# New Delay-Dependent Robust Exponential Stability Criteria of LPD Neutral Systems with Mixed Time-Varying Delays and Nonlinear Perturbations

Authors: Sirada Pinjai; Kanit Mukdasai
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper is concerned with the problem of robust exponential stability for linear parameter-dependent (LPD) neutral systems with mixed time-varying delays and nonlinear perturbations. Based on a new parameter-dependent Lyapunov-Krasovskii functional, Leibniz-Newton formula, decomposition technique of coefficient matrix, free-weighting matrices, Cauchy’s inequality, modified version of Jensen’s inequality, model transformation, and linear matrix inequality technique, new delay-dependent robust exponential stability criteria are established in terms of linear matrix inequalities (LMIs). Numerical examples are given to show the effectiveness and less conservativeness of the proposed methods.


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## Body

## 1. Introduction

Over the past decades, the problem of stability for neutral differential systems, which have delays in both their state and the derivatives of their states, has been widely investigated by many researchers, especially in the last decade. It is well known that nonlinearities, as time delays, may cause instability and poor performance of practical systems such as engineering, biology, and economics [1]. The problems of various stability and stabilization for dynamical systems with or without state delays and nonlinear perturbations have been intensively studied in the past years by many researchers of mathematics and control communities [1–35]. Stability criteria for dynamical systems with time delay are generally divided into two classes: delay-independent one and delay-dependent one. Delay-independent stability criteria tend to be more conservative, especially for small size delay; such criteria do not give any information on the size of the delay. On the other hand, delay-dependent stability criteria are concerned with the size of the delay and usually provide a maximal delay size.Recently, many researchers have studied the stability problem for neutral systems with time-varying delays and nonlinear perturbations have appeared [29, 31]. Furthermore, the convergence rates are essential for the practical system; then the exponential stability analysis of time delay systems has been favorably approved in the past decades; see, for example, [3, 9, 10, 14, 18–21, 25–28].In addition, many researchers have paid attention to the problem of stability for linear systems with polytope uncertainties. The linear systems with polytopic-type uncertainties are called linear parameter-dependent (LPD) systems. That is, the uncertain state matrices are in the polytope consisting of all convex combination of known matrices. Most of sufficient (or necessary and sufficient) conditions have been obtained via Lyapunov-Krasovskii theory approaches in which parameter-dependent Lyapunov-Krasovskii functional has been employed. These conditions are always expressed in terms of linear matrix inequalities (LMIs). The results have been obtained for robust stability for LPD systems in which time-delay occurs in state variable; for example, [17, 18] presented sufficient conditions for robust stability of LPD discrete-time systems with delays. Moreover, robust stability of LPD continuous-time systems with delays was studied in [6, 19, 22, 30].In consequence, it is important and interesting to study the problem of robust exponential stability for neutral systems with parametric uncertainties. This paper investigates the robust exponential stability analysis for LPD neutral systems with mixed time-varying delays and nonlinear perturbations. Based on combination of Leibniz-Newton formula, free-weighting matrices, Cauchy’s inequality, modified version of Jensen's inequality, decomposition technique of coefficient matrix, the use of suitable parameter-dependent Lyapunov-Krasovskii functional, model transformation, and linear matrix inequality technique, new delay-dependent robust exponential stability criteria for these systems will be obtained in terms of LMIs. Finally, numerical examples will be given to show the effectiveness of the obtained results.


## 2. Problem Formulation and Preliminaries

We introduce some notations, a definition, and lemmas 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)}; λmax(A(α))= max{λmax(Ai):i=1,2,…,N}; λmin(A(α))= min{λmin(Ai):i=1,2,…,N}; C([-b,0],Rn) denotes the space of all continuous vector functions mapping [-b,0] into Rn, where b=max{h,r}, h,r∈R+; * represents the elements below the main diagonal of a symmetric matrix.Consider the system described by the following state equations of the form:(1)x˙(t)=A(α)x(t)+B(α)x(t-h(t))+C(α)x˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t)))+w(t,x˙(t-r(t))),t>0;x(t+t0)=ϕ(t),x˙(t+t0)=ψ(t),t∈[-b,0],
where x(t)∈Rn is the state variable and A(α), B(α), C(α)∈Rn×n are uncertain matrices belonging to the polytope
(2)A(α)=∑i=1N‍αiAi,B(α)=∑i=1N‍αiBi,C(α)=∑i=1N‍αiCi,A(α)=∑i=1N‍αi=1,αi≥0,Ai,Bi,Ci∈Rn×n,i=1,…,N.h(t) and r(t) are discrete and neutral time-varying delays, respectively,
(3)0≤h(t)≤h,h˙(t)≤hd,(4)0≤r(t)≤r,r˙(t)≤rd,
where h, r, hd, and rd are given positive real constants. Consider the initial functions ϕ(t), ψ(t)∈ C([-b,0],Rn) with the norm ∥ϕ∥=supt∈[-b,0]∥ϕ(t)∥ and ∥ψ∥=supt∈[-b,0]∥ψ(t)∥. The uncertainties f(t,x(t)), g(t,x(t-h(t))), and w(t,x˙(t-r(t))) are the nonlinear perturbations with respect to current state x(t), discrete delayed state x(t-h(t)), and neutral delayed state x˙(t-r(t)), respectively, and are bounded in magnitude:
(5)fT(t,x(t))f(t,x(t))≤η2xT(t)x(t),gT(t,x(t-h(t)))g(t,x(t-h(t)))≤ρ2xT(t-h(t))x(t-h(t)),wT(t,x˙(t-r(t)))w(t,x˙(t-r(t)))≤γ2x˙T(t-r(t))x˙(t-r(t)),
where η,ρ, and γ are given positive real constants.In order to improve the bound of the discrete time-varying delayedh(t) in system (1), let us decompose the constant matrix B(α) as
(6)B(α)=B1(α)+B2(α),
where B1(α)=∑i=1N‍αiBi1, B2(α)=∑i=1N‍αiBi2, and ∑i=1N‍αi=1, αi≥0 with Bi1,Bi2∈Rn×n, i=1,…,N being real constant matrices. By Leibniz-Newton formula, we have
(7)0=x(t)-x(t-h(t))-∫t-h(t)t‍x˙(s)ds.By utilizing the following zero equation, we obtain(8)0=Gx(t)-Gx(t-βh(t))-G∫t-βh(t)t‍x˙(s)ds,
where β is a given positive real constant and G∈Rn×n will be chosen to guarantee the robust exponential stability of system (1). By (6), (7), and (8), system (1) can be represented by the form
(9)x˙(t)=[A(α)+B1(α)+G]x(t)+B2(α)x(t-h(t))-Gx(t-βh(t))+f(t,x(t))+g(t,x(t-h(t)))+w(t,x˙(t-r(t)))+C(α)x˙(t-r(t))-B1(α)∫t-h(t)t‍x˙(s)ds-G∫t-βh(t)t‍x˙(s)ds.Definition 1.
The system (1) is robustly exponentially stable, if there exist positive real constants k and M such that, for each ϕ(t),ψ(t)∈C([-b,0],Rn), the solution x(t,ϕ,ψ) of the system (1) satisfies
(10)∥x(t,ϕ,ψ)∥≤Mmax{∥ϕ∥,∥ψ∥}e-kt,∀t∈R+.Lemma 2 (Cauchy inequality).
For any constant symmetric positive definite matrixP∈Rn×n and a,b∈Rn, one has
(11)±2aTb≤aTPa+bTP-1b.Lemma 3 (see [15]).
The following inequality holds for anya∈Rn, b∈Rm, N,Y∈Rn×m, X∈Rn×n, and Z∈Rm×m:
(12)-2aTNb≤[ab]T[XY-N*Z][ab],
where [XY*Z]≥0.Lemma 4.
For any constant symmetric positive definite matrixQ∈Rn×n and h(t) which is discrete time-varying delays with (3), vector function ω:[-h,0]→Rn such that the integrations concerned are well defined; then
(13)h∫-h0‍ωT(s)Qω(s)ds≥∫-h(t)0‍ωT(s)dsQ∫-h(t)0‍ω(s)ds.Proof.
From Lemma2, it is easy to see that
(14)h∫-h0‍ωT(s)Qω(s)ds=h∫-h(t)0‍ωT(s)Qω(s)ds+h∫-h-h(t)‍ωT(s)Qω(s)ds≥h(t)∫-h(t)0‍ωT(s)Qω(s)ds=12∬-h(t)0[ωT(s)Qω(s)=12∬-h(t)0[+ωT(ξ)Qω(ξ)]dsdξ≥12∬-h(t)02ωT(s)Q1/2T12∫-h(t)0‍∫-h(t)0‍×Q1/2ω(ξ)dsdξ=∫-h(t)0‍ωT(s)dsQ∫-h(t)0‍ω(s)ds.Lemma 5.
Letx(t)∈Rn be a vector-valued function with first order continuous derivative entries. Then, the following integral inequality holds for any matrices Mi∈Rn×n, i=1,2,…,5, and h(t) is discrete time-varying delays with (3) and symmetric positive definite matrix X∈Rn×n:
(15)-∫t-ht‍x˙T(s)Xx˙(s)ds≤[x(t)x(t-h(t))]T×[M1+M1T-M1T+M2*-M2-M2T]×[x(t)x(t-h(t))]+h[x(t)x(t-h(t))]T[M3M4*M5]×[x(t)x(t-h(t))],
where
(16)[XM1M2*M3M4**M5]≥0.Proof.
From the Leibniz-Newton formula, one has(17)0=x(t)-x(t-h(t))-∫t-h(t)t‍x˙(t)ds.
Therefore, for any H1, H2∈Rn×n, the following equation is true:
(18)0=2[xT(t)-xT(t-h(t))-∫t-h(t)t‍x˙T(t)ds]×[H1x(t)+H2x(t-h(t))]=2xT(t)H1x(t)+2xT(t)H2x(t-h(t))-2xT(t-h(t))H1x(t)-2xT(t-h(t))H2Tx(t-h(t))-2∫t-h(t)t‍x˙T(s)dsH1x(t)-2∫t-h(t)t‍x˙T(s)dsH2x(t-h(t))=[x(t)x(t-h(t))]T[H1+H1T-H1T+H2*-H2-H2T]×[x(t)x(t-h(t))]-2∫t-h(t)t‍x˙T(s)[H1H2][x(t)x(t-h(t))]ds.
Using Lemma3 with a=x˙(s), b=[x(t)x(t-h(t))], N=[H1H2], Y=[M1M2], and Z=[M3M4*M5], we obtain
(19)-2∫t-h(t)t‍x˙T(s)[H1H2][x(t)x(t-h(t))]ds≤∫t-h(t)t‍[x˙(s)x(t)x(t-h(t))]T[XM1-H1M2-H2*M3M4**M5]≤∫t-h(t)t‍[×[x˙(s)x(t)x(t-h(t))]ds=∫t-h(t)t‍x˙T(s)Xx˙(s)ds+[x(t)x(t-h(t))]T×[M1+M1T-H1-H1T-M1T+M2+H1T-H2*-M2-M2T+H2+H2T]×[x(t)x(t-h(t))]+h(t)[x(t)x(t-h(t))]T[M3M4*M5]×[x(t)x(t-h(t))]≤∫t-h(t)t‍x˙T(s)Xx˙(s)ds+[x(t)x(t-h(t))]T×([M1+M1T-M1T+M2*-M2-M2T]×22-[H1+H1T-H1T+H2*-H2-H2T])[x(t)x(t-h(t))]+h[x(t)x(t-h(t))]T[M3M4*M5][x(t)x(t-h(t))].
Substituting (19) into (18), we obtain
(20)-∫t-h(t)t‍x˙T(s)Xx˙(s)ds≤[x(t)x(t-h(t))]T×[H1+H1T-H1T+H2*-H2-H2T]×[x(t)x(t-h(t))]+[x(t)x(t-h(t))]T×([M1+M1T-M1T+M2*-M2-M2T]+h22-[H1+H1T-H1T+H2*-H2-H2T][M1+M1T-M1T+M2*-M2-M2T])×[x(t)x(t-h(t))]+h[x(t)x(t-h(t))]T[M3M4*M5]×[x(t)x(t-h(t))]=[x(t)x(t-h(t))]T×[M1+M1T-M1T+M2*-M2-M2T]×[x(t)x(t-h(t))]+h[x(t)x(t-h(t))]T[M3M4*M5]×[x(t)x(t-h(t))].
From (3), it is clear that
(21)-∫t-ht‍x˙T(s)Xx˙(s)ds≤-∫t-h(t)t‍x˙T(s)Xx˙(s)ds.
From (20) and (21), the integral inequality becomes
(22)-∫t-ht‍x˙T(s)Xx˙(s)ds≤[x(t)x(t-h(t))]T×[M1+M1T-M1T+M2*-M2-M2T]×[x(t)x(t-h(t))]+h[x(t)x(t-h(t))]T[M3M4*M5]×[x(t)x(t-h(t))].
The proof of the theorem is complete.Remark 6.
In Lemma4 and Lemma 5, we have modified the method from [8, 33], respectively.


## 3. Main results

### 3.1. Robust Exponential Stability Criteria

In this section, robust exponential stability criteria dependent on mixed time-varying delays of LPD neutral delayed system (1) with nonlinear perturbations via linear matrix inequality (LMI) approach will be presented. We introduce the following notations for later use:
(23)Pj(α)=∑i=1N‍αiPij,Z1(α)=∑i=1N‍αiZi1,Rp(α)=∑i=1N‍αiRip,Nj(α)=∑i=1N‍αiNij,Oj(α)=∑i=1N‍αiOij,Wj(α)=∑i=1N‍αiWij,Mj(α)=∑i=1N‍αiMij,∑i=1N‍αi=1,αi≥0,Pij,Zi1,Rip,Wij,Nij,Oij,Mij∈Rn×n,j=1,2,…,10,p=1,2,…,6,i=1,2,…,N;∑i,j=[Σi,j1,1Σi,j1,2Σi,j1,3Σi,j1,4Σi,j1,5Σi,j1,6Σi,j1,7Σi,j1,8Σi,j1,9Σi,j1,10*Σi,j2,2Σi,j2,3Σi,j2,4Σi,j2,5Σi,j2,6Σi,j2,7Σi,j2,8Σi,j2,9Σi,j2,10**Σi3,3Σi,j3,4Σi3,5Σi3,6Σi,j3,7Σi3,8Σi3,9Σi3,10***Σi,j4,4Σi,j4,5Σi,j4,6Σi,j4,7Σi4,8Σi4,9Σi4,10****Σi5,5Σi5,6Σi,j5,7Σi5,8Σi5,9Σi5,10*****Σi6,6Σi,j6,7Σi6,8Σi6,9Σi6,10******Σi,j7,7Σi,j7,8Σi,j7,9Σi,j7,10*******Σi8,8Σi8,9Σi8,10********Σi9,9Σi9,10*********Σi10,10],
where
(24)Σi,j1,1=AiTPj1+Bi1TPj1+Pi1Aj+Pi1Bj1+Zi1+Zi1T+Pi3+Pi4+hMi1T+hMi1-e-2khPi9-e-2kβhPi10+h2Mi3+βhMi6+βhMi6T+β2h2Mi8+Ni1T+Ni1+Oi1T+Oi1+AiTWj1+Wi1TAj+Bi1TWj1+Wi1TBj1+ϵ1η2I+2kPi1,Σi,j1,2=Pi1Bj2-hMi1T+hMi2+h2Mi4+e-2khPi9+hRi2-Ni1T+Ni2+Oi2+Wi1TBj2+AiTWj2+Bi1TWj2,Σi,j1,3=-Zi1-βhMi6T+βhMi7+βh2Mi9+βhRi5+e-2kβhPi10+Ni3-Oi1T+Oi3+AiTWj3+Bi1TWj3,Σi,j1,4=-Pi1Bj1-Ni1T+Ni4+Oi4-Wi1TBj1+AiTWj4+Bi1TWj4,Σi,j1,5=-Zi1+Ni5-Oi1T+Oi5+AiTWj5+Bi1TWj5,Σi,j1,6=Ni6+Oi6-Wi1T+AiTWj6+Bi1TWj6,Σi,j1,7=Pi1Cj+Ni7+Oi7+Wi1TCj+AiTWj7+Bi1TWj7,Σi,j1,8=Pi1+Ni8+Oi8+Wi1T+AiTWj8+Bi1TWj8,Σi,j1,9=Pi1+Ni9+Oi9+Wi1T+AiTWj9+Bi1TWj9,Σi,j1,10=Pi1+Ni10+Oi10+Wi1T+AiTWj10+Bi1TWj10,Σi,j2,2=-(1-hd)e-2khPi3-hMi2T-hMi2+h2Mi5+h2Ri1-2hRi2-e-2khPi9+ϵ2ρ2I-Ni2T-Ni2+Wi2TBj2+Bi2TWj2,Σi,j2,3=-Ni3-Oi2T+Bi2TWj3,Σi,j2,4=-Ni2T-Ni4-Wi2TBj1+Bi2TWj4,Σi,j2,5=-Ni5-Oi2T+Bi2TWj5,Σi,j2,6=-Ni6-Wi2T+Bi2TWj6,Σi,j2,7=-Ni7+Wi2TCj+Bi2TWj7,Σi,j2,8=-Ni8+Wi2T+Bi2TWj8,Σi,j2,9=-Ni9T-Wi2T+Bi2TWj9,Σi,j2,10=-Ni10+Wi2T+Bi2TWj10,Σi3,3=-(1-βhd)e-2βkhPi4-e-2βkhPi10+β2h2Ri4-2βhRi5-βhMi7T-βhMi7+β2h2Mi10-Oi3T-Oi3,Σi,j3,4=-Ni3T-Oi4-Wi3TBj1,Σi3,5=-Oi3T-Oi5,Σi3,6=-Oi9+Wi3T,Σi3,7=-Oi7+Wi3TCj,Σi3,8=-Oi8+Wi3T,Σi3,9=-Oi9+Wi3T,Σi3,10=-Oi10+Wi3T,Σi,j4,4=-e2khPi7-Ni4T-Ni4+Wi4TBj1-Bi1TWj4,Σi,j4,5=-Ni5-Oi4T-Bi1TWj5,Σi,j4,6=-Ni6-Wi4T-Bi1TWj6,Σi,j4,7=-Ni7+Wi4TCj-Bi1TWj7,Σi4,8=-Oi8+Wi3T,Σi4,9=-Oi9+Wi3T,Σi4,10=-Oi10+Wi3T,Σi5,5=-e2βkhPi8-Oi5T-Oi5,Σi5,6=-Oi6-Wi5T,Σi,j5,7=-Oi7+Wi5TCj,Σi5,8=-Oi8+Wi5T,Σi5,9=-Oi9+Wi5T,Σi5,10=-Oi10+Wi5T,Σi6,6=Pi2+h2Pi5+β2h2Pi6+h2Pi7+β2h2Pi8+h2Pi9+β2h2Pi10-Wi6T-Wi6,Σi,j6,7=Wi6TCj-Wi7,Σi6,8=Wi6T-Wi8,Σi6,9=Wi6T-Wi9,Σi6,10=Wi6T-Wi10,Σi,j7,7=-(1-rd)e-2krPi2+ϵ3γ2I+Wi7TCj+CiTWj7,Σi,j7,8=Wi7T-CiTWj8,Σi,j7,9=Wi7T-CiTWj9,Σi,j7,10=Wi7T-CiTWj10,Σi8,8=Wi8T+Wi8-ϵ1I,Σi8,9=Wi8T+Wi9,Σi8,10=Wi8T+Wi10,Σi9,9=Wi9T+Wi9-ϵ2I,Σi9,10=Wi9T+Wi10,Σi10,10=Wi10T+Wi10-ϵ3I,Σi7,7^=-(1-rd)e2krPi2+Wi7TCj+CiTWj7,Zi1=Pi1G.Theorem 7.
For∥Ci∥+γ<1,  i=1,2,…,N and given positive real constants h, hd, r, rd, η, ρ, γ, and β, system (1) is robustly exponentially stable with a decay rate k, if there exist symmetric positive definite matrices Pij, any appropriate dimensional matrices Rip, Nij, Oij, Wij, Mij, G, p=1,2,…6, j=1,2,…,10, i=1,2,…,N, and positive real constants ϵ1, ϵ2, and ϵ3 such that the following symmetric linear matrix inequalities hold:
(25)[Ri1Ri2*Ri3]>0,i=1,2,…,N,[Ri4Ri5*Ri6]>0,i=1,2,…,N,[e-2khPi3-Ri3Mi1Mi2*Mi3Mi4**Mi5]≥0,i=1,2,…,N,[e-2βkhPi6-Ri6Mi6Mi7*Mi8Mi9**Mi10]≥0,i=1,2,…,N,∑i,i<-I,i=1,2,…,N,∑i,j+∑j,i<2(N-1)I,i=1,2,…,N-1,<2(N-1)Ij=i+1,i+2,…,N.
Moreover, the solutionx(t,ϕ,ψ) satisfies the inequality
(26)∥x(t,ϕ,ψ)∥≤Lλmin(P1(α))max[∥ϕ∥,∥ψ∥]e-kt,≤Lλmin(P1(α))max∥ϕ∥hlh∀t∈R+,
where L=λmax(P1(α))+rλmax(P2(α))+hλmax(P3(α)) + βhλmax(P4(α))+h3λmax(P5(α) + P7(α) + P9(α)) + (βh)3λmax(P6(α) + P8(α)+P10(α))+h3λmax([R1(α)R2(α)*R3(α)]) + (βh)3λmax([R4(α)R5(α)*R6(α)]).Proof.
Choose a parameter-dependent Lyapunov-Krasovskii functional candidate for system (9) as
(27)V(t)=∑i=17‍Vi(t),
where
(28)V1(t)=xT(t)P1(α)x(t),V2(t)=∫t-r(t)t‍e2k(s-t)x˙T(s)P2(α)x˙(s)ds,V3(t)=∫t-h(t)t‍e2k(s-t)xT(s)P3(α)x(s)ds+∫t-βh(t)t‍e2k(s-t)xT(s)P4(α)x(s)ds,V4(t)=h∫-h0‍∫t+θt‍e2k(s-t)x˙T(s)P5(α)x˙(s)dsdθ+βh∫-βh0‍∫t+θt‍e2k(s-t)x˙T(s)P6(α)x˙(s)dsdθ,V5(t)=h∫-h0‍∫t+θt‍e2k(s-t)x˙T(s)P7(α)x˙(s)dsdθ+βh∫-βh0‍∫t+θt‍e2k(s-t)x˙T(s)P8(α)x˙(s)dsdθ,V6(t)=h∫-h0‍∫t+θt‍e2k(s-t)x˙T(s)P9(α)x˙(s)dsdθ+βh∫-βh0‍∫t+θt‍e2k(s-t)x˙T(s)P10(α)x˙(s)dsdθ,V7(t)=h∫-ht‍∫θ-h(θ)θ‍e2k(θ-t)[x(θ-h(θ))x˙(s)]T=h∫-ht‍∫θ-h(θ)θ‍×[R1(α)R2(α)*R3(α)]=h∫-ht‍∫θ-h(θ)θ‍×[x(θ-h(θ))x˙(s)]dsdθ+βh∫-βht‍∫θ-βh(θ)θ‍e2k(θ-t)[x(θ-βh(θ))x˙(s)]T+βh∫-βht‍∫θ-βh(θ)θ‍×[R4(α)R5(α)*R6(α)]+βh∫-βht‍∫θ-βh(θ)θ‍×[x(θ-βh(θ))x˙(s)]dsdθ.
Calculating the time derivatives ofVi(t), i=1,2,3,…,7, along the trajectory of (9), yields
(29)V˙1(t)=2xT(t)P1(α)x˙(t)=2xTP1(α)[∫t-βh(t)t‍(A(α)+B1(α)+G)x(t)=2xTP1(α)h+B2(α)x(t-h(t))-Gx(t-βh(t))=2xTP1(α)h+f(t,x(t))+g(t,x(t-h(t)))=2xTP1(α)h+w(t,x˙(t-r(t)))+C(α)x˙(t-r(t))=2xTP1(α)h-B1(α)∫t-h(t)t‍x˙(s)ds=2xTP1(α)h-G∫t-βh(t)t‍x˙(s)ds]+2kxT(t)P1(α)x(t)-2kV1(t),V˙2(t)=x˙T(t)P2(α)x˙(t)-(1-r˙(t))e-2kr(t)x˙T(t-r(t))×P2(α)x˙(t-r(t))-2kV2(t)≤x˙T(t)P2(α)x˙(t)-e-2krx˙T(t-r(t))×P2(α)x˙(t-r(t))+rdx˙T(t-r(t))P2(α)x˙(t-r(t))-2kV2(t).
The time derivative ofV3(t) is
(30)V˙3(t)=xT(t)P3(α)x(t)-(1-h˙(t))e-2kh(t)xT(t-h(t))×P3(α)x(t-h(t))+xT(t)P4(α)x(t)-(1-βh˙(t))e-2kβh(t)xT(t-βh(t))×P4(α)x(t-βh(t))-2kV3(t)≤xT(t)P3(α)x(t)-e-2khxT(t-h(t))×P3(α)x(t-h(t))+hde-2khxT(t-h(t))×P3(α)x(t-h(t))+xT(t)P4(α)x(t)-e-2αβhxT(t-βh(t))P4(α)x(t-βh(t))+βhde-2kβhxT(t-βh(t))P4(α)x(t-βh(t))-2kV3(t).
Obviously, for any scalars∈[t-h,t], we get e-2kh≤e-2k(s-t)≤1 and for any scalar s∈[t-βh,t], we obtain e-2βkh≤e-2βk(s-t)≤1. Together with Lemma 4, we obtain
(31)V˙4(t)=h2x˙T(t)P5(α)x˙(t)-h∫-h0‍e2ksx˙T(t+s)P5(α)x˙(t+s)ds+β2h2x˙T(t)P6(α)x˙(t)-βh∫-βh0‍e2βksx˙T(t+s)P6(α)x˙(t+s)ds-2kV4(t)≤h2x˙T(t)P5(α)x˙(t)-h∫t-ht‍e2k(s-t)x˙T(s)P5(α)x˙(s)ds+β2h2x˙T(t)P6(α)x˙(t)-βh∫t-βht‍e2βk(s-t)x˙T(s)P6(α)x˙(s)ds-2kV4(t)≤h2x˙T(t)P5(α)x˙(t)-he-2kh∫t-ht‍x˙T(s)P5(α)x˙(s)ds+β2h2x˙T(t)P6(α)x˙(t)-βhe-2βkh∫t-βht‍x˙T(s)P6(α)x˙(s)ds-2kV4(t),V˙5(t)≤h2x˙T(t)P7(α)x˙(t)-he-2kh∫t-ht‍x˙T(s)P7(α)x˙(s)ds+β2h2x˙T(t)P8(α)x˙(t)-βhe-2βkh∫t-βht‍x˙T(s)P8(α)x˙(s)ds-2kV5(t)≤h2x˙T(t)P7(α)x˙(t)-e-2kh∫t-ht‍x˙T(s)dsP7(α)∫t-ht‍x˙(s)ds+β2h2x˙T(t)P8(α)x˙(t)-e-2βkh∫t-βht‍x˙T(s)dsP8(α)∫t-βht‍x˙(s)ds-2kV5(t)≤h2x˙T(t)P7(α)x˙(t)-e-2kh∫t-h(t)t‍x˙T(s)dsP7(α)∫t-h(t)t‍x˙(s)ds+β2h2x˙T(t)P8(α)x˙(t)-e-2βkh∫t-βh(t)t‍x˙T(s)dsP8(α)∫t-βh(t)t‍x˙(s)ds-2αV5(t),V˙6(t)≤h2x˙T(t)P9(α)x˙(t)-e-2kh∫t-h(t)t‍x˙T(s)dsP9(α)∫t-h(t)t‍x˙(s)ds+β2h2x˙T(t)P10(α)x˙(t)-e-2βkh∫t-βh(t)t‍x˙T(s)dsP10(α)∫t-βh(t)t‍x˙(s)ds-2kV6(t)=h2x˙T(t)P9(α)x˙(t)-e-2kh[xT(t)-xT(t-h(t))]×P9(α)[x(t)-x(t-h(t))]+βh2x˙T(t)P10(α)x˙(t)-e-2βkh[xT(t)-xT(t-βh(t))]×P10(α)[x(t)-x(t-βh(t))]-2kV6(t).
Taking the time derivative ofV7(t), we obtain
(32)V˙7(t)=hh(t)xT(t-h(t))R1(α)x(t-h(t))+2hxT(t-h(t))R2(α)x(t)-2hxT(t-h(t))R2(α)x(t-h(t))+h∫t-ht‍x˙T(s)R3(α)x˙(s)ds+β2hh(t)xT(t-βh(t))R4(α)x(t-βh(t))+2βhxT(t-h(t))R5(α)x(t)-2βhxT(t-βh(t))R5(α)x(t-βh(t))+βh∫t-βht‍x˙T(s)R6(α)x˙(s)ds-2kV7(t)≤h2xT(t-h(t))R1(α)x(t-h(t))+2hxT(t-h(t))R2(α)x(t)-2hxT(t-h(t))R2(α)x(t-h(t))+h∫t-ht‍x˙T(s)R3(α)x˙(s)ds+β2h2xT(t-βh(t))R4(α)x(t-βh(t))+2βhxT(t-h(t))R5(α)x(t)-2βhxT(t-βh(t))R5(α)x(t-βh(t))+βh∫t-βht‍x˙T(s)R6(α)x˙(s)ds-2kV7(t).
By Lemma5 and the integral term of the right hand side of V˙4(t) and V˙7(t), we obtain
(33)-h∫t-ht‍x˙T(s)[e-2khP5(α)-R3(α)]x˙(s)ds-βh∫t-βht‍x˙T(s)[e-2βkhP6(α)-R6(α)]x˙(s)ds≤h[x(t)xt-h(t)]T×[M1T(α)+M1(α)-M1T(α)+M2(α)*-M2T(α)-M2(α)]×[x(t)xt-h(t)]+h2[x(t)xt-h(t)]T[M3(α)M4(α)*M5(α)]×[x(t)xt-h(t)]+βh[x(t)xt-βh(t)]T×[M6T(α)+M6(α)-M6T(α)+M7(α)*-M7T(α)-M7(α)]×[x(t)xt-βh(t)]+β2h2[x(t)xt-βh(t)]T[M8(α)M9(α)*M10(α)]×[x(t)xt-βh(t)].
From the Leibniz-Newton formula, the following equations are true for any parameter-dependent real matricesNi(α), Oi(α), i=1,2,…,10 with appropriate dimensions:
(34)2[∫t-h(t)t‍xT(t)N1T(α)+xT(t-h(t))N2T(α)+xT(t-βh(t))N3T(α)+∫t-h(t)t‍x˙T(s)dsN4T(α)+∫t-βh(t)t‍x˙T(s)dsN5T(α)+x˙T(t)N6T(α)+x˙T(t-r(t))N7T(α)+fT(t,x(t))N8T(α)+gT(t,x(t-h(t)))N9T(α)+wT(t,x˙(t-r(t)))N10T(α)∫t-h(t)t‍]×[x(t)-x(t-h(t))-∫t-h(t)t‍x˙(s)ds]=0,2[∫t-h(t)t‍xT(t)O1T(α)+xT(t-h(t))O2T(α)+xT(t-βh(t))O3T(α)+∫t-h(t)t‍x˙T(s)dsO4T(α)+∫t-βh(t)t‍x˙T(s)dsO5T(α)+x˙T(t)O6T(α)+x˙T(t-r(t))O7T(α)+fT(t,x(t))O8T(α)+gT(t,x(t-h(t)))O9T(α)+wT(t,x˙(t-r(t)))O10T(α)∫t-h(t)t‍]×[x(t)-x(t-βh(t))-∫t-βh(t)t‍x˙(s)ds]=0.
From the utilization of zero equation, the following equation is true for any parameter-dependent real matricesWi, i=1,2,…,10 with appropriate dimensions:
(35)2[∫t-h(t)t‍xT(t)W1T(α)+xT(t-h(t))W2T(α)+xT(t-βh(t))W3T(α)+∫t-h(t)t‍x˙T(s)dsW4T(α)+∫t-βh(t)t‍x˙T(s)dsW5T(α)+x˙T(t)W6T(α)+x˙T(t-r(t))W7T(α)+fT(t,x(t))W8(α)+gT(t,x(t-h(t)))W9T(α)+wT(t,x˙(t-r(t)))W10T(α)∫t-h(t)t‍]×[∫t-h(t)t‍(A(α)+B1(α))x(t)+B2(α)x(t-h(t))×hh22+C(α)x˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t)))×hh22+w(t,x˙(t-r(t)))-B1(α)∫t-h(t)t‍x˙(s)ds-x˙(t)]=0.
From (5), we obtain, for any positive real constants ϵ1, ϵ2, and ϵ3,
(36)0≤ϵ1η2xT(t)x(t)-ϵ1fT(t,x(t))f(t,x(t)),0≤ϵ2ρ2xT(t-h(t))x(t-h(t))-ϵ2gT(t,x(t-h(t)))g(t,x(t-h(t))),0≤ϵ3γ2x˙T(t-r(t))x˙(t-r(t))-ϵ3wT(t,x˙(t-r(t)))w(t,x˙(t-r(t))).
According to (29)–(36), it is straightforward to see that
(37)V˙(t)≤ζT(t)∑i=1N‍∑j=1N‍αiαj∑i,jζ(t)-2kV(t),
where ζT(t)=[xT(t), xT(t-h(t)), xT(t-βh(t)), ∫t-h(t)t‍x˙T(s)ds,∫t-βh(t)t‍x˙T(s)ds,x˙T(t), x˙T(t-r(t)), fT(t,x(t)), gT(t,x(t-h(t))), wT(t,x˙(t-r(t))] and ∑‍i,j is defined in (23). From the fact that ∑i=1N‍αi=1,
(38)∑i=1N‍αiAi∑i=1N‍αiBi=∑i=1N‍αi2AiBi+∑i=1N-1‍∑j=i+1N‍αiαj[AiBj+AjBi],(N-1)∑i=1N‍αi2-2∑i=1N-1‍∑j=i+1N‍αiαj=∑i=1N-1‍∑j=i+1N‍[αi-αj]2≥0.
It is true that if conditions (25) hold, then
(39)V˙(t)+2kV(t)≤0,∀t∈R+,
which gives
(40)V(t)≤V(0)e-2kt,∀t∈R+.
From (40), it is easy to see that
(41)λmin(P1(α))∥x(t)∥2≤V(t)≤V(0)e-2kt,(42)V(0)=∑i=17‍Vi(0),
where
(43)V1(0)=xT(0)P1(α)x(0),V2(0)=∫-r(0)0‍e2ksx˙T(s)P2(α)x˙(s)ds,V3(0)=∫-h(0)0‍e2ksxT(s)P3(α)x(s)ds+∫-βh(0)0‍e2ksxT(s)P4(α)x(s)ds,V4(0)=h∫-h0‍∫θ0‍e2ksx˙T(s)P5(α)x˙(s)dsdθ+βh∫-βh0‍∫θ0‍e2ksx˙T(s)P6(α)x˙(s)dsdθ,V5(0)=h∫-h0‍∫θ0‍e2ksx˙T(s)P7(α)x˙(s)dsdθ+βh∫-βh0‍∫θ0‍e2ksx˙T(s)P8(α)x˙(s)dsdθ,V6(0)=h∫-h0‍∫θt‍e2ksx˙T(s)P9(α)x˙(s)dsdθ+βh∫-βh0‍∫θ0‍e2ksx˙T(s)P10(α)x˙(s)dsdθ,V7(0)=h∫-h0‍∫θ-h(θ)θ‍e2kθ[x(θ-h(θ))x˙(s)]T=h∫-h0‍∫θ-h(θ)θ‍×[R1(α)R2(α)*R3(α)]=h∫-h0‍∫θ-h(θ)θ‍×[x(θ-h(θ))x˙(s)]dsdθ+βh∫-βh0‍∫θ-βh(θ)θ‍e2kθ[x(θ-βh(θ))x˙(s)]T+βh∫-βh0‍∫θ-βh(θ)θ‍×[R4(α)R5(α)*R6(α)]+βh∫-βh0‍∫θ-βh(θ)θ‍×[x(θ-βh(θ))x˙(s)]dsdθ.
From (41), we conclude that
(44)λmin(P1(α))∥x(t)∥2≤V(0)e-2kt≤Lmax[∥ϕ∥,∥φ∥]2e-2kt,
where L=λmax(P1(α))+rλmax(P2(α)) + hλmax(P3(α))+βhλmax(P4(α)) + h3λmax(P5(α)+P7(α) + P9(α))+(βh)3λmax(P6(α)+P8(α)+P10(α)) +h3λmax([R1(α)R2(α)*R3(α)]) +(βh)3λmax([R4(α)R5(α)*R6(α)]). From (44), this means that the system (1) is robustly exponentially stable. The proof of the theorem is complete.Next, we consider the following system:(45)x˙(t)=A(α)x(t)+B(α)x(t-h(t))+C(α)x˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t))),t>0;x(t)=ϕ(t),x˙(t)=ψ(t),t∈[-b,0].We introduce the following notations for later use:(46)∏i,j=[Σi,j1,1Σi,j1,2Σi,j1,3Σi,j1,4Σi,j1,5Σi,j1,6Σi,j1,7Σi,j1,8Σi,j1,9*Σi,j2,2Σi,j2,3Σi,j2,4Σi,j2,5Σi,j2,6Σi,j2,7Σi,j2,8Σi,j2,9**Σi3,3Σi,j3,4Σi3,5Σi3,6Σi,j3,7Σi3,8Σi3,9***Σi,j4,4Σi,j4,5Σi,j4,6Σi,j4,7Σi4,8Σi4,9****Σi5,5Σi5,6Σi,j5,7Σi5,8Σi5,9*****Σi6,6Σi,j6,7Σi6,8Σi6,9******Σ^i,j7,7Σi,j7,8Σi,j7,9*******Σi8,8Σi8,9********Σi9,9].Corollary 8.
For∥Ci∥<1, i=1,2,…,N and given positive real constants h, hd, r, rd, η, ρ, and β, system (45) is robustly exponentially stable with a decay rate k, if there exist symmetric positive definite matrices Pij, any appropriate dimensional matrices Rip, Mil, Nij, Oij, Wij, p=1,2,…6, l=1,2,…,10, j=1,2,…,9, i=1,2,…,N, and positive real constants ϵ1, ϵ2 such that the following symmetric linear matrix inequalities hold:
(47)[Ri1Ri2*Ri3]>0,i=1,2,…,N,[Ri4Ri5*Ri6]>0,i=1,2,…,N,[e-2khPi3-Ri3Mi1Mi2*Mi3Mi4**Mi5]≥0,i=1,2,…,N,[e-2βkhPi6-Ri6Mi6Mi7*Mi8Mi9**Mi10]≥0,i=1,2,…,N,∏i,i<-I,i=1,2,…,N,∏i,j+∏j,i<2(N-1)I,i=1,2,…,N-1,2(N-1)hiIj=i+1,i+2,…,N.
Moreover, the solutionx(t,ϕ,ψ) satisfies the inequality
(48)∥x(t,ϕ,ψ)∥≤Lλmin(P1(α))max[∥ϕ∥,∥ψ∥]e-kt,≤Lλmin(P1(α))llmax∥ϕ∥1d∀t∈R+,
where L=λmax(P1(α))+rλmax(P2(α)) +hλmax(P3(α))+βhλmax(P4(α)) +h3λmax(P5(α)+P7(α) +P9(α)) + (βh)3λmax(P6(α) +P8(α) +P10(α)) + h3λmax([R1(α)R2(α)*R3(α)]) +(βh)3λmax([R4(α)R5(α)*R6(α)]).


### 3.2.  Exponential Stability Criteria

In this section, we study the exponential stability criteria for neutral systems with time-varying delays by using the combination of linear matrix inequality (LMI) technique and Lyapunov theory method. We introduce the following notations for later use:(49)∑=[Σ11Σ12Σ13Σ14Σ15Σ16Σ17Σ18Σ19Σ1,10*Σ22Σ23Σ24Σ25Σ26Σ27Σ28Σ29Σ2,10**Σ33Σ34Σ35Σ36Σ37Σ38Σ39Σ3,10***Σ44Σ45Σ46Σ47Σ48Σ49Σ4,10****Σ55Σ56Σ57Σ58Σ59Σ5,10*****Σ66Σ67Σ68Σ69Σ6,10******Σ77Σ78Σ79Σ7,10*******Σ88Σ89Σ8,10********Σ99Σ9,10*********Σ10,10],
where
(50)Σ1,1=ATP1+B1TP1+Z1T+P1A+P1B1+Z1+P3+P4+hM1T+hM1-e-2khP9-e-2kβhP10+h2M3+βhM6+βhM6T+β2h2M8+N1T+N1+O1T+O1+ATW1+W1TA+B1TW1+W1TB1+ϵ1η2I+2kP1,Σ1,2=P1B2-hM1T+hM2+h2M4+e-2khP9+hR2-N1T+N2+O2+W1TB2+ATW2+B1TW2,Σ1,3=-Z1-βhM6T+βhM7+βh2M9+βhR5+e-2kβhP10+N3-O1T+O3+ATW3+B1TW3,Σ1,4=-P1B1-N1T+N4+O4-W1TB1+ATW4+B1TW4,Σ1,5=-Z1+N5-O1T+O5+ATW5+B1TW5,Σ1,6=N6+O6-W1T+ATW6+B1TW6,Σ1,7=P1C+N7+O7+W1TC+ATW7+B1TW7,Σ1,8=P1+N8+O8+W1T+ATW8+B1TW8,Σ1,9=P1+N9+O9+W1T+ATW9+B1TW9,Σ1,10=P1+N10+O10+W1T+ATW10+B1TW10,Σ2,2=-(1-hd)e-2khP3-hM2T-hM2+h2M5+h2R1-2hR2-e-2khP9+ϵ2ρ2I-N2T-N2+W2TB2+B2TW2,Σ2,3=-N3-O2T+B2TW3,Σ2,4=-N2T-N4-W2TB1+B2TW4,Σ2,5=-N5-O2T+B2TW5,Σ2,6=-N6-W2T+B2TW6,Σ2,7=-N7+W2TC+B2TW7,Σ2,8=-N8+W2T+B2TW8,Σ2,9=-N9T-W2T+B2TW9,Σ2,10=-N10+W2T+B2TW10,Σ3,3=-(1-βhd)e-2βkhP4-e-2βkhP10+β2h2R4-2βhR5-βhM7T-βhM7+β2h2M10-O3T-O3,Σ3,4=-N3T-O4-W3TB1,Σ3,5=-O3T-O5,Σ3,6=-O9+W3T,Σ3,7=-O7+W3TC,Σ3,8=-O8+W3T,Σ3,9=-O9+W3T,Σ3,10=-O10+W3T,Σ4,4=-e2khP7-N4T-N4+W4TB1-B1TW4,Σ4,5=-N5-O4T-B1TW5,Σ4,6=-N6-W4T-B1TW6,Σ4,7=-N7+W4TC-B1TW7,Σ4,8=-O8+W3T,Σ4,9=-O9+W3T,Σ4,10=-O10+W3T,Σ5,5=-e2βkhP8-O5T-O5,Σ5,6=-O6-W5T,Σ5,7=-O7+W5TC,Σ5,8=-O8+W5T,Σ5,9=-O9+W5T,Σ5,10=-O10+W5T,Σ6,6=P2+h2P5+β2h2P6+h2P7+β2h2P8+h2P9+β2h2P10-W6T-W6,Σ6,7=W6TC-W7,Σ6,8=W6T-W8,Σ6,9=W6T-W9,Σ6,10=W6T-W10,Σ7,7=-(1-rd)e-2krP2+ϵ3γ2I+W7TC+CTW7,Σ7,8=W7T-CTW8,Σ7,9=W7T-CTW9,Σ7,10=W7T-CTW10Σ8,8=W8T+W8-ϵ1I,Σ8,9=W8T+W9,Σ8,10=W8T+W10,Σ9,9=W9T+W9-ϵ2I,Σ9,10=W9T+W10,Σ10,10=W10T+W10-ϵ3I,Σ7,7^=-(1-rd)e-2krP2+W7TC+CTW7,Z1=P1G.IfA(α)=A, B(α)=B, and C(α)=C, where A,B,C∈Rn×n are real constant matrices, then system (1) reduces to the following system:
(51)x˙(t)=Ax(t)+Bx(t-h(t))+Cx˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t)))+w(t,x˙(t-r(t))),t>0;x(t)=ϕ(t),x˙(t)=ψ(t),t∈[-b,0].Corollary 9.
For∥C∥+γ<1 and given positive real constants h, hd, r, rd,η, ρ, γ, and β, system (51) is exponentially stable with a decay rate k, if there exist symmetric positive definite matrices Pi, any appropriate dimensional matrices Rs, Ni, Oi, Wi, Mi, G,   s=1,2,…6, i=1,2,…,10, and positive real constants ϵ1, ϵ2, and ϵ3 such that the following symmetric linear matrix inequalities hold:
(52)[R1R2*R3]>0,[R4R5*R6]>0,[e-2khP3-R3M1M2*M3M4**M5]≥0,[e-2βkhP6-R6M6M7*M8M9**M10]≥0,∑<0.
Moreover, the solutionx(t,ϕ,ψ) satisfies the inequality
(53)∥x(t,ϕ,ψ)∥≤Dλmin(P1)max[∥ϕ∥,∥ψ∥]e-kt,∀t∈R+,
where D=λmax(P1) +rλmax(P2)+ hλmax(P3)+βhλmax(P4) +h3λmax(P5+P7+P9) +(βh)3λmax(P6+P8 +P10)+h3λmax([R1R2*R3]) +(βh)3λmax([R4R5*R6]).IfA(α)=A, B(α)=B, C(α)=C, and w(t,x˙(t-r(t)))=0, where A,B,C∈Rn×n are real constant matrices, then system (1) reduces to the following system:
(54)x˙(t)=Ax(t)+Bx(t-h(t))+Cx˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t))),t>0;x(t)=ϕ(t),x˙(t)=ψ(t),t∈[-b,0].Corollary 10.
For∥C∥<1 and given positive real constants h, hd, r, rd, η, ρ, and β, system (54) is exponentially stable with a decay rate k, if there exist symmetric positive definite matrices Pj, any appropriate dimensional matrices Rs, Ni, Oi, Wi, Mj, and G, where s=1,2,…6, i=1,2,…,9, and j=1,2,…10, and positive real constants ϵ1, ϵ2 such that the following symmetric linear matrix inequalities hold:
(55)[R1R2*R3]>0,[R4R5*R6]>0,[e-2khP3-R3M1M2*M3M4**M5]≥0,[e-2βkhP6-R6M6M7*M8M9**M10]≥0,∏<0,
where
(56)∏=[Σ11Σ12Σ13Σ14Σ15Σ16Σ17Σ18Σ19*Σ22Σ23Σ24Σ25Σ26Σ27Σ28Σ29**Σ33Σ34Σ35Σ36Σ37Σ38Σ39***Σ44Σ45Σ46Σ47Σ48Σ49****Σ55Σ56Σ57Σ58Σ59*****Σ66Σ67Σ68Σ69******Σ~77Σ78Σ79*******Σ88Σ89********Σ99].
Moreover, the solutionx(t,ϕ,ψ) satisfies the inequality
(57)∥x(t,ϕ,ψ)∥≤Dλmin(P1)max[∥ϕ∥,∥ψ∥]e-kt,∀t∈R+,
where D=λmax(P1) +rλmax(P2)+hλmax(P3) +βhλmax(P4) +h3λmax(P5+ P7+P9) +(βh)3 λmax(P6 +P8+ P10)+h3λmax([R1R2*R3]) +(βh)3λmax([R4R5*R6]).


## 3.1. Robust Exponential Stability Criteria

In this section, robust exponential stability criteria dependent on mixed time-varying delays of LPD neutral delayed system (1) with nonlinear perturbations via linear matrix inequality (LMI) approach will be presented. We introduce the following notations for later use:
(23)Pj(α)=∑i=1N‍αiPij,Z1(α)=∑i=1N‍αiZi1,Rp(α)=∑i=1N‍αiRip,Nj(α)=∑i=1N‍αiNij,Oj(α)=∑i=1N‍αiOij,Wj(α)=∑i=1N‍αiWij,Mj(α)=∑i=1N‍αiMij,∑i=1N‍αi=1,αi≥0,Pij,Zi1,Rip,Wij,Nij,Oij,Mij∈Rn×n,j=1,2,…,10,p=1,2,…,6,i=1,2,…,N;∑i,j=[Σi,j1,1Σi,j1,2Σi,j1,3Σi,j1,4Σi,j1,5Σi,j1,6Σi,j1,7Σi,j1,8Σi,j1,9Σi,j1,10*Σi,j2,2Σi,j2,3Σi,j2,4Σi,j2,5Σi,j2,6Σi,j2,7Σi,j2,8Σi,j2,9Σi,j2,10**Σi3,3Σi,j3,4Σi3,5Σi3,6Σi,j3,7Σi3,8Σi3,9Σi3,10***Σi,j4,4Σi,j4,5Σi,j4,6Σi,j4,7Σi4,8Σi4,9Σi4,10****Σi5,5Σi5,6Σi,j5,7Σi5,8Σi5,9Σi5,10*****Σi6,6Σi,j6,7Σi6,8Σi6,9Σi6,10******Σi,j7,7Σi,j7,8Σi,j7,9Σi,j7,10*******Σi8,8Σi8,9Σi8,10********Σi9,9Σi9,10*********Σi10,10],
where
(24)Σi,j1,1=AiTPj1+Bi1TPj1+Pi1Aj+Pi1Bj1+Zi1+Zi1T+Pi3+Pi4+hMi1T+hMi1-e-2khPi9-e-2kβhPi10+h2Mi3+βhMi6+βhMi6T+β2h2Mi8+Ni1T+Ni1+Oi1T+Oi1+AiTWj1+Wi1TAj+Bi1TWj1+Wi1TBj1+ϵ1η2I+2kPi1,Σi,j1,2=Pi1Bj2-hMi1T+hMi2+h2Mi4+e-2khPi9+hRi2-Ni1T+Ni2+Oi2+Wi1TBj2+AiTWj2+Bi1TWj2,Σi,j1,3=-Zi1-βhMi6T+βhMi7+βh2Mi9+βhRi5+e-2kβhPi10+Ni3-Oi1T+Oi3+AiTWj3+Bi1TWj3,Σi,j1,4=-Pi1Bj1-Ni1T+Ni4+Oi4-Wi1TBj1+AiTWj4+Bi1TWj4,Σi,j1,5=-Zi1+Ni5-Oi1T+Oi5+AiTWj5+Bi1TWj5,Σi,j1,6=Ni6+Oi6-Wi1T+AiTWj6+Bi1TWj6,Σi,j1,7=Pi1Cj+Ni7+Oi7+Wi1TCj+AiTWj7+Bi1TWj7,Σi,j1,8=Pi1+Ni8+Oi8+Wi1T+AiTWj8+Bi1TWj8,Σi,j1,9=Pi1+Ni9+Oi9+Wi1T+AiTWj9+Bi1TWj9,Σi,j1,10=Pi1+Ni10+Oi10+Wi1T+AiTWj10+Bi1TWj10,Σi,j2,2=-(1-hd)e-2khPi3-hMi2T-hMi2+h2Mi5+h2Ri1-2hRi2-e-2khPi9+ϵ2ρ2I-Ni2T-Ni2+Wi2TBj2+Bi2TWj2,Σi,j2,3=-Ni3-Oi2T+Bi2TWj3,Σi,j2,4=-Ni2T-Ni4-Wi2TBj1+Bi2TWj4,Σi,j2,5=-Ni5-Oi2T+Bi2TWj5,Σi,j2,6=-Ni6-Wi2T+Bi2TWj6,Σi,j2,7=-Ni7+Wi2TCj+Bi2TWj7,Σi,j2,8=-Ni8+Wi2T+Bi2TWj8,Σi,j2,9=-Ni9T-Wi2T+Bi2TWj9,Σi,j2,10=-Ni10+Wi2T+Bi2TWj10,Σi3,3=-(1-βhd)e-2βkhPi4-e-2βkhPi10+β2h2Ri4-2βhRi5-βhMi7T-βhMi7+β2h2Mi10-Oi3T-Oi3,Σi,j3,4=-Ni3T-Oi4-Wi3TBj1,Σi3,5=-Oi3T-Oi5,Σi3,6=-Oi9+Wi3T,Σi3,7=-Oi7+Wi3TCj,Σi3,8=-Oi8+Wi3T,Σi3,9=-Oi9+Wi3T,Σi3,10=-Oi10+Wi3T,Σi,j4,4=-e2khPi7-Ni4T-Ni4+Wi4TBj1-Bi1TWj4,Σi,j4,5=-Ni5-Oi4T-Bi1TWj5,Σi,j4,6=-Ni6-Wi4T-Bi1TWj6,Σi,j4,7=-Ni7+Wi4TCj-Bi1TWj7,Σi4,8=-Oi8+Wi3T,Σi4,9=-Oi9+Wi3T,Σi4,10=-Oi10+Wi3T,Σi5,5=-e2βkhPi8-Oi5T-Oi5,Σi5,6=-Oi6-Wi5T,Σi,j5,7=-Oi7+Wi5TCj,Σi5,8=-Oi8+Wi5T,Σi5,9=-Oi9+Wi5T,Σi5,10=-Oi10+Wi5T,Σi6,6=Pi2+h2Pi5+β2h2Pi6+h2Pi7+β2h2Pi8+h2Pi9+β2h2Pi10-Wi6T-Wi6,Σi,j6,7=Wi6TCj-Wi7,Σi6,8=Wi6T-Wi8,Σi6,9=Wi6T-Wi9,Σi6,10=Wi6T-Wi10,Σi,j7,7=-(1-rd)e-2krPi2+ϵ3γ2I+Wi7TCj+CiTWj7,Σi,j7,8=Wi7T-CiTWj8,Σi,j7,9=Wi7T-CiTWj9,Σi,j7,10=Wi7T-CiTWj10,Σi8,8=Wi8T+Wi8-ϵ1I,Σi8,9=Wi8T+Wi9,Σi8,10=Wi8T+Wi10,Σi9,9=Wi9T+Wi9-ϵ2I,Σi9,10=Wi9T+Wi10,Σi10,10=Wi10T+Wi10-ϵ3I,Σi7,7^=-(1-rd)e2krPi2+Wi7TCj+CiTWj7,Zi1=Pi1G.Theorem 7.
For∥Ci∥+γ<1,  i=1,2,…,N and given positive real constants h, hd, r, rd, η, ρ, γ, and β, system (1) is robustly exponentially stable with a decay rate k, if there exist symmetric positive definite matrices Pij, any appropriate dimensional matrices Rip, Nij, Oij, Wij, Mij, G, p=1,2,…6, j=1,2,…,10, i=1,2,…,N, and positive real constants ϵ1, ϵ2, and ϵ3 such that the following symmetric linear matrix inequalities hold:
(25)[Ri1Ri2*Ri3]>0,i=1,2,…,N,[Ri4Ri5*Ri6]>0,i=1,2,…,N,[e-2khPi3-Ri3Mi1Mi2*Mi3Mi4**Mi5]≥0,i=1,2,…,N,[e-2βkhPi6-Ri6Mi6Mi7*Mi8Mi9**Mi10]≥0,i=1,2,…,N,∑i,i<-I,i=1,2,…,N,∑i,j+∑j,i<2(N-1)I,i=1,2,…,N-1,<2(N-1)Ij=i+1,i+2,…,N.
Moreover, the solutionx(t,ϕ,ψ) satisfies the inequality
(26)∥x(t,ϕ,ψ)∥≤Lλmin(P1(α))max[∥ϕ∥,∥ψ∥]e-kt,≤Lλmin(P1(α))max∥ϕ∥hlh∀t∈R+,
where L=λmax(P1(α))+rλmax(P2(α))+hλmax(P3(α)) + βhλmax(P4(α))+h3λmax(P5(α) + P7(α) + P9(α)) + (βh)3λmax(P6(α) + P8(α)+P10(α))+h3λmax([R1(α)R2(α)*R3(α)]) + (βh)3λmax([R4(α)R5(α)*R6(α)]).Proof.
Choose a parameter-dependent Lyapunov-Krasovskii functional candidate for system (9) as
(27)V(t)=∑i=17‍Vi(t),
where
(28)V1(t)=xT(t)P1(α)x(t),V2(t)=∫t-r(t)t‍e2k(s-t)x˙T(s)P2(α)x˙(s)ds,V3(t)=∫t-h(t)t‍e2k(s-t)xT(s)P3(α)x(s)ds+∫t-βh(t)t‍e2k(s-t)xT(s)P4(α)x(s)ds,V4(t)=h∫-h0‍∫t+θt‍e2k(s-t)x˙T(s)P5(α)x˙(s)dsdθ+βh∫-βh0‍∫t+θt‍e2k(s-t)x˙T(s)P6(α)x˙(s)dsdθ,V5(t)=h∫-h0‍∫t+θt‍e2k(s-t)x˙T(s)P7(α)x˙(s)dsdθ+βh∫-βh0‍∫t+θt‍e2k(s-t)x˙T(s)P8(α)x˙(s)dsdθ,V6(t)=h∫-h0‍∫t+θt‍e2k(s-t)x˙T(s)P9(α)x˙(s)dsdθ+βh∫-βh0‍∫t+θt‍e2k(s-t)x˙T(s)P10(α)x˙(s)dsdθ,V7(t)=h∫-ht‍∫θ-h(θ)θ‍e2k(θ-t)[x(θ-h(θ))x˙(s)]T=h∫-ht‍∫θ-h(θ)θ‍×[R1(α)R2(α)*R3(α)]=h∫-ht‍∫θ-h(θ)θ‍×[x(θ-h(θ))x˙(s)]dsdθ+βh∫-βht‍∫θ-βh(θ)θ‍e2k(θ-t)[x(θ-βh(θ))x˙(s)]T+βh∫-βht‍∫θ-βh(θ)θ‍×[R4(α)R5(α)*R6(α)]+βh∫-βht‍∫θ-βh(θ)θ‍×[x(θ-βh(θ))x˙(s)]dsdθ.
Calculating the time derivatives ofVi(t), i=1,2,3,…,7, along the trajectory of (9), yields
(29)V˙1(t)=2xT(t)P1(α)x˙(t)=2xTP1(α)[∫t-βh(t)t‍(A(α)+B1(α)+G)x(t)=2xTP1(α)h+B2(α)x(t-h(t))-Gx(t-βh(t))=2xTP1(α)h+f(t,x(t))+g(t,x(t-h(t)))=2xTP1(α)h+w(t,x˙(t-r(t)))+C(α)x˙(t-r(t))=2xTP1(α)h-B1(α)∫t-h(t)t‍x˙(s)ds=2xTP1(α)h-G∫t-βh(t)t‍x˙(s)ds]+2kxT(t)P1(α)x(t)-2kV1(t),V˙2(t)=x˙T(t)P2(α)x˙(t)-(1-r˙(t))e-2kr(t)x˙T(t-r(t))×P2(α)x˙(t-r(t))-2kV2(t)≤x˙T(t)P2(α)x˙(t)-e-2krx˙T(t-r(t))×P2(α)x˙(t-r(t))+rdx˙T(t-r(t))P2(α)x˙(t-r(t))-2kV2(t).
The time derivative ofV3(t) is
(30)V˙3(t)=xT(t)P3(α)x(t)-(1-h˙(t))e-2kh(t)xT(t-h(t))×P3(α)x(t-h(t))+xT(t)P4(α)x(t)-(1-βh˙(t))e-2kβh(t)xT(t-βh(t))×P4(α)x(t-βh(t))-2kV3(t)≤xT(t)P3(α)x(t)-e-2khxT(t-h(t))×P3(α)x(t-h(t))+hde-2khxT(t-h(t))×P3(α)x(t-h(t))+xT(t)P4(α)x(t)-e-2αβhxT(t-βh(t))P4(α)x(t-βh(t))+βhde-2kβhxT(t-βh(t))P4(α)x(t-βh(t))-2kV3(t).
Obviously, for any scalars∈[t-h,t], we get e-2kh≤e-2k(s-t)≤1 and for any scalar s∈[t-βh,t], we obtain e-2βkh≤e-2βk(s-t)≤1. Together with Lemma 4, we obtain
(31)V˙4(t)=h2x˙T(t)P5(α)x˙(t)-h∫-h0‍e2ksx˙T(t+s)P5(α)x˙(t+s)ds+β2h2x˙T(t)P6(α)x˙(t)-βh∫-βh0‍e2βksx˙T(t+s)P6(α)x˙(t+s)ds-2kV4(t)≤h2x˙T(t)P5(α)x˙(t)-h∫t-ht‍e2k(s-t)x˙T(s)P5(α)x˙(s)ds+β2h2x˙T(t)P6(α)x˙(t)-βh∫t-βht‍e2βk(s-t)x˙T(s)P6(α)x˙(s)ds-2kV4(t)≤h2x˙T(t)P5(α)x˙(t)-he-2kh∫t-ht‍x˙T(s)P5(α)x˙(s)ds+β2h2x˙T(t)P6(α)x˙(t)-βhe-2βkh∫t-βht‍x˙T(s)P6(α)x˙(s)ds-2kV4(t),V˙5(t)≤h2x˙T(t)P7(α)x˙(t)-he-2kh∫t-ht‍x˙T(s)P7(α)x˙(s)ds+β2h2x˙T(t)P8(α)x˙(t)-βhe-2βkh∫t-βht‍x˙T(s)P8(α)x˙(s)ds-2kV5(t)≤h2x˙T(t)P7(α)x˙(t)-e-2kh∫t-ht‍x˙T(s)dsP7(α)∫t-ht‍x˙(s)ds+β2h2x˙T(t)P8(α)x˙(t)-e-2βkh∫t-βht‍x˙T(s)dsP8(α)∫t-βht‍x˙(s)ds-2kV5(t)≤h2x˙T(t)P7(α)x˙(t)-e-2kh∫t-h(t)t‍x˙T(s)dsP7(α)∫t-h(t)t‍x˙(s)ds+β2h2x˙T(t)P8(α)x˙(t)-e-2βkh∫t-βh(t)t‍x˙T(s)dsP8(α)∫t-βh(t)t‍x˙(s)ds-2αV5(t),V˙6(t)≤h2x˙T(t)P9(α)x˙(t)-e-2kh∫t-h(t)t‍x˙T(s)dsP9(α)∫t-h(t)t‍x˙(s)ds+β2h2x˙T(t)P10(α)x˙(t)-e-2βkh∫t-βh(t)t‍x˙T(s)dsP10(α)∫t-βh(t)t‍x˙(s)ds-2kV6(t)=h2x˙T(t)P9(α)x˙(t)-e-2kh[xT(t)-xT(t-h(t))]×P9(α)[x(t)-x(t-h(t))]+βh2x˙T(t)P10(α)x˙(t)-e-2βkh[xT(t)-xT(t-βh(t))]×P10(α)[x(t)-x(t-βh(t))]-2kV6(t).
Taking the time derivative ofV7(t), we obtain
(32)V˙7(t)=hh(t)xT(t-h(t))R1(α)x(t-h(t))+2hxT(t-h(t))R2(α)x(t)-2hxT(t-h(t))R2(α)x(t-h(t))+h∫t-ht‍x˙T(s)R3(α)x˙(s)ds+β2hh(t)xT(t-βh(t))R4(α)x(t-βh(t))+2βhxT(t-h(t))R5(α)x(t)-2βhxT(t-βh(t))R5(α)x(t-βh(t))+βh∫t-βht‍x˙T(s)R6(α)x˙(s)ds-2kV7(t)≤h2xT(t-h(t))R1(α)x(t-h(t))+2hxT(t-h(t))R2(α)x(t)-2hxT(t-h(t))R2(α)x(t-h(t))+h∫t-ht‍x˙T(s)R3(α)x˙(s)ds+β2h2xT(t-βh(t))R4(α)x(t-βh(t))+2βhxT(t-h(t))R5(α)x(t)-2βhxT(t-βh(t))R5(α)x(t-βh(t))+βh∫t-βht‍x˙T(s)R6(α)x˙(s)ds-2kV7(t).
By Lemma5 and the integral term of the right hand side of V˙4(t) and V˙7(t), we obtain
(33)-h∫t-ht‍x˙T(s)[e-2khP5(α)-R3(α)]x˙(s)ds-βh∫t-βht‍x˙T(s)[e-2βkhP6(α)-R6(α)]x˙(s)ds≤h[x(t)xt-h(t)]T×[M1T(α)+M1(α)-M1T(α)+M2(α)*-M2T(α)-M2(α)]×[x(t)xt-h(t)]+h2[x(t)xt-h(t)]T[M3(α)M4(α)*M5(α)]×[x(t)xt-h(t)]+βh[x(t)xt-βh(t)]T×[M6T(α)+M6(α)-M6T(α)+M7(α)*-M7T(α)-M7(α)]×[x(t)xt-βh(t)]+β2h2[x(t)xt-βh(t)]T[M8(α)M9(α)*M10(α)]×[x(t)xt-βh(t)].
From the Leibniz-Newton formula, the following equations are true for any parameter-dependent real matricesNi(α), Oi(α), i=1,2,…,10 with appropriate dimensions:
(34)2[∫t-h(t)t‍xT(t)N1T(α)+xT(t-h(t))N2T(α)+xT(t-βh(t))N3T(α)+∫t-h(t)t‍x˙T(s)dsN4T(α)+∫t-βh(t)t‍x˙T(s)dsN5T(α)+x˙T(t)N6T(α)+x˙T(t-r(t))N7T(α)+fT(t,x(t))N8T(α)+gT(t,x(t-h(t)))N9T(α)+wT(t,x˙(t-r(t)))N10T(α)∫t-h(t)t‍]×[x(t)-x(t-h(t))-∫t-h(t)t‍x˙(s)ds]=0,2[∫t-h(t)t‍xT(t)O1T(α)+xT(t-h(t))O2T(α)+xT(t-βh(t))O3T(α)+∫t-h(t)t‍x˙T(s)dsO4T(α)+∫t-βh(t)t‍x˙T(s)dsO5T(α)+x˙T(t)O6T(α)+x˙T(t-r(t))O7T(α)+fT(t,x(t))O8T(α)+gT(t,x(t-h(t)))O9T(α)+wT(t,x˙(t-r(t)))O10T(α)∫t-h(t)t‍]×[x(t)-x(t-βh(t))-∫t-βh(t)t‍x˙(s)ds]=0.
From the utilization of zero equation, the following equation is true for any parameter-dependent real matricesWi, i=1,2,…,10 with appropriate dimensions:
(35)2[∫t-h(t)t‍xT(t)W1T(α)+xT(t-h(t))W2T(α)+xT(t-βh(t))W3T(α)+∫t-h(t)t‍x˙T(s)dsW4T(α)+∫t-βh(t)t‍x˙T(s)dsW5T(α)+x˙T(t)W6T(α)+x˙T(t-r(t))W7T(α)+fT(t,x(t))W8(α)+gT(t,x(t-h(t)))W9T(α)+wT(t,x˙(t-r(t)))W10T(α)∫t-h(t)t‍]×[∫t-h(t)t‍(A(α)+B1(α))x(t)+B2(α)x(t-h(t))×hh22+C(α)x˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t)))×hh22+w(t,x˙(t-r(t)))-B1(α)∫t-h(t)t‍x˙(s)ds-x˙(t)]=0.
From (5), we obtain, for any positive real constants ϵ1, ϵ2, and ϵ3,
(36)0≤ϵ1η2xT(t)x(t)-ϵ1fT(t,x(t))f(t,x(t)),0≤ϵ2ρ2xT(t-h(t))x(t-h(t))-ϵ2gT(t,x(t-h(t)))g(t,x(t-h(t))),0≤ϵ3γ2x˙T(t-r(t))x˙(t-r(t))-ϵ3wT(t,x˙(t-r(t)))w(t,x˙(t-r(t))).
According to (29)–(36), it is straightforward to see that
(37)V˙(t)≤ζT(t)∑i=1N‍∑j=1N‍αiαj∑i,jζ(t)-2kV(t),
where ζT(t)=[xT(t), xT(t-h(t)), xT(t-βh(t)), ∫t-h(t)t‍x˙T(s)ds,∫t-βh(t)t‍x˙T(s)ds,x˙T(t), x˙T(t-r(t)), fT(t,x(t)), gT(t,x(t-h(t))), wT(t,x˙(t-r(t))] and ∑‍i,j is defined in (23). From the fact that ∑i=1N‍αi=1,
(38)∑i=1N‍αiAi∑i=1N‍αiBi=∑i=1N‍αi2AiBi+∑i=1N-1‍∑j=i+1N‍αiαj[AiBj+AjBi],(N-1)∑i=1N‍αi2-2∑i=1N-1‍∑j=i+1N‍αiαj=∑i=1N-1‍∑j=i+1N‍[αi-αj]2≥0.
It is true that if conditions (25) hold, then
(39)V˙(t)+2kV(t)≤0,∀t∈R+,
which gives
(40)V(t)≤V(0)e-2kt,∀t∈R+.
From (40), it is easy to see that
(41)λmin(P1(α))∥x(t)∥2≤V(t)≤V(0)e-2kt,(42)V(0)=∑i=17‍Vi(0),
where
(43)V1(0)=xT(0)P1(α)x(0),V2(0)=∫-r(0)0‍e2ksx˙T(s)P2(α)x˙(s)ds,V3(0)=∫-h(0)0‍e2ksxT(s)P3(α)x(s)ds+∫-βh(0)0‍e2ksxT(s)P4(α)x(s)ds,V4(0)=h∫-h0‍∫θ0‍e2ksx˙T(s)P5(α)x˙(s)dsdθ+βh∫-βh0‍∫θ0‍e2ksx˙T(s)P6(α)x˙(s)dsdθ,V5(0)=h∫-h0‍∫θ0‍e2ksx˙T(s)P7(α)x˙(s)dsdθ+βh∫-βh0‍∫θ0‍e2ksx˙T(s)P8(α)x˙(s)dsdθ,V6(0)=h∫-h0‍∫θt‍e2ksx˙T(s)P9(α)x˙(s)dsdθ+βh∫-βh0‍∫θ0‍e2ksx˙T(s)P10(α)x˙(s)dsdθ,V7(0)=h∫-h0‍∫θ-h(θ)θ‍e2kθ[x(θ-h(θ))x˙(s)]T=h∫-h0‍∫θ-h(θ)θ‍×[R1(α)R2(α)*R3(α)]=h∫-h0‍∫θ-h(θ)θ‍×[x(θ-h(θ))x˙(s)]dsdθ+βh∫-βh0‍∫θ-βh(θ)θ‍e2kθ[x(θ-βh(θ))x˙(s)]T+βh∫-βh0‍∫θ-βh(θ)θ‍×[R4(α)R5(α)*R6(α)]+βh∫-βh0‍∫θ-βh(θ)θ‍×[x(θ-βh(θ))x˙(s)]dsdθ.
From (41), we conclude that
(44)λmin(P1(α))∥x(t)∥2≤V(0)e-2kt≤Lmax[∥ϕ∥,∥φ∥]2e-2kt,
where L=λmax(P1(α))+rλmax(P2(α)) + hλmax(P3(α))+βhλmax(P4(α)) + h3λmax(P5(α)+P7(α) + P9(α))+(βh)3λmax(P6(α)+P8(α)+P10(α)) +h3λmax([R1(α)R2(α)*R3(α)]) +(βh)3λmax([R4(α)R5(α)*R6(α)]). From (44), this means that the system (1) is robustly exponentially stable. The proof of the theorem is complete.Next, we consider the following system:(45)x˙(t)=A(α)x(t)+B(α)x(t-h(t))+C(α)x˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t))),t>0;x(t)=ϕ(t),x˙(t)=ψ(t),t∈[-b,0].We introduce the following notations for later use:(46)∏i,j=[Σi,j1,1Σi,j1,2Σi,j1,3Σi,j1,4Σi,j1,5Σi,j1,6Σi,j1,7Σi,j1,8Σi,j1,9*Σi,j2,2Σi,j2,3Σi,j2,4Σi,j2,5Σi,j2,6Σi,j2,7Σi,j2,8Σi,j2,9**Σi3,3Σi,j3,4Σi3,5Σi3,6Σi,j3,7Σi3,8Σi3,9***Σi,j4,4Σi,j4,5Σi,j4,6Σi,j4,7Σi4,8Σi4,9****Σi5,5Σi5,6Σi,j5,7Σi5,8Σi5,9*****Σi6,6Σi,j6,7Σi6,8Σi6,9******Σ^i,j7,7Σi,j7,8Σi,j7,9*******Σi8,8Σi8,9********Σi9,9].Corollary 8.
For∥Ci∥<1, i=1,2,…,N and given positive real constants h, hd, r, rd, η, ρ, and β, system (45) is robustly exponentially stable with a decay rate k, if there exist symmetric positive definite matrices Pij, any appropriate dimensional matrices Rip, Mil, Nij, Oij, Wij, p=1,2,…6, l=1,2,…,10, j=1,2,…,9, i=1,2,…,N, and positive real constants ϵ1, ϵ2 such that the following symmetric linear matrix inequalities hold:
(47)[Ri1Ri2*Ri3]>0,i=1,2,…,N,[Ri4Ri5*Ri6]>0,i=1,2,…,N,[e-2khPi3-Ri3Mi1Mi2*Mi3Mi4**Mi5]≥0,i=1,2,…,N,[e-2βkhPi6-Ri6Mi6Mi7*Mi8Mi9**Mi10]≥0,i=1,2,…,N,∏i,i<-I,i=1,2,…,N,∏i,j+∏j,i<2(N-1)I,i=1,2,…,N-1,2(N-1)hiIj=i+1,i+2,…,N.
Moreover, the solutionx(t,ϕ,ψ) satisfies the inequality
(48)∥x(t,ϕ,ψ)∥≤Lλmin(P1(α))max[∥ϕ∥,∥ψ∥]e-kt,≤Lλmin(P1(α))llmax∥ϕ∥1d∀t∈R+,
where L=λmax(P1(α))+rλmax(P2(α)) +hλmax(P3(α))+βhλmax(P4(α)) +h3λmax(P5(α)+P7(α) +P9(α)) + (βh)3λmax(P6(α) +P8(α) +P10(α)) + h3λmax([R1(α)R2(α)*R3(α)]) +(βh)3λmax([R4(α)R5(α)*R6(α)]).


## 3.2.  Exponential Stability Criteria

In this section, we study the exponential stability criteria for neutral systems with time-varying delays by using the combination of linear matrix inequality (LMI) technique and Lyapunov theory method. We introduce the following notations for later use:(49)∑=[Σ11Σ12Σ13Σ14Σ15Σ16Σ17Σ18Σ19Σ1,10*Σ22Σ23Σ24Σ25Σ26Σ27Σ28Σ29Σ2,10**Σ33Σ34Σ35Σ36Σ37Σ38Σ39Σ3,10***Σ44Σ45Σ46Σ47Σ48Σ49Σ4,10****Σ55Σ56Σ57Σ58Σ59Σ5,10*****Σ66Σ67Σ68Σ69Σ6,10******Σ77Σ78Σ79Σ7,10*******Σ88Σ89Σ8,10********Σ99Σ9,10*********Σ10,10],
where
(50)Σ1,1=ATP1+B1TP1+Z1T+P1A+P1B1+Z1+P3+P4+hM1T+hM1-e-2khP9-e-2kβhP10+h2M3+βhM6+βhM6T+β2h2M8+N1T+N1+O1T+O1+ATW1+W1TA+B1TW1+W1TB1+ϵ1η2I+2kP1,Σ1,2=P1B2-hM1T+hM2+h2M4+e-2khP9+hR2-N1T+N2+O2+W1TB2+ATW2+B1TW2,Σ1,3=-Z1-βhM6T+βhM7+βh2M9+βhR5+e-2kβhP10+N3-O1T+O3+ATW3+B1TW3,Σ1,4=-P1B1-N1T+N4+O4-W1TB1+ATW4+B1TW4,Σ1,5=-Z1+N5-O1T+O5+ATW5+B1TW5,Σ1,6=N6+O6-W1T+ATW6+B1TW6,Σ1,7=P1C+N7+O7+W1TC+ATW7+B1TW7,Σ1,8=P1+N8+O8+W1T+ATW8+B1TW8,Σ1,9=P1+N9+O9+W1T+ATW9+B1TW9,Σ1,10=P1+N10+O10+W1T+ATW10+B1TW10,Σ2,2=-(1-hd)e-2khP3-hM2T-hM2+h2M5+h2R1-2hR2-e-2khP9+ϵ2ρ2I-N2T-N2+W2TB2+B2TW2,Σ2,3=-N3-O2T+B2TW3,Σ2,4=-N2T-N4-W2TB1+B2TW4,Σ2,5=-N5-O2T+B2TW5,Σ2,6=-N6-W2T+B2TW6,Σ2,7=-N7+W2TC+B2TW7,Σ2,8=-N8+W2T+B2TW8,Σ2,9=-N9T-W2T+B2TW9,Σ2,10=-N10+W2T+B2TW10,Σ3,3=-(1-βhd)e-2βkhP4-e-2βkhP10+β2h2R4-2βhR5-βhM7T-βhM7+β2h2M10-O3T-O3,Σ3,4=-N3T-O4-W3TB1,Σ3,5=-O3T-O5,Σ3,6=-O9+W3T,Σ3,7=-O7+W3TC,Σ3,8=-O8+W3T,Σ3,9=-O9+W3T,Σ3,10=-O10+W3T,Σ4,4=-e2khP7-N4T-N4+W4TB1-B1TW4,Σ4,5=-N5-O4T-B1TW5,Σ4,6=-N6-W4T-B1TW6,Σ4,7=-N7+W4TC-B1TW7,Σ4,8=-O8+W3T,Σ4,9=-O9+W3T,Σ4,10=-O10+W3T,Σ5,5=-e2βkhP8-O5T-O5,Σ5,6=-O6-W5T,Σ5,7=-O7+W5TC,Σ5,8=-O8+W5T,Σ5,9=-O9+W5T,Σ5,10=-O10+W5T,Σ6,6=P2+h2P5+β2h2P6+h2P7+β2h2P8+h2P9+β2h2P10-W6T-W6,Σ6,7=W6TC-W7,Σ6,8=W6T-W8,Σ6,9=W6T-W9,Σ6,10=W6T-W10,Σ7,7=-(1-rd)e-2krP2+ϵ3γ2I+W7TC+CTW7,Σ7,8=W7T-CTW8,Σ7,9=W7T-CTW9,Σ7,10=W7T-CTW10Σ8,8=W8T+W8-ϵ1I,Σ8,9=W8T+W9,Σ8,10=W8T+W10,Σ9,9=W9T+W9-ϵ2I,Σ9,10=W9T+W10,Σ10,10=W10T+W10-ϵ3I,Σ7,7^=-(1-rd)e-2krP2+W7TC+CTW7,Z1=P1G.IfA(α)=A, B(α)=B, and C(α)=C, where A,B,C∈Rn×n are real constant matrices, then system (1) reduces to the following system:
(51)x˙(t)=Ax(t)+Bx(t-h(t))+Cx˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t)))+w(t,x˙(t-r(t))),t>0;x(t)=ϕ(t),x˙(t)=ψ(t),t∈[-b,0].Corollary 9.
For∥C∥+γ<1 and given positive real constants h, hd, r, rd,η, ρ, γ, and β, system (51) is exponentially stable with a decay rate k, if there exist symmetric positive definite matrices Pi, any appropriate dimensional matrices Rs, Ni, Oi, Wi, Mi, G,   s=1,2,…6, i=1,2,…,10, and positive real constants ϵ1, ϵ2, and ϵ3 such that the following symmetric linear matrix inequalities hold:
(52)[R1R2*R3]>0,[R4R5*R6]>0,[e-2khP3-R3M1M2*M3M4**M5]≥0,[e-2βkhP6-R6M6M7*M8M9**M10]≥0,∑<0.
Moreover, the solutionx(t,ϕ,ψ) satisfies the inequality
(53)∥x(t,ϕ,ψ)∥≤Dλmin(P1)max[∥ϕ∥,∥ψ∥]e-kt,∀t∈R+,
where D=λmax(P1) +rλmax(P2)+ hλmax(P3)+βhλmax(P4) +h3λmax(P5+P7+P9) +(βh)3λmax(P6+P8 +P10)+h3λmax([R1R2*R3]) +(βh)3λmax([R4R5*R6]).IfA(α)=A, B(α)=B, C(α)=C, and w(t,x˙(t-r(t)))=0, where A,B,C∈Rn×n are real constant matrices, then system (1) reduces to the following system:
(54)x˙(t)=Ax(t)+Bx(t-h(t))+Cx˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t))),t>0;x(t)=ϕ(t),x˙(t)=ψ(t),t∈[-b,0].Corollary 10.
For∥C∥<1 and given positive real constants h, hd, r, rd, η, ρ, and β, system (54) is exponentially stable with a decay rate k, if there exist symmetric positive definite matrices Pj, any appropriate dimensional matrices Rs, Ni, Oi, Wi, Mj, and G, where s=1,2,…6, i=1,2,…,9, and j=1,2,…10, and positive real constants ϵ1, ϵ2 such that the following symmetric linear matrix inequalities hold:
(55)[R1R2*R3]>0,[R4R5*R6]>0,[e-2khP3-R3M1M2*M3M4**M5]≥0,[e-2βkhP6-R6M6M7*M8M9**M10]≥0,∏<0,
where
(56)∏=[Σ11Σ12Σ13Σ14Σ15Σ16Σ17Σ18Σ19*Σ22Σ23Σ24Σ25Σ26Σ27Σ28Σ29**Σ33Σ34Σ35Σ36Σ37Σ38Σ39***Σ44Σ45Σ46Σ47Σ48Σ49****Σ55Σ56Σ57Σ58Σ59*****Σ66Σ67Σ68Σ69******Σ~77Σ78Σ79*******Σ88Σ89********Σ99].
Moreover, the solutionx(t,ϕ,ψ) satisfies the inequality
(57)∥x(t,ϕ,ψ)∥≤Dλmin(P1)max[∥ϕ∥,∥ψ∥]e-kt,∀t∈R+,
where D=λmax(P1) +rλmax(P2)+hλmax(P3) +βhλmax(P4) +h3λmax(P5+ P7+P9) +(βh)3 λmax(P6 +P8+ P10)+h3λmax([R1R2*R3]) +(βh)3λmax([R4R5*R6]).


## 4. Numerical Examples

In order to show the effectiveness of the approaches presented in Section3, three numerical examples are provided.Example 1.
Consider the robust exponential stability of system (1) with
(58)A(α)=α1[-200-1]+α2[-200-0.8],B(α)=α1[-10-1-0.8]+α2[-10-1-1],C(α)=α1[0.1000.1]+α2[0.2000.1],f(t,x(t))=[0.05sin(t)x1(t)0.05cos(t)x2(t)],g(t,x(t-h(t)))=[0.1cos(t)2x1(t-h(t))0.1sin(t)2x2(t-h(t))],g(t,x(t-h(t)))hhhhh=h(t)=0.9sin2(t2),w(t,x˙(t-r(t)))=[0.1cos(t)x˙1(t-r(t))0.1sin(t)x˙2(t-r(t))],w(t,x˙(t-r(t)))hhhhhh1=r(t)=cos2(t4),
where x(t)∈R2. It is easy to see that h=0.9, hd=0.45, r=1, rd=0.25, η=0.05, ρ=0.1, γ=0.1, and given rate of convergence k=0.1. Decompose matrix B(α) as follows: B(α)=B1(α)+B2(α), where
(59)B1(α)=α1[-0.830-1-0.63]+α2[-0.830-1-0.83],B2(α)=α1[-0.1700-0.17]+α2[-0.1700-0.17].
The numerical solutionsx1(t) and x2(t) of system (1) with (58)-(59) are plotted in Figure 1 where the states x1(t) and x2(t) are attracted to the stable origin.
Solution. By using the LMI Toolbox in MATLAB (with accuracy 0.01), we use conditions (25) in Theorem 7 for system (1) with (58)-(59). The solutions of LMIs verify as follows:
(60)P11=103×[3.3091-0.0455-0.04550.7941],P21=103×[3.3091-0.0455-0.04550.7941],P12=103×[4.3739-0.3929-0.39292.7610],P22=103×[4.7896-0.4618-0.46182.8307],P13=104×[1.59760.13840.13840.3189],P23=104×[1.59760.13840.13840.3189],P14=103×[9.8670-0.0635-0.06354.5726],P24=103×[8.8868-0.2421-0.24213.2473],P15=103×[1.3009-0.1372-0.13720.6505],P25=103×[1.0898-0.0980-0.09800.3986],P16=104×[2.8885-0.0338-0.03382.6935],P26=104×[2.8526-0.0449-0.04492.4922],P17=103×[2.6064-0.3045-0.30452.5259],P27=103×[2.3663-0.1722-0.17222.5747],P18=104×[1.4801-0.0277-0.02771.4063],P28=104×[1.4740-0.0862-0.08621.4207],P19=103×[2.6064-0.3045-0.30452.5259],P29=103×[2.3663-0.1722-0.17222.5747],P110=104×[1.4570-0.0000-0.00001.2074],P210=104×[1.43540.00190.00191.0978],M11=103×[-9.1839-1.2537-1.2537-2.0706],M21=104×[-1.0642-0.1844-0.1844-0.2293],M12=103×[8.84591.15031.15032.0066],M22=104×[1.02880.17630.17630.2238],M13=104×[1.04990.10900.10900.2928],M23=104×[1.18380.17660.17660.2881],M14=103×[-8.8058-0.9604-0.9604-2.6545],M24=104×[-1.0454-0.1677-0.1677-0.2680],M15=104×[1.03030.10180.10180.3190],M25=104×[1.15900.16220.16220.3078],M16=[-34.9655-18.7513-18.751335.4129],M26=[-39.8158-24.8476-24.847613.4547],M17=[2.09042.77172.7717-7.8109],M27=[3.28035.29295.2929-4.0693],M18=104×[1.46790.00030.00031.2678],M28=104×[1.4438-0.0080-0.00801.1830],M19=103×[-0.89050.00580.0058-2.9028],M29=103×[-1.1307-0.0760-0.0760-3.7474],M110=104×[1.46840.00050.00051.2673],M210=104×[1.4444-0.0077-0.00771.1828],R11=103×[2.91310.17480.17480.7782],R21=103×[2.34630.03530.03530.5806],R12=103×[1.13930.05310.05310.1251],R22=[876.562732.555032.555098.5599],R13=103×[2.6387-0.0759-0.07590.5866],R23=103×[2.0346-0.0762-0.07620.3589],R14=103×[5.6600-0.0415-0.04155.0541],R24=103×[5.5782-0.0710-0.07104.6031],R15=104×[-7.90710.05510.0551-3.8199],R25=104×[-7.39120.45760.4576-3.2145],R16=104×[1.4893-0.0162-0.01621.3778],R26=104×[1.4703-0.0223-0.02231.2713],N11=103×[8.74933.85343.8534-3.1296],N21=104×[0.66910.26860.2686-1.0299],N12=103×[-3.8698-0.3606-0.36061.4605],N22=103×[-4.5981-0.9753-0.97531.2772],N13=[-114.8946-186.6948-186.6948-113.8034],N23=[210.0606-115.9077-115.9077-174.4056],N14=104×[-1.1296-0.0266-0.02660.1446],N24=104×[-1.00620.10410.10410.8782],N15=103×[4.5589-2.3847-2.38470.8561],N25=103×[5.3887-2.4363-2.43631.1619],N16=103×[5.9564-1.9530-1.95301.5688],N26=103×[6.7696-1.9958-1.99582.0384],N17=103×[5.9602-0.2891-0.28912.9332],N27=103×[6.3929-0.0855-0.08553.1668],N18=103×[-9.6508-0.4780-0.47803.5901],N28=104×[-0.85050.05740.05741.1344],N19=103×[5.5789-2.0704-2.07041.3737],N29=103×[6.3923-2.1169-2.11691.8008],N110=103×[-3.65953.20323.2032-0.1648],N210=103×[-4.37813.21203.21200.1271],O11=104×[1.5923-0.7669-0.76691.6350],O21=104×[1.8134-0.6324-0.63242.3112],O12=104×[1.0371-0.0001-0.0001-0.3635],O22=104×[0.9152-0.1464-0.1464-1.0978],O13=103×[-1.03550.15150.1515-0.3626],O23=103×[-1.49900.31480.3148-0.4988],O14=[-51.179771.716771.7167-390.4844],O24=[-22.7818-177.9764-177.9764237.5644],O15=104×[-1.45690.23760.2376-0.4763],O25=104×[-1.50780.22770.2277-0.4312],O16=104×[-1.36670.24750.2475-0.4459],O26=104×[-1.41690.24410.2441-0.4015],O17=103×[5.50380.79000.7900-4.7294],O27=104×[0.4206-0.0170-0.0170-1.2249],O18=[244.568488.459688.4596481.2263],O28=[272.5078401.6259401.6259-0.2695],O19=104×[-1.39110.24470.2447-0.4543],O29=104×[-1.44150.23940.2394-0.4095],O110=104×[1.3674-0.2535-0.25350.3868],O210=104×[1.4277-0.2479-0.24790.3580],W11=104×[1.5106-0.0468-0.04680.7929],W21=104×[1.4523-0.0341-0.03410.7069],W12=103×[5.9653-2.4134-2.41341.7851],W22=103×[6.9091-2.5666-2.56661.9597],W13=[-289.708524.420024.4200-461.3886],W23=[-223.519393.656693.6566-234.3346],W14=104×[-1.27460.26530.2653-0.3694],W24=104×[-1.32420.26360.2636-0.3371],W15=103×[4.8778-0.0916-0.09163.8521],W25=103×[4.8245-0.0681-0.06813.9058],W16=103×[7.9418-0.4415-0.44156.9527],W26=103×[7.9864-0.4713-0.47137.2384],W17=103×[4.5999-2.3966-2.39660.1608],W27=103×[5.2537-2.3858-2.38580.0285],W18=104×[-1.38440.25770.2577-0.4094],W28=104×[-1.44260.24990.2499-0.3763],W19=103×[7.1086-0.3474-0.34746.1090],W29=103×[7.1257-0.3627-0.36276.3346],W110=103×[8.0468-0.7400-0.74006.5133],W210=103×[8.1365-0.8020-0.80206.8109],G=[-2.15271.43896.5441-11.7139],ϵ1=4.2164×104,ϵ2=5.7269×104,ϵ3=5.3163×104.Figure 1
The simulation solutionsx1(t) and x2(t) are presented for system (1) with (58)-(59) in Example 1, α1=α2=1/2, and initial conditions x1(t)=2+3cos(t), x2(t)=1+2sin(t), t∈[-1,0], by using the Runge-Kutta fourth order method with Matlab.Example 2.
Consider the following neutral system (54), which is considered in [3, 25, 26]:
(61)x˙(t)=Ax(t)+Bx(t-h(t))+Cx˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t))),
with
(62)A=[-200-0.9],B=[-10-1-1],C=[0.1000.1].
Decompose matrixB as follows: B=B1+B2, where
(63)B1=[-0.830-1-0.83],B2=[-0.1700-0.17],∥f(t,x(t))∥≤η∥x(t)∥, and ∥g(t,x(t-h(t)))∥≤ρ∥x(t-h(t))∥. The maximum value h for exponential stability of system (61) with (62)-(63) is listed in the comparison in Table 1, for different values of hd and rd. In Table 1, we let η=0.05, ρ=0.1, β=0.1 and h(t)=r(t). We can see that our results in Corollary 8 are much less conservative than in [3, 25, 26].Table 1
The maximum allowed time delayh for k=0.1, η=0.05, ρ=0.1, and β=0.1.
h
d
=
r
d
0
0.5
0.9
Chen et al. (2008) [3]
1.2999
0.9442
0.5471
Qiu and Cui (2010) [26]
1.4008
1.0120
0.6438
Pinjai and Mukdasai (2011) [25]
1.6237
1.1052
0.6205
Corollary10
6.4417
5.2362
2.5666Example 3.
Consider the following neutral system (51), which is considered in [14, 28]:
(64)x˙(t)=Ax(t)+Bx(t-h(t))+Cx˙(t-r(t))+f(t,x(t))+g(t,x(t-h(t)))+w(t,x˙(t-r(t))),
with
(65)A=[-200-2],B=[00.40.40],C=[0.1000.1].
Decompose matrixB as follows: B=B1+B2, where
(66)B1=[00.30.20],B2=[00.10.20],∥f(t,x(t))∥≤η∥x(t)∥, ∥g(t,x(t-h(t)))∥≤ρ∥x(t-h(t))∥, and ∥w(t,x˙(t-r(t)))∥≤γ∥x˙(t-r(t))∥. By Corollary 9 to system (64) with (65)-(66), one can obtain the maximum upper bounds of the time delay with different convergence rate k as listed in Table 2. In Table 2, we let h=0.1, η=0.1, ρ=0.05, γ=0.05, and β=0.1. It is clear that the results in Corollary 9 give larger delay bounds than the recent results in [14, 28].Table 2
The maximum allowed time delayh.
h
d
=
r
d
=
0
k
=
0.1
k
=
0.3
k
=
0.5
k
=
0.7
k
=
0.9
Syed Ali (2012) [28]
10.2180
2.9481
1.4126
0.7232
0.3045
Liu et al. (2013) [14]
12.2475
3.7460
1.9563
1.1015
0.5957
Corollary9
14.1728
4.7242
2.8345
2.0246
1.5747
h
d
=
r
d
=
0.5
k
=
0.1
k
=
0.3
k
=
0.5
k
=
0.7
k
=
0.9
Syed Ali (2012) [28]
6.7523
1.7922
0.7308
0.3580
0.1027
Liu et al. (2013) [14]
10.8211
3.3202
1.7390
0.9662
0.4857
Corollary9
11.5872
3.8624
2.3174
1.6553
1.2874


## 5. Conclusions

The problem of robust exponential stability for LPD neutral systems with mixed time-varying delays and nonlinear uncertainties has been presented. Based on combination of Leibniz-Newton formula, free-weighting matrices, linear matrix inequality, Cauchy’s inequality, modified version of Jensen's inequality, model transformation, and the use of suitable parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent robust exponential stability criteria are formulated in terms of LMIs. Numerical examples have shown significant improvements over some existing results.



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*Source: 268905-2013-12-09.xml*