α=0) of system (5.1) by different methods. We can see from Table 1 that our results (Corollary 3.2) are superior to those in Lemma  4 [16], Lemma  1 [4], and Theorem  1 [18].Table 1
Comparison of the maximum allowed time delayhmax.
Methodshmax(δ=0.9)hmax(δ=1)hmax(δ=1.2)hmax(δ=1.5)Fridman and Shaked [16]1.180.990.990.99He et al. [4]1.371.341.341.34Zhu and Yang [18]1.431.391.361.35Our results2.001.511.511.51Example 5.2.
Consider the exponential stability of the neutral system (3.24) with time-varying delays and nonlinear perturbations with
(5.3)A=[-200-0.9],B=[-10-1-1],C=[0.1000.1],β1=0.05,β2=0.1,h(t)=r(t).
Decompose matrix B as follows which is the same as the decomposition in Example 5.1.
Table2 lists the comparison of the upper bounds delay for exponential stability of (5.3) by different methods. It is clear that our results (Corollary 3.3) are superior to those in Theorem  1 [1].Table 2
Comparison of the maximum allowed time delayhmax.
Methodshmax(δ=0)hmax(δ=0.5)hmax(δ=0.9)Chen et al. [1]  (α=0.1)1.290.940.54Our results (α=0.1)1.621.100.62Chen et al. [1] (α=0.5)0.690.600.46Our results (α=0.5)0.750.610.47Chen et al. [1] (α=0.7)0.570.530.43Our results (α=0.7)0.610.540.43Example 5.3.
Consider the asymptotic stability(α=0) of the uncertain neutral system (4.7) with time-varying delays with
(5.4)A=[-200-0.9],B=[-10-1-1],C=[c00c],0≤c<1,E=I,N1=N2=0.2I,h(t)=r(t).
Decompose matrix B as follows which is the same as the decomposition in Example 5.1.
Table3 shows the comparison of the upper bounds delay allowed obtained from Corollary 4.2 for asymptotic stability (α=0) of system (4.7) with (5.4) by other methods. It can be found from Table 3 that our results are significantly better than those in [2, 5, 9].Table 3
Comparison of the maximum allowed time delayhmax.
Methodshmax(c=0.1)hmax(c=0.15)hmax(c=0.2)hmax(c=0.3)hmax(c=0.4)Han [2]1.481.331.160.790.37He et al. [5]1.751.490.270.910.63Nian et al. [9]1.861.631.421.060.76Our results1.961.871.751.501.23


## 6. Conclusions

The problem of robust exponential stability for uncertain neutral systems with discrete and distributed time-varying delays and nonlinear perturbations was studied. Based on the combination of descriptor model transformation, decomposition technique of coefficient matrix, and utilization of zero equation and new Lyapunov functional, sufficient conditions for robust exponential stability were obtained and formulated in terms of linear matrix inequalities (LMIs). Numerical examples have shown significant improvements over some existing results.



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*Source: 463603-2011-08-13.xml*