research article new delay-dependent robust exponential ...new delay-dependent robust exponential...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 268905, 18 pageshttp://dx.doi.org/10.1155/2013/268905

Research ArticleNew Delay-Dependent Robust Exponential StabilityCriteria of LPD Neutral Systems with Mixed Time-VaryingDelays and Nonlinear Perturbations

Sirada Pinjai and Kanit Mukdasai

Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

Correspondence should be addressed to Kanit Mukdasai; [email protected]

Received 15 July 2013; Revised 30 September 2013; Accepted 2 October 2013

Academic Editor: Jitao Sun

Copyright © 2013 S. Pinjai and K. Mukdasai. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper is concernedwith the problemof robust exponential stability for linear parameter-dependent (LPD) neutral systemswithmixed 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, modifiedversion of Jensen’s inequality, model transformation, and linear matrix inequality technique, new delay-dependent robustexponential stability criteria are established in terms of linear matrix inequalities (LMIs). Numerical examples are given to showthe effectiveness and less conservativeness of the proposed methods.

1. Introduction

Over the past decades, the problem of stability for neutraldifferential systems, which have delays in both their state andthe derivatives of their states, has been widely investigatedby many researchers, especially in the last decade. It iswell known that nonlinearities, as time delays, may causeinstability and poor performance of practical systems suchas engineering, biology, and economics [1]. The problemsof various stability and stabilization for dynamical systemswith or without state delays and nonlinear perturbations havebeen intensively studied in the past years bymany researchersof mathematics and control communities [1–35]. Stabilitycriteria for dynamical systems with time delay are generallydivided into two classes: delay-independent one and delay-dependent one. Delay-independent stability criteria tend tobe more conservative, especially for small size delay; suchcriteria do not give any information on the size of the delay.On the other hand, delay-dependent stability criteria areconcerned with the size of the delay and usually provide amaximal delay size.

Recently, many researchers have studied the stabilityproblem for neutral systems with time-varying delays andnonlinear perturbations have appeared [29, 31]. Furthermore,the convergence rates are essential for the practical system;then the exponential stability analysis of time delay systemshas been favorably approved in the past decades; see, forexample, [3, 9, 10, 14, 18–21, 25–28].

In addition, many researchers have paid attention to theproblem of stability for linear systems with polytope uncer-tainties. The linear systems with polytopic-type uncertaintiesare called linear parameter-dependent (LPD) systems. Thatis, the uncertain state matrices are in the polytope con-sisting of all convex combination of known matrices. Mostof sufficient (or necessary and sufficient) conditions havebeen obtained via Lyapunov-Krasovskii theory approachesin which parameter-dependent Lyapunov-Krasovskii func-tional has been employed. These conditions are alwaysexpressed in terms of linear matrix inequalities (LMIs).The results have been obtained for robust stability for LPDsystems in which time-delay occurs in state variable; forexample, [17, 18] presented sufficient conditions for robust

2 Journal of Applied Mathematics

stability of LPD discrete-time systems with delays. Moreover,robust stability of LPD continuous-time systems with delayswas studied in [6, 19, 22, 30].

In consequence, it is important and interesting to studythe problem of robust exponential stability for neutral sys-tems with parametric uncertainties. This paper investigatesthe robust exponential stability analysis for LPD neutralsystems with mixed time-varying delays and nonlinear per-turbations. Based on combination of Leibniz-Newton for-mula, free-weighting matrices, Cauchy’s inequality, modifiedversion of Jensen’s inequality, decomposition technique ofcoefficient matrix, the use of suitable parameter-dependentLyapunov-Krasovskii functional, model transformation, andlinear matrix inequality technique, new delay-dependentrobust exponential stability criteria for these systems will beobtained in terms of LMIs. Finally, numerical examples willbe given to show the effectiveness of the obtained results.

2. Problem Formulation and Preliminaries

We introduce some notations, a definition, and lemmas thatwill be used throughout the paper. 𝑅+ denotes the set ofall real nonnegative numbers; 𝑅𝑛 denotes the 𝑛-dimensionalspace with the vector norm ‖ ⋅ ‖; ‖𝑥‖ denotes the Euclideanvector norm of 𝑥 ∈ 𝑅𝑛; 𝑅𝑛×𝑟 denotes the set of 𝑛 × 𝑟real 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{Re 𝜆 : 𝜆 ∈ 𝜆(𝐴)}; 𝜆min(𝐴) = min{Re 𝜆 : 𝜆 ∈ 𝜆(𝐴)};𝜆max(𝐴(𝛼)) = max{𝜆max(𝐴 𝑖) : 𝑖 = 1, 2, . . . , 𝑁}; 𝜆min(𝐴(𝛼)) =min{𝜆min(𝐴 𝑖) : 𝑖 = 1, 2, . . . , 𝑁}; 𝐶([−𝑏, 0], 𝑅

𝑛) denotes the

space of all continuous vector functions mapping [−𝑏, 0] into𝑅𝑛, 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:

�̇� (𝑡) = 𝐴 (𝛼) 𝑥 (𝑡) + 𝐵 (𝛼) 𝑥 (𝑡 − ℎ (𝑡))

+ 𝐶 (𝛼) �̇� (𝑡 − 𝑟 (𝑡)) + 𝑓 (𝑡, 𝑥 (𝑡))

+ 𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡))) + 𝑤 (𝑡, �̇� (𝑡 − 𝑟 (𝑡))) , 𝑡 > 0;

𝑥 (𝑡 + 𝑡0) = 𝜙 (𝑡) , �̇� (𝑡 + 𝑡

0) = 𝜓 (𝑡) , 𝑡 ∈ [−𝑏, 0] ,

(1)

where 𝑥(𝑡) ∈ 𝑅𝑛 is the state variable and 𝐴(𝛼), 𝐵(𝛼), 𝐶(𝛼) ∈𝑅𝑛×𝑛 are uncertain matrices belonging to the polytope

𝐴 (𝛼) =

𝑁

∑

𝑖=1

𝛼𝑖𝐴𝑖, 𝐵 (𝛼) =

𝑁

∑

𝑖=1

𝛼𝑖𝐵𝑖, 𝐶 (𝛼) =

𝑁

∑

𝑖=1

𝛼𝑖𝐶𝑖,

𝑁

∑

𝑖=1

𝛼𝑖= 1, 𝛼

𝑖≥ 0, 𝐴

𝑖, 𝐵𝑖, 𝐶𝑖∈ 𝑅𝑛×𝑛

, 𝑖 = 1, . . . , 𝑁.

(2)

ℎ(𝑡) and 𝑟(𝑡) are discrete and neutral time-varying delays,respectively,

0 ≤ ℎ (𝑡) ≤ ℎ, ℎ̇ (𝑡) ≤ ℎ𝑑, (3)

0 ≤ 𝑟 (𝑡) ≤ 𝑟, ̇𝑟 (𝑡) ≤ 𝑟𝑑, (4)

where ℎ, 𝑟, ℎ𝑑, and 𝑟

𝑑are given positive real constants.

Consider the initial functions 𝜙(𝑡),𝜓(𝑡) ∈ 𝐶([−𝑏, 0], 𝑅𝑛)withthe norm ‖𝜙‖ = sup

𝑡∈[−𝑏,0]‖𝜙(𝑡)‖ and ‖𝜓‖ = sup

𝑡∈[−𝑏,0]‖𝜓(𝑡)‖.

The uncertainties 𝑓(𝑡, 𝑥(𝑡)), 𝑔(𝑡, 𝑥(𝑡 − ℎ(𝑡))), and 𝑤(𝑡, �̇�(𝑡 −𝑟(𝑡))) are the nonlinear perturbations with respect to currentstate 𝑥(𝑡), discrete delayed state 𝑥(𝑡 − ℎ(𝑡)), and neutraldelayed state �̇�(𝑡 − 𝑟(𝑡)), respectively, and are bounded inmagnitude:

𝑓𝑇

(𝑡, 𝑥 (𝑡)) 𝑓 (𝑡, 𝑥 (𝑡)) ≤ 𝜂2

𝑥𝑇

(𝑡) 𝑥 (𝑡) ,

𝑔𝑇

(𝑡, 𝑥 (𝑡 − ℎ (𝑡))) 𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡)))

≤ 𝜌2

𝑥𝑇

(𝑡 − ℎ (𝑡)) 𝑥 (𝑡 − ℎ (𝑡)) ,

𝑤𝑇

(𝑡, �̇� (𝑡 − 𝑟 (𝑡))) 𝑤 (𝑡, �̇� (𝑡 − 𝑟 (𝑡)))

≤ 𝛾2

�̇�𝑇

(𝑡 − 𝑟 (𝑡)) �̇� (𝑡 − 𝑟 (𝑡)) ,

(5)

where 𝜂, 𝜌, and 𝛾 are given positive real constants.In order to improve the bound of the discrete time-

varying delayed ℎ(𝑡) in system (1), let us decompose theconstant matrix 𝐵(𝛼) as

𝐵 (𝛼) = 𝐵1(𝛼) + 𝐵

2(𝛼) , (6)

where 𝐵1(𝛼) = ∑

𝑁

𝑖=1𝛼𝑖𝐵1

𝑖, 𝐵2(𝛼) = ∑

𝑁

𝑖=1𝛼𝑖𝐵2

𝑖, and ∑𝑁

𝑖=1𝛼𝑖= 1,

𝛼𝑖≥ 0 with 𝐵1

𝑖, 𝐵2

𝑖∈ 𝑅𝑛×𝑛, 𝑖 = 1, . . . , 𝑁 being real constant

matrices. By Leibniz-Newton formula, we have

0 = 𝑥 (𝑡) − 𝑥 (𝑡 − ℎ (𝑡)) − ∫

𝑡

𝑡−ℎ(𝑡)

�̇� (𝑠) 𝑑𝑠. (7)

By utilizing the following zero equation, we obtain

0 = 𝐺𝑥 (𝑡) − 𝐺𝑥 (𝑡 − 𝛽ℎ (𝑡)) − 𝐺∫

𝑡

𝑡−𝛽ℎ(𝑡)

�̇� (𝑠) 𝑑𝑠, (8)

where 𝛽 is a given positive real constant and 𝐺 ∈ 𝑅𝑛×𝑛 will bechosen to guarantee the robust exponential stability of system(1). By (6), (7), and (8), system (1) can be represented by theform

�̇� (𝑡) = [𝐴 (𝛼) + 𝐵1(𝛼) + 𝐺] 𝑥 (𝑡) + 𝐵

2(𝛼) 𝑥 (𝑡 − ℎ (𝑡))

− 𝐺𝑥 (𝑡 − 𝛽ℎ (𝑡)) + 𝑓 (𝑡, 𝑥 (𝑡))

+ 𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡))) + 𝑤 (𝑡, �̇� (𝑡 − 𝑟 (𝑡)))

+ 𝐶 (𝛼) �̇� (𝑡 − 𝑟 (𝑡)) − 𝐵1(𝛼) ∫

𝑡

𝑡−ℎ(𝑡)

�̇� (𝑠) 𝑑𝑠

− 𝐺∫

𝑡


�̇� (𝑠) 𝑑𝑠.

(9)

Journal of Applied Mathematics 3

Definition 1. The system (1) is robustly exponentially stable,if there exist positive real constants 𝑘 and 𝑀 such that, foreach 𝜙(𝑡), 𝜓(𝑡) ∈ 𝐶([−𝑏, 0], 𝑅𝑛), the solution 𝑥(𝑡, 𝜙, 𝜓) of thesystem (1) satisfies

𝑥 (𝑡, 𝜙, 𝜓) ≤ 𝑀max {

𝜙 ,𝜓

} 𝑒−𝑘𝑡

, ∀𝑡 ∈ 𝑅+

. (10)

Lemma 2 (Cauchy inequality). For any constant symmetricpositive definite matrix 𝑃 ∈ 𝑅𝑛×𝑛 and 𝑎, 𝑏 ∈ 𝑅𝑛, one has

±2𝑎𝑇

𝑏 ≤ 𝑎𝑇

𝑃𝑎 + 𝑏𝑇

𝑃−1

𝑏. (11)

Lemma 3 (see [15]). The following inequality holds for any 𝑎 ∈𝑅𝑛, 𝑏 ∈ 𝑅𝑚,𝑁,𝑌 ∈ 𝑅𝑛×𝑚,𝑋 ∈ 𝑅𝑛×𝑛, and 𝑍 ∈ 𝑅𝑚×𝑚:

−2𝑎𝑇

𝑁𝑏 ≤ [𝑎

𝑏]

𝑇

[𝑋 𝑌 − 𝑁

∗ 𝑍][

𝑎

𝑏] , (12)

where [𝑋 𝑌∗ 𝑍

] ≥ 0.

Lemma4. For any constant symmetric positive definitematrix𝑄 ∈ 𝑅

𝑛×𝑛 and ℎ(𝑡) which is discrete time-varying delayswith (3), vector function 𝜔 : [−ℎ, 0] → 𝑅𝑛 such that theintegrations concerned are well defined; then

ℎ∫

0

−ℎ

𝜔𝑇

(𝑠) 𝑄𝜔 (𝑠) 𝑑𝑠 ≥ ∫

0

−ℎ(𝑡)

𝜔𝑇

(𝑠) 𝑑𝑠𝑄∫

0

−ℎ(𝑡)

𝜔 (𝑠) 𝑑𝑠.

(13)

Proof. From Lemma 2, it is easy to see that

ℎ∫

0

−ℎ

𝜔𝑇

(𝑠) 𝑄𝜔 (𝑠) 𝑑𝑠 = ℎ∫

0

−ℎ(𝑡)

𝜔𝑇

(𝑠) 𝑄𝜔 (𝑠) 𝑑𝑠

+ ℎ∫

−ℎ(𝑡)

−ℎ

𝜔𝑇


≥ ℎ (𝑡) ∫

0

−ℎ(𝑡)

𝜔𝑇


=1

2∬

0

−ℎ(𝑡)

[𝜔𝑇

(𝑠) 𝑄𝜔 (𝑠)

+ 𝜔𝑇

(𝜉) 𝑄𝜔 (𝜉)] 𝑑𝑠𝑑𝜉

≥1

2∬

0

−ℎ(𝑡)

2𝜔𝑇

(𝑠) 𝑄1/2𝑇

× 𝑄1/2

𝜔 (𝜉) 𝑑𝑠 𝑑𝜉

= ∫

0

−ℎ(𝑡)

𝜔𝑇

(𝑠) 𝑑𝑠𝑄∫

0

−ℎ(𝑡)

𝜔 (𝑠) 𝑑𝑠.

(14)

Lemma 5. Let 𝑥(𝑡) ∈ 𝑅𝑛 be a vector-valued function with firstorder continuous derivative entries.Then, the following integralinequality holds for any matrices 𝑀

𝑖∈ 𝑅𝑛×𝑛

, 𝑖 = 1, 2, . . . , 5,

and ℎ(𝑡) is discrete time-varying delays with (3) and symmetricpositive definite matrix 𝑋 ∈ 𝑅𝑛×𝑛:

−∫

𝑡

𝑡−ℎ

�̇�𝑇

(𝑠)𝑋�̇� (𝑠) 𝑑𝑠 ≤ [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

× [𝑀1+𝑀𝑇

1−𝑀𝑇

1+𝑀2

∗ −𝑀2−𝑀𝑇

2

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

+ ℎ[𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

[𝑀3

𝑀4

∗ 𝑀5

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))] ,

(15)

where

[

[

𝑋 𝑀1

𝑀2

∗ 𝑀3

𝑀4

∗ ∗ 𝑀5

]

]

≥ 0. (16)

Proof. From the Leibniz-Newton formula, one has

0 = 𝑥 (𝑡) − 𝑥 (𝑡 − ℎ (𝑡)) − ∫

𝑡

𝑡−ℎ(𝑡)

�̇� (𝑡) 𝑑𝑠. (17)

Therefore, for any 𝐻1, 𝐻2∈ 𝑅𝑛×𝑛, the following equation is

true:

0 = 2 [𝑥𝑇

(𝑡) − 𝑥𝑇

(𝑡 − ℎ (𝑡)) − ∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑡) 𝑑𝑠]

× [𝐻1𝑥 (𝑡) + 𝐻

2𝑥 (𝑡 − ℎ (𝑡))]

= 2𝑥𝑇

(𝑡)𝐻1𝑥 (𝑡) + 2𝑥

𝑇

(𝑡)𝐻2𝑥 (𝑡 − ℎ (𝑡))

− 2𝑥𝑇

(𝑡 − ℎ (𝑡))𝐻1𝑥 (𝑡)

− 2𝑥𝑇

(𝑡 − ℎ (𝑡))𝐻𝑇

2𝑥 (𝑡 − ℎ (𝑡))

− 2∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠) 𝑑𝑠𝐻1𝑥 (𝑡)

− 2∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠) 𝑑𝑠𝐻2𝑥 (𝑡 − ℎ (𝑡))

= [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

[

[

𝐻1+ 𝐻𝑇

1−𝐻𝑇

1+ 𝐻2

∗ −𝐻2− 𝐻𝑇

2

]

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

− 2∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠) [𝐻1

𝐻2] [

𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))] 𝑑𝑠.

(18)


Using Lemma 3 with 𝑎 = �̇�(𝑠), 𝑏 = [ 𝑥(𝑡)𝑥(𝑡−ℎ(𝑡))

], 𝑁 = [𝐻1

𝐻2], 𝑌 = [𝑀

1𝑀2], and 𝑍 = [𝑀3 𝑀4

∗ 𝑀5], we obtain

− 2∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠) [𝐻1

𝐻2] [

𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))] 𝑑𝑠

≤ ∫

𝑡

𝑡−ℎ(𝑡)

[

[

�̇� (𝑠)

𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))

]

]

𝑇

[

[

𝑋 𝑀1− 𝐻1

𝑀2− 𝐻2

∗ 𝑀3

𝑀4

∗ ∗ 𝑀5

]

]

× [

[

�̇� (𝑠)

𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))

]

]

𝑑𝑠

= ∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠)𝑋�̇� (𝑠) 𝑑𝑠

+ [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

× [𝑀1+𝑀𝑇

1− 𝐻1− 𝐻𝑇

1−𝑀𝑇

1+𝑀2+ 𝐻𝑇

1− 𝐻2

∗ −𝑀2−𝑀𝑇

2+ 𝐻2+ 𝐻𝑇

2

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))] + ℎ (𝑡) [

𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

[𝑀3

𝑀4

∗ 𝑀5

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

≤ ∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠)𝑋�̇� (𝑠) 𝑑𝑠 + [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

× ([𝑀1+𝑀𝑇

1−𝑀𝑇

1+𝑀2

∗ −𝑀2−𝑀𝑇

2

]

− [𝐻1+ 𝐻𝑇

1−𝐻𝑇

1+ 𝐻2

∗ −𝐻2− 𝐻𝑇

2

]) [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

+ ℎ[𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

[𝑀3

𝑀4

∗ 𝑀5

] [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))] .

(19)

Substituting (19) into (18), we obtain

−∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠)𝑋�̇� (𝑠) 𝑑𝑠 ≤ [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

× [𝐻1+ 𝐻𝑇

1−𝐻𝑇

1+ 𝐻2

∗ −𝐻2− 𝐻𝑇

2

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

+ [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

× ([

[

𝑀1+𝑀𝑇

1−𝑀𝑇

1+𝑀2

∗ −𝑀2−𝑀𝑇

2

]

]

− [𝐻1+ 𝐻𝑇

1−𝐻𝑇

1+ 𝐻2

∗ −𝐻2− 𝐻𝑇

2

])

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

+ ℎ[𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

[𝑀3

𝑀4

∗ 𝑀5

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

= [𝑥(𝑡)

𝑥(𝑡 − ℎ(𝑡))]

𝑇

× [𝑀1+𝑀𝑇

1−𝑀𝑇

1+𝑀2

∗ −𝑀2−𝑀𝑇

2

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

+ ℎ[𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

[𝑀3

𝑀4

∗ 𝑀5

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))] .

(20)

From (3), it is clear that

−∫

𝑡

𝑡−ℎ

�̇�𝑇

(𝑠)𝑋�̇� (𝑠) 𝑑𝑠 ≤ −∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠)𝑋�̇� (𝑠) 𝑑𝑠. (21)

From (20) and (21), the integral inequality becomes

−∫

𝑡

𝑡−ℎ

�̇�𝑇

(𝑠)𝑋�̇� (𝑠) 𝑑𝑠 ≤ [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

× [

[

𝑀1+𝑀𝑇

1−𝑀𝑇

1+𝑀2

∗ −𝑀2−𝑀𝑇

2

]

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

+ ℎ[𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))]

𝑇

[𝑀3

𝑀4

∗ 𝑀5

]

× [𝑥 (𝑡)

𝑥 (𝑡 − ℎ (𝑡))] .

(22)

The proof of the theorem is complete.


Remark 6. In Lemma 4 and Lemma 5, we have modified themethod from [8, 33], respectively.

3. Main results

3.1. Robust Exponential Stability Criteria. In this section,robust exponential stability criteria dependent on mixedtime-varying delays of LPD neutral delayed system (1) withnonlinear perturbations via linear matrix inequality (LMI)approach will be presented. We introduce the followingnotations for later use:

𝑃𝑗(𝛼) =

𝑁

∑

𝑖=1

𝛼𝑖𝑃𝑗

𝑖, 𝑍

1(𝛼) =

𝑁

∑

𝑖=1

𝛼𝑖𝑍1

𝑖,

𝑅𝑝(𝛼) =

𝑁

∑

𝑖=1

𝛼𝑖𝑅𝑝

𝑖, 𝑁

𝑗(𝛼) =

𝑁

∑

𝑖=1

𝛼𝑖𝑁𝑗

𝑖,

𝑂𝑗(𝛼) =

𝑁

∑

𝑖=1

𝛼𝑖𝑂𝑗

𝑖, 𝑊

𝑗(𝛼) =

𝑁

∑

𝑖=1

𝛼𝑖𝑊𝑗

𝑖,

𝑀𝑗(𝛼) =

𝑁

∑

𝑖=1

𝛼𝑖𝑀𝑗

𝑖,

𝑁

∑

𝑖=1

𝛼𝑖= 1, 𝛼

𝑖≥ 0,

𝑃𝑗

𝑖, 𝑍1

𝑖, 𝑅𝑝

𝑖,𝑊𝑗

𝑖, 𝑁𝑗

𝑖, 𝑂𝑗

𝑖,𝑀𝑗

𝑖∈ 𝑅𝑛×𝑛

,

𝑗 = 1, 2, . . . , 10, 𝑝 = 1, 2, . . . , 6, 𝑖 = 1, 2, . . . , 𝑁;

∑

𝑖,𝑗

=

[[[[[[[[[[[[[[[[[

[

Σ1,1

𝑖,𝑗Σ1,2

𝑖,𝑗Σ1,3

𝑖,𝑗Σ1,4

𝑖,𝑗Σ1,5

𝑖,𝑗Σ1,6

𝑖,𝑗Σ1,7

𝑖,𝑗Σ1,8

𝑖,𝑗Σ1,9

𝑖,𝑗Σ1,10

𝑖,𝑗

∗ Σ2,2

𝑖,𝑗Σ2,3

𝑖,𝑗Σ2,4

𝑖,𝑗Σ2,5

𝑖,𝑗Σ2,6

𝑖,𝑗Σ2,7

𝑖,𝑗Σ2,8

𝑖,𝑗Σ2,9

𝑖,𝑗Σ2,10

𝑖,𝑗

∗ ∗ Σ3,3

𝑖Σ3,4

𝑖,𝑗Σ3,5

𝑖Σ3,6

𝑖Σ3,7

𝑖,𝑗Σ3,8

𝑖Σ3,9

𝑖Σ3,10

𝑖

∗ ∗ ∗ Σ4,4

𝑖,𝑗Σ4,5

𝑖,𝑗Σ4,6

𝑖,𝑗Σ4,7

𝑖,𝑗Σ4,8

𝑖Σ4,9

𝑖Σ4,10

𝑖

∗ ∗ ∗ ∗ Σ5,5

𝑖Σ5,6

𝑖Σ5,7

𝑖,𝑗Σ5,8

𝑖Σ5,9

𝑖Σ5,10

𝑖

∗ ∗ ∗ ∗ ∗ Σ6,6

𝑖Σ6,7

𝑖,𝑗Σ6,8

𝑖Σ6,9

𝑖Σ6,10

𝑖

∗ ∗ ∗ ∗ ∗ ∗ Σ7,7

𝑖,𝑗Σ7,8

𝑖,𝑗Σ7,9

𝑖,𝑗Σ7,10

𝑖,𝑗

∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ8,8

𝑖Σ8,9

𝑖Σ8,10

𝑖

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ9,9

𝑖Σ9,10

𝑖

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ10,10

𝑖

]]]]]]]]]]]]]]]]]

]

,

(23)

where

Σ1,1

𝑖,𝑗= 𝐴𝑇

𝑖𝑃1

𝑗+ 𝐵1

𝑖

𝑇

𝑃1

𝑗+ 𝑃1

𝑖𝐴𝑗+ 𝑃1

𝑖𝐵1

𝑗+ 𝑍1

𝑖+ 𝑍1

𝑖

𝑇

+ 𝑃3

𝑖+ 𝑃4

𝑖+ ℎ𝑀

1

𝑖

𝑇

+ ℎ𝑀1

𝑖− 𝑒−2𝑘ℎ

𝑃9

𝑖

− 𝑒−2𝑘𝛽ℎ

𝑃10

𝑖+ ℎ2

𝑀3

𝑖+ 𝛽ℎ𝑀

6

𝑖+ 𝛽ℎ𝑀

6

𝑖

𝑇

+ 𝛽2

ℎ2

𝑀8

𝑖

+ 𝑁1

𝑖

𝑇

+ 𝑁1

𝑖+ 𝑂1

𝑖

𝑇

+ 𝑂1

𝑖+ 𝐴𝑇

𝑖𝑊1

𝑗+𝑊1

𝑖

𝑇

𝐴𝑗

+ 𝐵1

𝑖

𝑇

𝑊1

𝑗+𝑊1

𝑖

𝑇

𝐵1

𝑗+ 𝜖1𝜂2

𝐼 + 2𝑘𝑃1

𝑖,

Σ1,2

𝑖,𝑗= 𝑃1

𝑖𝐵2

𝑗− ℎ𝑀

1

𝑖

𝑇

+ ℎ𝑀2

𝑖+ ℎ2

𝑀4

𝑖+ 𝑒−2𝑘ℎ

𝑃9

𝑖

+ ℎ𝑅2

𝑖− 𝑁1

𝑖

𝑇

+ 𝑁2

𝑖+ 𝑂2

𝑖+𝑊1

𝑖

𝑇

𝐵2

𝑗

+ 𝐴𝑇

𝑖𝑊2

𝑗+ 𝐵1

𝑖

𝑇

𝑊2

𝑗,

Σ1,3

𝑖,𝑗= −𝑍1

𝑖− 𝛽ℎ𝑀

6

𝑖

𝑇

+ 𝛽ℎ𝑀7

𝑖+ 𝛽ℎ2

𝑀9

𝑖+ 𝛽ℎ𝑅

5

𝑖

+ 𝑒−2𝑘𝛽ℎ

𝑃10

𝑖+ 𝑁3

𝑖− 𝑂1

𝑖

𝑇

+ 𝑂3

𝑖

+ 𝐴𝑇

𝑖𝑊3

𝑗+ 𝐵1

𝑖

𝑇

𝑊3

𝑗,

Σ1,4

𝑖,𝑗= −𝑃1

𝑖𝐵1

𝑗− 𝑁1

𝑖

𝑇

+ 𝑁4

𝑖+ 𝑂4

𝑖

−𝑊1

𝑖

𝑇

𝐵1

𝑗+ 𝐴𝑇

𝑖𝑊4

𝑗+ 𝐵1

𝑖

𝑇

𝑊4

𝑗,

Σ1,5

𝑖,𝑗= −𝑍1

𝑖+ 𝑁5

𝑖− 𝑂1

𝑖

𝑇

+ 𝑂5

𝑖+ 𝐴𝑇

𝑖𝑊5

𝑗+ 𝐵1

𝑖

𝑇

𝑊5

𝑗,

Σ1,6

𝑖,𝑗= 𝑁6

𝑖+ 𝑂6

𝑖−𝑊1

𝑖

𝑇

+ 𝐴𝑇

𝑖𝑊6

𝑗+ 𝐵1

𝑖

𝑇

𝑊6

𝑗,

Σ1,7

𝑖,𝑗= 𝑃1

𝑖𝐶𝑗+ 𝑁7

𝑖+ 𝑂7

𝑖+𝑊1

𝑖

𝑇

𝐶𝑗+ 𝐴𝑇

𝑖𝑊7

𝑗+ 𝐵1

𝑖

𝑇

𝑊7

𝑗,

Σ1,8

𝑖,𝑗= 𝑃1

𝑖+ 𝑁8

𝑖+ 𝑂8

𝑖+𝑊1

𝑖

𝑇

+ 𝐴𝑇

𝑖𝑊8

𝑗+ 𝐵1

𝑖

𝑇

𝑊8

𝑗,

Σ1,9

𝑖,𝑗= 𝑃1

𝑖+ 𝑁9

𝑖+ 𝑂9

𝑖+𝑊1

𝑖

𝑇

+ 𝐴𝑇

𝑖𝑊9

𝑗+ 𝐵1

𝑖

𝑇

𝑊9

𝑗,

Σ1,10

𝑖,𝑗= 𝑃1

𝑖+ 𝑁10

𝑖+ 𝑂10

𝑖+𝑊1

𝑖

𝑇

+ 𝐴𝑇

𝑖𝑊10

𝑗+ 𝐵1

𝑖

𝑇

𝑊10

𝑗,

Σ2,2

𝑖,𝑗= − (1 − ℎ

𝑑) 𝑒−2𝑘ℎ

𝑃3

𝑖− ℎ𝑀

2

𝑖

𝑇

− ℎ𝑀2

𝑖+ ℎ2

𝑀5

𝑖

+ ℎ2

𝑅1

𝑖− 2ℎ𝑅

2

𝑖− 𝑒−2𝑘ℎ

𝑃9

𝑖+ 𝜖2𝜌2

𝐼 − 𝑁2

𝑖

𝑇

− 𝑁2

𝑖+𝑊2

𝑖

𝑇

𝐵2

𝑗+ 𝐵2

𝑖

𝑇

𝑊2

𝑗,

Σ2,3

𝑖,𝑗= −𝑁

3

𝑖− 𝑂2

𝑖

𝑇

+ 𝐵2

𝑖

𝑇

𝑊3

𝑗,

Σ2,4

𝑖,𝑗= −𝑁

2

𝑖

𝑇

− 𝑁4

𝑖−𝑊2

𝑖

𝑇

𝐵1

𝑗+ 𝐵2

𝑖

𝑇

𝑊4

𝑗,

Σ2,5

𝑖,𝑗= −𝑁

5

𝑖− 𝑂2

𝑖

𝑇

+ 𝐵2

𝑖

𝑇

𝑊5

𝑗,

Σ2,6

𝑖,𝑗= −𝑁

6

𝑖−𝑊2

𝑖

𝑇

+ 𝐵2

𝑖

𝑇

𝑊6

𝑗,

Σ2,7

𝑖,𝑗= −𝑁

7

𝑖+𝑊2

𝑖

𝑇

𝐶𝑗+ 𝐵2

𝑖

𝑇

𝑊7

𝑗,

Σ2,8

𝑖,𝑗= −𝑁

8

𝑖+𝑊2

𝑖

𝑇

+ 𝐵2

𝑖

𝑇

𝑊8

𝑗,

Σ2,9

𝑖,𝑗= −𝑁

9

𝑖

𝑇

−𝑊2

𝑖

𝑇

+ 𝐵2

𝑖

𝑇

𝑊9

𝑗,

Σ2,10

𝑖,𝑗= −𝑁

10

𝑖+𝑊2

𝑖

𝑇

+ 𝐵2

𝑖

𝑇

𝑊10

𝑗,

Σ3,3

𝑖= − (1 − 𝛽ℎ

𝑑) 𝑒−2𝛽𝑘ℎ

𝑃4

𝑖− 𝑒−2𝛽𝑘ℎ

𝑃10

𝑖+ 𝛽2

ℎ2

𝑅4

𝑖

− 2𝛽ℎ𝑅5

𝑖− 𝛽ℎ𝑀

7

𝑖

𝑇

− 𝛽ℎ𝑀7

𝑖+ 𝛽2

ℎ2

𝑀10

𝑖

− 𝑂3

𝑖

𝑇

− 𝑂3

𝑖,


Σ3,4

𝑖,𝑗= −𝑁

3

𝑖

𝑇

− 𝑂4

𝑖−𝑊3

𝑖

𝑇

𝐵1

𝑗,

Σ3,5

𝑖= −𝑂3

𝑖

𝑇

− 𝑂5

𝑖, Σ

3,6

𝑖= −𝑂9

𝑖+𝑊3

𝑖

𝑇

,

Σ3,7

𝑖= −𝑂7

𝑖+𝑊3

𝑖

𝑇

𝐶𝑗, Σ

3,8

𝑖= −𝑂8

𝑖+𝑊3

𝑖

𝑇

,

Σ3,9

𝑖= −𝑂9

𝑖+𝑊3

𝑖

𝑇

, Σ3,10

𝑖= −𝑂10

𝑖+𝑊3

𝑖

𝑇

,

Σ4,4

𝑖,𝑗= −𝑒2𝑘ℎ

𝑃7

𝑖− 𝑁4

𝑖

𝑇

− 𝑁4

𝑖+𝑊4

𝑖

𝑇

𝐵1

𝑗− 𝐵1

𝑖

𝑇

𝑊4

𝑗,

Σ4,5

𝑖,𝑗= −𝑁

5

𝑖− 𝑂4

𝑖

𝑇

− 𝐵1

𝑖

𝑇

𝑊5

𝑗,

Σ4,6

𝑖,𝑗= −𝑁

6

𝑖−𝑊4

𝑖

𝑇

− 𝐵1

𝑖

𝑇

𝑊6

𝑗,

Σ4,7

𝑖,𝑗= −𝑁

7

𝑖+𝑊4

𝑖

𝑇

𝐶𝑗− 𝐵1

𝑖

𝑇

𝑊7

𝑗, Σ

4,8

𝑖= −𝑂8

𝑖+𝑊3

𝑖

𝑇

,

Σ4,9

𝑖= −𝑂9

𝑖+𝑊3

𝑖

𝑇

, Σ4,10

𝑖= −𝑂10

𝑖+𝑊3

𝑖

𝑇

,

Σ5,5

𝑖= −𝑒2𝛽𝑘ℎ

𝑃8

𝑖− 𝑂5

𝑖

𝑇

− 𝑂5

𝑖, Σ

5,6

𝑖= −𝑂6

𝑖−𝑊5

𝑖

𝑇

,

Σ5,7

𝑖,𝑗= −𝑂7

𝑖+𝑊5

𝑖

𝑇

𝐶𝑗, Σ

5,8

𝑖= −𝑂8

𝑖+𝑊5

𝑖

𝑇

,

Σ5,9

𝑖= −𝑂9

𝑖+𝑊5

𝑖

𝑇

, Σ5,10

𝑖= −𝑂10

𝑖+𝑊5

𝑖

𝑇

,

Σ6,6

𝑖= 𝑃2

𝑖+ ℎ2

𝑃5

𝑖+ 𝛽2

ℎ2

𝑃6

𝑖+ ℎ2

𝑃7

𝑖+ 𝛽2

ℎ2

𝑃8

𝑖

+ ℎ2

𝑃9

𝑖+ 𝛽2

ℎ2

𝑃10

𝑖−𝑊6

𝑖

𝑇

−𝑊6

𝑖,

Σ6,7

𝑖,𝑗= 𝑊6

𝑖

𝑇

𝐶𝑗−𝑊7

𝑖, Σ

6,8

𝑖= 𝑊6

𝑖

𝑇

−𝑊8

𝑖,

Σ6,9

𝑖= 𝑊6

𝑖

𝑇

−𝑊9

𝑖, Σ

6,10

𝑖= 𝑊6

𝑖

𝑇

−𝑊10

𝑖,

Σ7,7

𝑖,𝑗= − (1 − 𝑟

𝑑) 𝑒−2𝑘𝑟

𝑃2

𝑖+ 𝜖3𝛾2

𝐼 + 𝑊7

𝑖

𝑇

𝐶𝑗+ 𝐶𝑇

𝑖𝑊7

𝑗,

Σ7,8

𝑖,𝑗= 𝑊7

𝑖

𝑇

− 𝐶𝑇

𝑖𝑊8

𝑗, Σ

7,9

𝑖,𝑗= 𝑊7

𝑖

𝑇

− 𝐶𝑇

𝑖𝑊9

𝑗,

Σ7,10

𝑖,𝑗= 𝑊7

𝑖

𝑇

− 𝐶𝑇

𝑖𝑊10

𝑗, Σ

8,8

𝑖= 𝑊8

𝑖

𝑇

+𝑊8

𝑖− 𝜖1𝐼,

Σ8,9

𝑖= 𝑊8

𝑖

𝑇

+𝑊9

𝑖, Σ

8,10

𝑖= 𝑊8

𝑖

𝑇

+𝑊10

𝑖,

Σ9,9

𝑖= 𝑊9

𝑖

𝑇

+𝑊9

𝑖− 𝜖2𝐼, Σ

9,10

𝑖= 𝑊9

𝑖

𝑇

+𝑊10

𝑖,

Σ10,10

𝑖= 𝑊10

𝑖

𝑇

+𝑊10

𝑖− 𝜖3𝐼,

̂Σ7,7

𝑖= − (1 − 𝑟𝑑) 𝑒

2𝑘𝑟

𝑃2

𝑖+𝑊7

𝑖

𝑇

𝐶𝑗+ 𝐶𝑇

𝑖𝑊7

𝑗,

𝑍1

𝑖= 𝑃1

𝑖𝐺.

(24)

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 thereexist symmetric positive definite matrices 𝑃𝑗

𝑖, any appropriate

dimensional matrices 𝑅𝑝𝑖, 𝑁𝑗𝑖, 𝑂𝑗𝑖, 𝑊𝑗𝑖, 𝑀𝑗𝑖, 𝐺, 𝑝 = 1, 2, . . . 6,

𝑗 = 1, 2, . . . , 10, 𝑖 = 1, 2, . . . , 𝑁, and positive real constants

𝜖1, 𝜖2, and 𝜖

3such that the following symmetric linear matrix

inequalities hold:

[

[

𝑅1

𝑖𝑅2

𝑖

∗ 𝑅3

𝑖

]

]

> 0, 𝑖 = 1, 2, . . . , 𝑁,

[

[

𝑅4

𝑖𝑅5

𝑖

∗ 𝑅6

𝑖

]

]

> 0, 𝑖 = 1, 2, . . . , 𝑁,

[

[

𝑒−2𝑘ℎ

𝑃3

𝑖− 𝑅3

𝑖𝑀1

𝑖𝑀2

𝑖

∗ 𝑀3

𝑖𝑀4

𝑖

∗ ∗ 𝑀5

𝑖

]

]

≥ 0, 𝑖 = 1, 2, . . . , 𝑁,

[

[

𝑒−2𝛽𝑘ℎ

𝑃6

𝑖− 𝑅6

𝑖𝑀6

𝑖𝑀7

𝑖

∗ 𝑀8

𝑖𝑀9

𝑖

∗ ∗ 𝑀10

𝑖

]

]

≥ 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,

𝑗 = 𝑖 + 1, 𝑖 + 2, . . . , 𝑁.

(25)

Moreover, the solution 𝑥(𝑡, 𝜙, 𝜓) satisfies the inequality

𝑥 (𝑡, 𝜙, 𝜓) ≤ √

𝐿

𝜆min (𝑃1 (𝛼))max [𝜙

,𝜓

] 𝑒−𝑘𝑡

,

∀𝑡 ∈ 𝑅+

,

(26)

where 𝐿 = 𝜆max(𝑃1(𝛼)) + 𝑟𝜆max(𝑃2(𝛼)) + ℎ𝜆max(𝑃3(𝛼))+𝛽ℎ𝜆max(𝑃4(𝛼)) + ℎ

3𝜆max(𝑃5(𝛼)+𝑃7(𝛼) + 𝑃9(𝛼)) +

(𝛽ℎ)3

𝜆max(𝑃6(𝛼)+𝑃8(𝛼) + 𝑃10(𝛼)) + ℎ3𝜆max ([𝑅1(𝛼) 𝑅2(𝛼)

∗ 𝑅3(𝛼)])+

(𝛽ℎ)3

𝜆max ([𝑅4(𝛼) 𝑅5(𝛼)

∗ 𝑅6(𝛼)]).

Proof. Choose a parameter-dependent Lyapunov-Krasovskiifunctional candidate for system (9) as

𝑉 (𝑡) =

7

∑

𝑖=1

𝑉𝑖(𝑡) , (27)

where

𝑉1(𝑡) = 𝑥

𝑇

(𝑡) 𝑃1(𝛼) 𝑥 (𝑡) ,

𝑉2(𝑡) = ∫

𝑡

𝑡−𝑟(𝑡)

𝑒2𝑘(𝑠−𝑡)

�̇�𝑇

(𝑠) 𝑃2(𝛼) �̇� (𝑠) 𝑑𝑠,

𝑉3(𝑡) = ∫

𝑡

𝑡−ℎ(𝑡)


𝑥𝑇

(𝑠) 𝑃3(𝛼) 𝑥 (𝑠) 𝑑𝑠

+ ∫

𝑡



𝑥𝑇

(𝑠) 𝑃4(𝛼) 𝑥 (𝑠) 𝑑𝑠,


𝑉4(𝑡) = ℎ∫

0

−ℎ

∫

𝑡

𝑡+𝜃


�̇�𝑇

(𝑠) 𝑃5(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃

+ 𝛽ℎ∫

0

−𝛽ℎ

∫

𝑡

𝑡+𝜃


�̇�𝑇

(𝑠) 𝑃6(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃,

𝑉5(𝑡) = ℎ∫

0

−ℎ

∫

𝑡

𝑡+𝜃


�̇�𝑇

(𝑠) 𝑃7(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃

+ 𝛽ℎ∫

0

−𝛽ℎ

∫

𝑡

𝑡+𝜃


�̇�𝑇

(𝑠) 𝑃8(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃,

𝑉6(𝑡) = ℎ∫

0

−ℎ

∫

𝑡

𝑡+𝜃


�̇�𝑇

(𝑠) 𝑃9(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃

+ 𝛽ℎ∫

0

−𝛽ℎ

∫

𝑡

𝑡+𝜃


�̇�𝑇

(𝑠) 𝑃10(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃,

𝑉7(𝑡) = ℎ∫

𝑡

−ℎ

∫

𝜃

𝜃−ℎ(𝜃)

𝑒2𝑘(𝜃−𝑡)

[𝑥 (𝜃 − ℎ (𝜃))

�̇� (𝑠)]

𝑇

× [𝑅1(𝛼) 𝑅

2(𝛼)

∗ 𝑅3(𝛼)

]

× [𝑥 (𝜃 − ℎ (𝜃))

�̇� (𝑠)] 𝑑𝑠 𝑑𝜃

+ 𝛽ℎ∫

𝑡

−𝛽ℎ

∫

𝜃

𝜃−𝛽ℎ(𝜃)

𝑒2𝑘(𝜃−𝑡)

[𝑥 (𝜃 − 𝛽ℎ (𝜃))

�̇� (𝑠)]

𝑇

× [𝑅4(𝛼) 𝑅

5(𝛼)

∗ 𝑅6(𝛼)

]

× [𝑥 (𝜃 − 𝛽ℎ (𝜃))

�̇� (𝑠)] 𝑑𝑠 𝑑𝜃.

(28)

Calculating the time derivatives of 𝑉𝑖(𝑡), 𝑖 = 1, 2, 3, . . . , 7,

along the trajectory of (9), yields

�̇�1(𝑡) = 2𝑥

𝑇

(𝑡) 𝑃1(𝛼) �̇� (𝑡)

= 2𝑥𝑇

𝑃1(𝛼) [ (𝐴 (𝛼) + 𝐵

1(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, �̇� (𝑡 − 𝑟 (𝑡))) + 𝐶 (𝛼) �̇� (𝑡 − 𝑟 (𝑡))

− 𝐵1(𝛼) ∫

𝑡

𝑡−ℎ(𝑡)

�̇� (𝑠) 𝑑𝑠

−𝐺∫

𝑡


�̇� (𝑠) 𝑑𝑠]

+ 2𝑘𝑥𝑇

(𝑡) 𝑃1(𝛼) 𝑥 (𝑡) − 2𝑘𝑉

1(𝑡) ,

�̇�2(𝑡) = �̇�

𝑇

(𝑡) 𝑃2(𝛼) �̇� (𝑡)

− (1 − ̇𝑟 (𝑡)) 𝑒−2𝑘𝑟(𝑡)

�̇�𝑇

(𝑡 − 𝑟 (𝑡))

× 𝑃2(𝛼) �̇� (𝑡 − 𝑟 (𝑡)) − 2𝑘𝑉

2(𝑡)

≤ �̇�𝑇

(𝑡) 𝑃2(𝛼) �̇� (𝑡) − 𝑒

−2𝑘𝑟

�̇�𝑇

(𝑡 − 𝑟 (𝑡))

× 𝑃2(𝛼) �̇� (𝑡 − 𝑟 (𝑡))

+ 𝑟𝑑�̇�𝑇

(𝑡 − 𝑟 (𝑡)) 𝑃2(𝛼) �̇� (𝑡 − 𝑟 (𝑡))

− 2𝑘𝑉2(𝑡) .

(29)

The time derivative of 𝑉3(𝑡) is

�̇�3(𝑡) = 𝑥

𝑇

(𝑡) 𝑃3(𝛼) 𝑥 (𝑡) − (1 − ℎ̇ (𝑡)) 𝑒

−2𝑘ℎ(𝑡)

𝑥𝑇

(𝑡 − ℎ (𝑡))

× 𝑃3(𝛼) 𝑥 (𝑡 − ℎ (𝑡)) + 𝑥

𝑇

(𝑡) 𝑃4(𝛼) 𝑥 (𝑡)

− (1 − 𝛽ℎ̇ (𝑡)) 𝑒−2𝑘𝛽ℎ(𝑡)

𝑥𝑇

(𝑡 − 𝛽ℎ (𝑡))

× 𝑃4(𝛼) 𝑥 (𝑡 − 𝛽ℎ (𝑡)) − 2𝑘𝑉

3(𝑡)

≤ 𝑥𝑇

(𝑡) 𝑃3(𝛼) 𝑥 (𝑡) − 𝑒

−2𝑘ℎ

𝑥𝑇

(𝑡 − ℎ (𝑡))

× 𝑃3(𝛼) 𝑥 (𝑡 − ℎ (𝑡)) + ℎ

𝑑𝑒−2𝑘ℎ

𝑥𝑇

(𝑡 − ℎ (𝑡))

× 𝑃3(𝛼) 𝑥 (𝑡 − ℎ (𝑡)) + 𝑥

𝑇

(𝑡) 𝑃4(𝛼) 𝑥 (𝑡)

− 𝑒−2𝛼𝛽ℎ

𝑥𝑇

(𝑡 − 𝛽ℎ (𝑡)) 𝑃4(𝛼) 𝑥 (𝑡 − 𝛽ℎ (𝑡))

+ 𝛽ℎ𝑑𝑒−2𝑘𝛽ℎ

𝑥𝑇

(𝑡 − 𝛽ℎ (𝑡)) 𝑃4(𝛼) 𝑥 (𝑡 − 𝛽ℎ (𝑡))

− 2𝑘𝑉3(𝑡) .

(30)

Obviously, for any scalar 𝑠 ∈ [𝑡 − ℎ, 𝑡], we get 𝑒−2𝑘ℎ ≤𝑒−2𝑘(𝑠−𝑡)

≤ 1 and for any scalar 𝑠 ∈ [𝑡 − 𝛽ℎ, 𝑡], we obtain𝑒−2𝛽𝑘ℎ

≤ 𝑒−2𝛽𝑘(𝑠−𝑡)

≤ 1. Together with Lemma 4, we obtain

�̇�4(𝑡) = ℎ

2

�̇�𝑇

(𝑡) 𝑃5(𝛼) �̇� (𝑡)

− ℎ∫

0

−ℎ

𝑒2𝑘𝑠

�̇�𝑇

(𝑡 + 𝑠) 𝑃5(𝛼) �̇� (𝑡 + 𝑠) 𝑑𝑠

+ 𝛽2

ℎ2

�̇�𝑇

(𝑡) 𝑃6(𝛼) �̇� (𝑡)

− 𝛽ℎ∫

0

−𝛽ℎ

𝑒2𝛽𝑘𝑠

�̇�𝑇

(𝑡 + 𝑠) 𝑃6(𝛼) �̇� (𝑡 + 𝑠) 𝑑𝑠

− 2𝑘𝑉4(𝑡)


≤ ℎ2

�̇�𝑇

(𝑡) 𝑃5(𝛼) �̇� (𝑡)

− ℎ∫

𝑡

𝑡−ℎ


�̇�𝑇

(𝑠) 𝑃5(𝛼) �̇� (𝑠) 𝑑𝑠

+ 𝛽2

ℎ2

�̇�𝑇

(𝑡) 𝑃6(𝛼) �̇� (𝑡)

− 𝛽ℎ∫

𝑡

𝑡−𝛽ℎ

𝑒2𝛽𝑘(𝑠−𝑡)

�̇�𝑇

(𝑠) 𝑃6(𝛼) �̇� (𝑠) 𝑑𝑠

− 2𝑘𝑉4(𝑡)

≤ ℎ2

�̇�𝑇

(𝑡) 𝑃5(𝛼) �̇� (𝑡)

− ℎ𝑒−2𝑘ℎ

∫

𝑡

𝑡−ℎ

�̇�𝑇

(𝑠) 𝑃5(𝛼) �̇� (𝑠) 𝑑𝑠

+ 𝛽2

ℎ2

�̇�𝑇

(𝑡) 𝑃6(𝛼) �̇� (𝑡)

− 𝛽ℎ𝑒−2𝛽𝑘ℎ

∫

𝑡

𝑡−𝛽ℎ

�̇�𝑇

(𝑠) 𝑃6(𝛼) �̇� (𝑠) 𝑑𝑠

− 2𝑘𝑉4(𝑡) ,

�̇�5(𝑡) ≤ ℎ

2

�̇�𝑇

(𝑡) 𝑃7(𝛼) �̇� (𝑡)

− ℎ𝑒−2𝑘ℎ

∫

𝑡

𝑡−ℎ

�̇�𝑇

(𝑠) 𝑃7(𝛼) �̇� (𝑠) 𝑑𝑠

+ 𝛽2

ℎ2

�̇�𝑇

(𝑡) 𝑃8(𝛼) �̇� (𝑡)

− 𝛽ℎ𝑒−2𝛽𝑘ℎ

∫

𝑡

𝑡−𝛽ℎ

�̇�𝑇

(𝑠) 𝑃8(𝛼) �̇� (𝑠) 𝑑𝑠

− 2𝑘𝑉5(𝑡)

≤ ℎ2

�̇�𝑇

(𝑡) 𝑃7(𝛼) �̇� (𝑡)

− 𝑒−2𝑘ℎ

∫

𝑡

𝑡−ℎ

�̇�𝑇

(𝑠) 𝑑𝑠𝑃7(𝛼)∫

𝑡

𝑡−ℎ

�̇� (𝑠) 𝑑𝑠

+ 𝛽2

ℎ2

�̇�𝑇

(𝑡) 𝑃8(𝛼) �̇� (𝑡)

− 𝑒−2𝛽𝑘ℎ

∫

𝑡

𝑡−𝛽ℎ

�̇�𝑇

(𝑠) 𝑑𝑠𝑃8(𝛼) ∫

𝑡

𝑡−𝛽ℎ

�̇� (𝑠) 𝑑𝑠

− 2𝑘𝑉5(𝑡)

≤ ℎ2

�̇�𝑇

(𝑡) 𝑃7(𝛼) �̇� (𝑡)

− 𝑒−2𝑘ℎ

∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠) 𝑑𝑠𝑃7(𝛼) ∫

𝑡

𝑡−ℎ(𝑡)

�̇� (𝑠) 𝑑𝑠

+ 𝛽2

ℎ2

�̇�𝑇

(𝑡) 𝑃8(𝛼) �̇� (𝑡)


∫

𝑡


�̇�𝑇

(𝑠) 𝑑𝑠𝑃8(𝛼) ∫

𝑡


�̇� (𝑠) 𝑑𝑠

− 2𝛼𝑉5(𝑡) ,

�̇�6(𝑡) ≤ ℎ

2

�̇�𝑇

(𝑡) 𝑃9(𝛼) �̇� (𝑡)

− 𝑒−2𝑘ℎ

∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠) 𝑑𝑠𝑃9(𝛼) ∫

𝑡

𝑡−ℎ(𝑡)

�̇� (𝑠) 𝑑𝑠

+ 𝛽2

ℎ2

�̇�𝑇

(𝑡) 𝑃10(𝛼) �̇� (𝑡)


∫

𝑡


�̇�𝑇

(𝑠) 𝑑𝑠𝑃10(𝛼)∫

𝑡


�̇� (𝑠) 𝑑𝑠

− 2𝑘𝑉6(𝑡)

= ℎ2

�̇�𝑇

(𝑡) 𝑃9(𝛼) �̇� (𝑡)

− 𝑒−2𝑘ℎ

[𝑥𝑇

(𝑡) − 𝑥𝑇

(𝑡 − ℎ (𝑡))]

× 𝑃9(𝛼) [𝑥 (𝑡) − 𝑥 (𝑡 − ℎ (𝑡))]

+ 𝛽ℎ2

�̇�𝑇

(𝑡) 𝑃10(𝛼) �̇� (𝑡)


[𝑥𝑇

(𝑡) − 𝑥𝑇

(𝑡 − 𝛽ℎ (𝑡))]

× 𝑃10(𝛼) [𝑥 (𝑡) − 𝑥 (𝑡 − 𝛽ℎ (𝑡))] − 2𝑘𝑉

6(𝑡) .

(31)

Taking the time derivative of 𝑉7(𝑡), we obtain

�̇�7(𝑡) = ℎℎ (𝑡) 𝑥

𝑇

(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

𝑡−ℎ

�̇�𝑇

(𝑠) 𝑅3(𝛼) �̇� (𝑠) 𝑑𝑠

+ 𝛽2

ℎℎ (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

𝑡−𝛽ℎ

�̇�𝑇

(𝑠) 𝑅6(𝛼) �̇� (𝑠) 𝑑𝑠 − 2𝑘𝑉

7(𝑡)

≤ ℎ2

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

𝑡−ℎ

�̇�𝑇

(𝑠) 𝑅3(𝛼) �̇� (𝑠) 𝑑𝑠

+ 𝛽2

ℎ2

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

𝑡−𝛽ℎ

�̇�𝑇

(𝑠) 𝑅6(𝛼) �̇� (𝑠) 𝑑𝑠 − 2𝑘𝑉

7(𝑡) .

(32)


By Lemma 5 and the integral term of the right hand sideof �̇�4(𝑡) and �̇�

7(𝑡), we obtain

− ℎ∫

𝑡

𝑡−ℎ

�̇�𝑇

(𝑠) [𝑒−2𝑘ℎ

𝑃5(𝛼) − 𝑅

3(𝛼)] �̇� (𝑠) 𝑑𝑠

− 𝛽ℎ∫

𝑡

𝑡−𝛽ℎ

�̇�𝑇

(𝑠) [𝑒−2𝛽𝑘ℎ

𝑃6(𝛼) − 𝑅

6(𝛼)] �̇� (𝑠) 𝑑𝑠

≤ ℎ[𝑥 (𝑡)

𝑥𝑡 − ℎ (𝑡)]

𝑇

× [𝑀𝑇

1(𝛼) + 𝑀

1(𝛼) −𝑀

𝑇

1(𝛼) + 𝑀

2(𝛼)

∗ −𝑀𝑇

2(𝛼) − 𝑀

2(𝛼)

]

× [𝑥 (𝑡)

𝑥𝑡 − ℎ (𝑡)]

+ ℎ2

[𝑥 (𝑡)

𝑥𝑡 − ℎ (𝑡)]

𝑇

[𝑀3(𝛼) 𝑀

4(𝛼)

∗ 𝑀5(𝛼)

]

× [𝑥 (𝑡)

𝑥𝑡 − ℎ (𝑡)] + 𝛽ℎ[

𝑥 (𝑡)

𝑥𝑡 − 𝛽ℎ (𝑡)]

𝑇

× [𝑀𝑇

6(𝛼) + 𝑀

6(𝛼) −𝑀

𝑇

6(𝛼) + 𝑀

7(𝛼)

∗ −𝑀𝑇

7(𝛼) − 𝑀

7(𝛼)

]

× [𝑥 (𝑡)


+ 𝛽2

ℎ2

[𝑥 (𝑡)


𝑇

[𝑀8(𝛼) 𝑀

9(𝛼)

∗ 𝑀10(𝛼)

]

× [𝑥 (𝑡)

𝑥𝑡 − 𝛽ℎ (𝑡)] .

(33)

From the Leibniz-Newton formula, the following equa-tions are true for any parameter-dependent real matrices𝑁𝑖(𝛼), 𝑂

𝑖(𝛼), 𝑖 = 1, 2, . . . , 10 with appropriate dimensions:

2 [𝑥𝑇

(𝑡)𝑁𝑇

1(𝛼) + 𝑥

𝑇

(𝑡 − ℎ (𝑡))𝑁𝑇

2(𝛼)

+ 𝑥𝑇

(𝑡 − 𝛽ℎ (𝑡))𝑁𝑇

3(𝛼) + ∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠) 𝑑𝑠𝑁𝑇

4(𝛼)

+ ∫

𝑡


�̇�𝑇

(𝑠) 𝑑𝑠𝑁𝑇

5(𝛼) + �̇�

𝑇

(𝑡)𝑁𝑇

6(𝛼)

+ �̇�𝑇

(𝑡 − 𝑟 (𝑡))𝑁𝑇

7(𝛼) + 𝑓

𝑇

(𝑡, 𝑥 (𝑡))𝑁𝑇

8(𝛼)

+𝑔𝑇

(𝑡, 𝑥 (𝑡 − ℎ (𝑡)))𝑁𝑇

9(𝛼) + 𝑤

𝑇

(𝑡, �̇� (𝑡 − 𝑟 (𝑡)))𝑁𝑇

10(𝛼) ]

× [𝑥 (𝑡) − 𝑥 (𝑡 − ℎ (𝑡)) − ∫

𝑡

𝑡−ℎ(𝑡)

�̇� (𝑠) 𝑑𝑠] = 0,

2 [𝑥𝑇

(𝑡) 𝑂𝑇

1(𝛼) + 𝑥

𝑇

(𝑡 − ℎ (𝑡)) 𝑂𝑇

2(𝛼)

+ 𝑥𝑇

(𝑡 − 𝛽ℎ (𝑡)) 𝑂𝑇

3(𝛼) + ∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠) 𝑑𝑠𝑂𝑇

4(𝛼)

+ ∫

𝑡


�̇�𝑇

(𝑠) 𝑑𝑠𝑂𝑇

5(𝛼) + �̇�

𝑇

(𝑡) 𝑂𝑇

6(𝛼)

+ �̇�𝑇

(𝑡 − 𝑟 (𝑡)) 𝑂𝑇

7(𝛼) + 𝑓

𝑇

(𝑡, 𝑥 (𝑡)) 𝑂𝑇

8(𝛼)

+ 𝑔𝑇

(𝑡, 𝑥 (𝑡 − ℎ (𝑡))) 𝑂𝑇

9(𝛼) +𝑤

𝑇

(𝑡, �̇� (𝑡 − 𝑟 (𝑡))) 𝑂𝑇

10(𝛼) ]

× [𝑥 (𝑡) − 𝑥 (𝑡 − 𝛽ℎ (𝑡)) − ∫

𝑡


�̇� (𝑠) 𝑑𝑠] = 0.

(34)

From the utilization of zero equation, the followingequation is true for any parameter-dependent real matrices𝑊𝑖, 𝑖 = 1, 2, . . . , 10 with appropriate dimensions:

2 [𝑥𝑇

(𝑡)𝑊𝑇

1(𝛼) + 𝑥

𝑇

(𝑡 − ℎ (𝑡))𝑊𝑇

2(𝛼)

+ 𝑥𝑇

(𝑡 − 𝛽ℎ (𝑡))𝑊𝑇

3(𝛼) + ∫

𝑡

𝑡−ℎ(𝑡)

�̇�𝑇

(𝑠) 𝑑𝑠𝑊𝑇

4(𝛼)

+ ∫

𝑡


�̇�𝑇

(𝑠) 𝑑𝑠𝑊𝑇

5(𝛼) + �̇�

𝑇

(𝑡)𝑊𝑇

6(𝛼)

+ �̇�𝑇

(𝑡 − 𝑟 (𝑡))𝑊𝑇

7(𝛼) + 𝑓

𝑇

(𝑡, 𝑥 (𝑡))𝑊8(𝛼)

+ 𝑔𝑇

(𝑡, 𝑥 (𝑡 − ℎ (𝑡)))𝑊𝑇

9(𝛼)

+𝑤𝑇

(𝑡, �̇� (𝑡 − 𝑟 (𝑡)))𝑊𝑇

10(𝛼) ]

× [ (𝐴 (𝛼) + 𝐵1(𝛼)) 𝑥 (𝑡) + 𝐵

2(𝛼) 𝑥 (𝑡 − ℎ (𝑡))

+ 𝐶 (𝛼) �̇� (𝑡 − 𝑟 (𝑡)) + 𝑓 (𝑡, 𝑥 (𝑡)) + 𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡)))

+ 𝑤 (𝑡, �̇� (𝑡 − 𝑟 (𝑡))) − 𝐵1(𝛼) ∫

𝑡

𝑡−ℎ(𝑡)

�̇� (𝑠) 𝑑𝑠 − �̇� (𝑡)]

= 0.

(35)

From (5), we obtain, for any positive real constants 𝜖1, 𝜖2,

and 𝜖3,

0 ≤ 𝜖1𝜂2

𝑥𝑇

(𝑡) 𝑥 (𝑡) − 𝜖1𝑓𝑇

(𝑡, 𝑥 (𝑡)) 𝑓 (𝑡, 𝑥 (𝑡)) ,

0 ≤ 𝜖2𝜌2

𝑥𝑇

(𝑡 − ℎ (𝑡)) 𝑥 (𝑡 − ℎ (𝑡))

− 𝜖2𝑔𝑇

(𝑡, 𝑥 (𝑡 − ℎ (𝑡))) 𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡))) ,

0 ≤ 𝜖3𝛾2

�̇�𝑇

(𝑡 − 𝑟 (𝑡)) �̇� (𝑡 − 𝑟 (𝑡))

− 𝜖3𝑤𝑇

(𝑡, �̇� (𝑡 − 𝑟 (𝑡))) 𝑤 (𝑡, �̇� (𝑡 − 𝑟 (𝑡))) .

(36)


According to (29)–(36), it is straightforward to see that

�̇� (𝑡) ≤ 𝜁𝑇

(𝑡)

𝑁

∑

𝑖=1

𝑁

∑

𝑗=1

𝛼𝑖𝛼𝑗∑

𝑖,𝑗

𝜁 (𝑡) − 2𝑘𝑉 (𝑡) , (37)

where 𝜁𝑇(𝑡) = [𝑥𝑇(𝑡),𝑥𝑇(𝑡−ℎ(𝑡)),𝑥𝑇(𝑡−𝛽ℎ(𝑡)),∫𝑡𝑡−ℎ(𝑡)

�̇�𝑇(𝑠)𝑑𝑠,

∫𝑡

𝑡−𝛽ℎ(𝑡)�̇�𝑇(𝑠)𝑑𝑠, �̇�

𝑇(𝑡), �̇�𝑇(𝑡 − 𝑟(𝑡)), 𝑓𝑇(𝑡, 𝑥(𝑡)), 𝑔𝑇(𝑡, 𝑥(𝑡 −

ℎ(𝑡))), 𝑤𝑇(𝑡, �̇�(𝑡 − 𝑟(𝑡))] and ∑𝑖,𝑗is defined in (23). From the

fact that ∑𝑁𝑖=1

𝛼𝑖= 1,

𝑁

∑

𝑖=1

𝛼𝑖𝐴𝑖

𝑁

∑

𝑖=1

𝛼𝑖𝐵𝑖=

𝑁

∑

𝑖=1

𝛼2

𝑖𝐴𝑖𝐵𝑖+

𝑁−1

∑

𝑖=1

𝑁

∑

𝑗=𝑖+1

𝛼𝑖𝛼𝑗[𝐴𝑖𝐵𝑗+ 𝐴𝑗𝐵𝑖] ,

(𝑁 − 1)

𝑁

∑

𝑖=1

𝛼2

𝑖− 2

𝑁−1

∑

𝑖=1

𝑁

∑

𝑗=𝑖+1

𝛼𝑖𝛼𝑗=

𝑁−1

∑

𝑖=1

𝑁

∑

𝑗=𝑖+1

[𝛼𝑖− 𝛼𝑗]2

≥ 0.

(38)It is true that if conditions (25) hold, then

�̇� (𝑡) + 2𝑘𝑉 (𝑡) ≤ 0, ∀𝑡 ∈ 𝑅+

, (39)which gives

𝑉 (𝑡) ≤ 𝑉 (0) 𝑒−2𝑘𝑡

, ∀𝑡 ∈ 𝑅+

. (40)From (40), it is easy to see that

𝜆min (𝑃1 (𝛼)) ‖𝑥 (𝑡)‖2

≤ 𝑉 (𝑡) ≤ 𝑉 (0) 𝑒−2𝑘𝑡

, (41)

𝑉 (0) =

7

∑

𝑖=1

𝑉𝑖(0) , (42)

where𝑉1(0) = 𝑥

𝑇

(0) 𝑃1(𝛼) 𝑥 (0) ,

𝑉2(0) = ∫

0

−𝑟(0)

𝑒2𝑘𝑠

�̇�𝑇

(𝑠) 𝑃2(𝛼) �̇� (𝑠) 𝑑𝑠,

𝑉3(0) = ∫

0

−ℎ(0)

𝑒2𝑘𝑠

𝑥𝑇

(𝑠) 𝑃3(𝛼) 𝑥 (𝑠) 𝑑𝑠

+ ∫

0

−𝛽ℎ(0)

𝑒2𝑘𝑠

𝑥𝑇

(𝑠) 𝑃4(𝛼) 𝑥 (𝑠) 𝑑𝑠,

𝑉4(0) = ℎ∫

0

−ℎ

∫

0

𝜃

𝑒2𝑘𝑠

�̇�𝑇

(𝑠) 𝑃5(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃

+ 𝛽ℎ∫

0

−𝛽ℎ

∫

0

𝜃

𝑒2𝑘𝑠

�̇�𝑇

(𝑠) 𝑃6(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃,

𝑉5(0) = ℎ∫

0

−ℎ

∫

0

𝜃

𝑒2𝑘𝑠

�̇�𝑇

(𝑠) 𝑃7(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃

+ 𝛽ℎ∫

0

−𝛽ℎ

∫

0

𝜃

𝑒2𝑘𝑠

�̇�𝑇

(𝑠) 𝑃8(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃,

𝑉6(0) = ℎ∫

0

−ℎ

∫

𝑡

𝜃

𝑒2𝑘𝑠

�̇�𝑇

(𝑠) 𝑃9(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃

+ 𝛽ℎ∫

0

−𝛽ℎ

∫

0

𝜃

𝑒2𝑘𝑠

�̇�𝑇

(𝑠) 𝑃10(𝛼) �̇� (𝑠) 𝑑𝑠 𝑑𝜃,

𝑉7(0) = ℎ∫

0

−ℎ

∫

𝜃

𝜃−ℎ(𝜃)

𝑒2𝑘𝜃

[𝑥 (𝜃 − ℎ (𝜃))

�̇� (𝑠)]

𝑇

× [𝑅1(𝛼) 𝑅

2(𝛼)

∗ 𝑅3(𝛼)

]

× [𝑥 (𝜃 − ℎ (𝜃))

�̇� (𝑠)] 𝑑𝑠 𝑑𝜃

+ 𝛽ℎ∫

0

−𝛽ℎ

∫

𝜃

𝜃−𝛽ℎ(𝜃)

𝑒2𝑘𝜃

[𝑥 (𝜃 − 𝛽ℎ (𝜃))

�̇� (𝑠)]

𝑇

× [𝑅4(𝛼) 𝑅

5(𝛼)

∗ 𝑅6(𝛼)

]

× [𝑥 (𝜃 − 𝛽ℎ (𝜃))

�̇� (𝑠)] 𝑑𝑠 𝑑𝜃.

(43)

From (41), we conclude that

𝜆min (𝑃1 (𝛼)) ‖𝑥 (𝑡)‖2

≤ 𝑉 (0) 𝑒−2𝑘𝑡

≤ 𝐿max [𝜙 ,𝜑

]2

𝑒−2𝑘𝑡

,

(44)

where 𝐿 = 𝜆max(𝑃1(𝛼)) + 𝑟𝜆max(𝑃2(𝛼))+ ℎ𝜆max(𝑃3(𝛼)) +𝛽ℎ𝜆max(𝑃4(𝛼))+ ℎ

3𝜆max(𝑃5(𝛼) + 𝑃7(𝛼)+𝑃9(𝛼)) + (𝛽ℎ)

3

𝜆max(𝑃6(𝛼) +𝑃8(𝛼) +𝑃10(𝛼)) + ℎ3𝜆max ([

𝑅1(𝛼) 𝑅2(𝛼)

∗ 𝑅3(𝛼)]) + (𝛽ℎ)

3


∗ 𝑅6(𝛼)]). From (44), this means that the system

(1) is robustly exponentially stable. The proof of the theoremis complete.

Next, we consider the following system:

�̇� (𝑡) = 𝐴 (𝛼) 𝑥 (𝑡) + 𝐵 (𝛼) 𝑥 (𝑡 − ℎ (𝑡)) + 𝐶 (𝛼) �̇� (𝑡 − 𝑟 (𝑡))

+ 𝑓 (𝑡, 𝑥 (𝑡)) + 𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡))) , 𝑡 > 0;

𝑥 (𝑡) = 𝜙 (𝑡) , �̇� (𝑡) = 𝜓 (𝑡) , 𝑡 ∈ [−𝑏, 0] .

(45)

We introduce the following notations for later use:

∏

𝑖,𝑗

=

[[[[[[[[[[[[[[[[

[

Σ1,1

𝑖,𝑗Σ1,2

𝑖,𝑗Σ1,3

𝑖,𝑗Σ1,4

𝑖,𝑗Σ1,5

𝑖,𝑗Σ1,6

𝑖,𝑗Σ1,7

𝑖,𝑗Σ1,8

𝑖,𝑗Σ1,9

𝑖,𝑗

∗ Σ2,2

𝑖,𝑗Σ2,3

𝑖,𝑗Σ2,4

𝑖,𝑗Σ2,5

𝑖,𝑗Σ2,6

𝑖,𝑗Σ2,7

𝑖,𝑗Σ2,8

𝑖,𝑗Σ2,9

𝑖,𝑗

∗ ∗ Σ3,3

𝑖Σ3,4

𝑖,𝑗Σ3,5

𝑖Σ3,6

𝑖Σ3,7

𝑖,𝑗Σ3,8

𝑖Σ3,9

𝑖

∗ ∗ ∗ Σ4,4

𝑖,𝑗Σ4,5

𝑖,𝑗Σ4,6

𝑖,𝑗Σ4,7

𝑖,𝑗Σ4,8

𝑖Σ4,9

𝑖

∗ ∗ ∗ ∗ Σ5,5

𝑖Σ5,6

𝑖Σ5,7

𝑖,𝑗Σ5,8

𝑖Σ5,9

𝑖

∗ ∗ ∗ ∗ ∗ Σ6,6

𝑖Σ6,7

𝑖,𝑗Σ6,8

𝑖Σ6,9

𝑖

∗ ∗ ∗ ∗ ∗ ∗ Σ̂7,7

𝑖,𝑗Σ7,8

𝑖,𝑗Σ7,9

𝑖,𝑗

∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ8,8

𝑖Σ8,9

𝑖

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ9,9

𝑖

]]]]]]]]]]]]]]]]

]

.

(46)

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 thereexist symmetric positive definite matrices 𝑃𝑗

𝑖, any appropriate

dimensional matrices 𝑅𝑝𝑖, 𝑀𝑙𝑖, 𝑁𝑗𝑖, 𝑂𝑗𝑖, 𝑊𝑗𝑖, 𝑝 = 1, 2, . . . 6,

𝑙 = 1, 2, . . . , 10, 𝑗 = 1, 2, . . . , 9, 𝑖 = 1, 2, . . . , 𝑁, and positive


real constants 𝜖1, 𝜖2such that the following symmetric linear

matrix inequalities hold:

[𝑅1

𝑖𝑅2

𝑖

∗ 𝑅3

𝑖

] > 0, 𝑖 = 1, 2, . . . , 𝑁,

[𝑅4

𝑖𝑅5

𝑖

∗ 𝑅6

𝑖

] > 0, 𝑖 = 1, 2, . . . , 𝑁,

[

[

𝑒−2𝑘ℎ

𝑃3

𝑖− 𝑅3

𝑖𝑀1

𝑖𝑀2

𝑖

∗ 𝑀3

𝑖𝑀4

𝑖

∗ ∗ 𝑀5

𝑖

]

]

≥ 0, 𝑖 = 1, 2, . . . , 𝑁,

[

[

𝑒−2𝛽𝑘ℎ

𝑃6

𝑖− 𝑅6

𝑖𝑀6

𝑖𝑀7

𝑖

∗ 𝑀8

𝑖𝑀9

𝑖

∗ ∗ 𝑀10

𝑖

]

]

≥ 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,

𝑗 = 𝑖 + 1, 𝑖 + 2, . . . , 𝑁.

(47)


𝑥 (𝑡, 𝜙, 𝜓) ≤ √

𝐿

𝜆min (𝑃1 (𝛼))max [𝜙

,𝜓

] 𝑒−𝑘𝑡

,

∀𝑡 ∈ 𝑅+

,

(48)

where 𝐿 = 𝜆max(𝑃1(𝛼)) + 𝑟𝜆max(𝑃2(𝛼)) + ℎ𝜆max(𝑃3(𝛼)) +𝛽ℎ𝜆max(𝑃4(𝛼)) + ℎ

3𝜆max(𝑃5(𝛼) + 𝑃7(𝛼) + 𝑃9(𝛼)) +

(𝛽ℎ)3

𝜆max(𝑃6(𝛼) + 𝑃8(𝛼) + 𝑃10(𝛼)) + ℎ3𝜆max ([

𝑅1(𝛼) 𝑅2(𝛼)

∗ 𝑅3(𝛼)]) +

(𝛽ℎ)3


∗ 𝑅6(𝛼)]).

3.2. Exponential Stability Criteria. In this section, we studythe exponential stability criteria for neutral systems withtime-varying delays by using the combination of linearmatrixinequality (LMI) technique and Lyapunov theory method.We introduce the following notations for later use:

∑ =

[[[[[[[[[[[[[[

[

Σ11

Σ12

Σ13

Σ14

Σ15

Σ16

Σ17

Σ18

Σ19

Σ1,10

∗ Σ22

Σ23

Σ24

Σ25

Σ26

Σ27

Σ28

Σ29

Σ2,10

∗ ∗ Σ33

Σ34

Σ35

Σ36

Σ37

Σ38

Σ39

Σ3,10

∗ ∗ ∗ Σ44

Σ45

Σ46

Σ47

Σ48

Σ49

Σ4,10

∗ ∗ ∗ ∗ Σ55

Σ56

Σ57

Σ58

Σ59

Σ5,10

∗ ∗ ∗ ∗ ∗ Σ66

Σ67

Σ68

Σ69

Σ6,10

∗ ∗ ∗ ∗ ∗ ∗ Σ77

Σ78

Σ79

Σ7,10

∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ88

Σ89

Σ8,10

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ99

Σ9,10

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ10,10

]]]]]]]]]]]]]]

]

,

(49)

where

Σ1,1

= 𝐴𝑇

𝑃1+ 𝐵𝑇

1𝑃1+ 𝑍𝑇

1+ 𝑃1𝐴 + 𝑃

1𝐵1+ 𝑍1+ 𝑃3

+ 𝑃4+ ℎ𝑀

𝑇

1+ ℎ𝑀

1− 𝑒−2𝑘ℎ

𝑃9− 𝑒−2𝑘𝛽ℎ

𝑃10

+ ℎ2

𝑀3+ 𝛽ℎ𝑀

6+ 𝛽ℎ𝑀

𝑇

6+ 𝛽2

ℎ2

𝑀8

+ 𝑁𝑇

1+ 𝑁1+ 𝑂𝑇

1+ 𝑂1+ 𝐴𝑇

𝑊1+𝑊𝑇

1𝐴

+ 𝐵𝑇

1𝑊1+𝑊𝑇

1𝐵1+ 𝜖1𝜂2

𝐼 + 2𝑘𝑃1,

Σ1,2

= 𝑃1𝐵2− ℎ𝑀

𝑇

1+ ℎ𝑀

2+ ℎ2

𝑀4+ 𝑒−2𝑘ℎ

𝑃9

+ ℎ𝑅2− 𝑁𝑇

1+ 𝑁2+ 𝑂2+𝑊𝑇

1𝐵2

+ 𝐴𝑇

𝑊2+ 𝐵𝑇

1𝑊2,

Σ1,3

= −𝑍1− 𝛽ℎ𝑀

𝑇

6+ 𝛽ℎ𝑀

7+ 𝛽ℎ2

𝑀9+ 𝛽ℎ𝑅

5

+ 𝑒−2𝑘𝛽ℎ

𝑃10+ 𝑁3− 𝑂𝑇

1+ 𝑂3

+ 𝐴𝑇

𝑊3+ 𝐵𝑇

1𝑊3,

Σ1,4

= −𝑃1𝐵1− 𝑁𝑇

1+ 𝑁4+ 𝑂4

−𝑊𝑇

1𝐵1+ 𝐴𝑇

𝑊4+ 𝐵𝑇

1𝑊4,

Σ1,5

= −𝑍1+ 𝑁5− 𝑂𝑇

1+ 𝑂5+ 𝐴𝑇

𝑊5+ 𝐵𝑇

1𝑊5,

Σ1,6

= 𝑁6+ 𝑂6−𝑊𝑇

1+ 𝐴𝑇

𝑊6+ 𝐵𝑇

1𝑊6,

Σ1,7

= 𝑃1𝐶 + 𝑁

7+ 𝑂7+𝑊𝑇

1𝐶 + 𝐴

𝑇

𝑊7+ 𝐵𝑇

1𝑊7,

Σ1,8

= 𝑃1+ 𝑁8+ 𝑂8+𝑊𝑇

1+ 𝐴𝑇

𝑊8+ 𝐵𝑇

1𝑊8,

Σ1,9

= 𝑃1+ 𝑁9+ 𝑂9+𝑊𝑇

1+ 𝐴𝑇

𝑊9+ 𝐵𝑇

1𝑊9,

Σ1,10

= 𝑃1+ 𝑁10+ 𝑂10+𝑊𝑇

1+ 𝐴𝑇

𝑊10+ 𝐵𝑇

1𝑊10,

Σ2,2

= − (1 − ℎ𝑑) 𝑒−2𝑘ℎ

𝑃3− ℎ𝑀

𝑇

2− ℎ𝑀

2

+ ℎ2

𝑀5+ ℎ2

𝑅1− 2ℎ𝑅

2− 𝑒−2𝑘ℎ

𝑃9

+ 𝜖2𝜌2

𝐼 − 𝑁𝑇

2− 𝑁2+𝑊𝑇

2𝐵2+ 𝐵𝑇

2𝑊2,

Σ2,3

= −𝑁3− 𝑂𝑇

2+ 𝐵𝑇

2𝑊3,

Σ2,4

= −𝑁𝑇

2− 𝑁4−𝑊𝑇

2𝐵1+ 𝐵𝑇

2𝑊4,

Σ2,5

= −𝑁5− 𝑂𝑇

2+ 𝐵𝑇

2𝑊5,

Σ2,6

= −𝑁6−𝑊𝑇

2+ 𝐵𝑇

2𝑊6,

Σ2,7

= −𝑁7+𝑊𝑇

2𝐶 + 𝐵

𝑇

2𝑊7,

Σ2,8

= −𝑁8+𝑊𝑇

2+ 𝐵𝑇

2𝑊8,

Σ2,9

= −𝑁𝑇

9−𝑊𝑇

2+ 𝐵𝑇

2𝑊9,

Σ2,10

= −𝑁10+𝑊𝑇

2+ 𝐵𝑇

2𝑊10,


Σ3,3

= − (1 − 𝛽ℎ𝑑) 𝑒−2𝛽𝑘ℎ

𝑃4− 𝑒−2𝛽𝑘ℎ

𝑃10+ 𝛽2

ℎ2

𝑅4

− 2𝛽ℎ𝑅5− 𝛽ℎ𝑀

𝑇

7− 𝛽ℎ𝑀

7+ 𝛽2

ℎ2

𝑀10

− 𝑂𝑇

3− 𝑂3,

Σ3,4

= −𝑁𝑇

3− 𝑂4−𝑊𝑇

3𝐵1,

Σ3,5

= −𝑂𝑇

3− 𝑂5,

Σ3,6

= −𝑂9+𝑊𝑇

3, Σ

3,7= −𝑂7+𝑊𝑇

3𝐶,

Σ3,8

= −𝑂8+𝑊𝑇

3, Σ

3,9= −𝑂9+𝑊𝑇

3,

Σ3,10

= −𝑂10+𝑊𝑇

3,

Σ4,4

= −𝑒2𝑘ℎ

𝑃7− 𝑁𝑇

4− 𝑁4+𝑊𝑇

4𝐵1− 𝐵𝑇

1𝑊4,

Σ4,5

= −𝑁5− 𝑂𝑇

4− 𝐵𝑇

1𝑊5,

Σ4,6

= −𝑁6−𝑊𝑇

4− 𝐵𝑇

1𝑊6,

Σ4,7

= −𝑁7+𝑊𝑇

4𝐶 − 𝐵

𝑇

1𝑊7,

Σ4,8

= −𝑂8+𝑊𝑇

3,

Σ4,9

= −𝑂9+𝑊𝑇

3, Σ

4,10= −𝑂10+𝑊𝑇

3,

Σ5,5

= −𝑒2𝛽𝑘ℎ

𝑃8− 𝑂𝑇

5− 𝑂5,

Σ5,6

= −𝑂6−𝑊𝑇

5,

Σ5,7

= −𝑂7+𝑊𝑇

5𝐶, Σ

5,8= −𝑂8+𝑊𝑇

5,

Σ5,9

= −𝑂9+𝑊𝑇

5,

Σ5,10

= −𝑂10+𝑊𝑇

5,

Σ6,6

= 𝑃2+ ℎ2

𝑃5+ 𝛽2

ℎ2

𝑃6+ ℎ2

𝑃7+ 𝛽2

ℎ2

𝑃8

+ ℎ2

𝑃9+ 𝛽2

ℎ2

𝑃10−𝑊𝑇

6−𝑊6,

Σ6,7

= 𝑊𝑇

6𝐶 −𝑊

7, Σ

6,8= 𝑊𝑇

6−𝑊8,

Σ6,9

= 𝑊𝑇

6−𝑊9, Σ

6,10= 𝑊𝑇

6−𝑊10,

Σ7,7

= − (1 − 𝑟𝑑) 𝑒−2𝑘𝑟

𝑃2+ 𝜖3𝛾2

𝐼

+ 𝑊𝑇

7𝐶 + 𝐶

𝑇

𝑊7,

Σ7,8

= 𝑊𝑇

7− 𝐶𝑇

𝑊8, Σ

7,9= 𝑊𝑇

7− 𝐶𝑇

𝑊9,

Σ7,10

= 𝑊𝑇

7− 𝐶𝑇

𝑊10

Σ8,8

= 𝑊𝑇

8+𝑊8− 𝜖1𝐼,

Σ8,9

= 𝑊𝑇

8+𝑊9, Σ

8,10= 𝑊𝑇

8+𝑊10,

Σ9,9

= 𝑊𝑇

9+𝑊9− 𝜖2𝐼, Σ

9,10= 𝑊𝑇

9+𝑊10,

Σ10,10

= 𝑊𝑇

10+𝑊10− 𝜖3𝐼,

Σ̂7,7

= − (1 − 𝑟𝑑) 𝑒−2𝑘𝑟

𝑃2+𝑊𝑇

7𝐶 + 𝐶

𝑇

𝑊7,

𝑍1= 𝑃1𝐺.

(50)

If 𝐴(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:

�̇� (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑥 (𝑡 − ℎ (𝑡)) + 𝐶�̇� (𝑡 − 𝑟 (𝑡))

+ 𝑓 (𝑡, 𝑥 (𝑡)) + 𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡)))

+ 𝑤 (𝑡, �̇� (𝑡 − 𝑟 (𝑡))) , 𝑡 > 0;

𝑥 (𝑡) = 𝜙 (𝑡) , �̇� (𝑡) = 𝜓 (𝑡) , 𝑡 ∈ [−𝑏, 0] .

(51)

Corollary 9. For ‖𝐶‖+𝛾 < 1 and given positive real constantsℎ, ℎ𝑑, 𝑟, 𝑟𝑑,𝜂,𝜌, 𝛾, and𝛽, system (51) is exponentially stablewith

a decay rate 𝑘, if there exist symmetric positive definitematrices𝑃𝑖, any appropriate dimensional matrices 𝑅

𝑠, 𝑁𝑖, 𝑂𝑖, 𝑊𝑖, 𝑀𝑖,

𝐺, 𝑠 = 1, 2, . . . 6, 𝑖 = 1, 2, . . . , 10, and positive real constants𝜖1, 𝜖2, and 𝜖

3such that the following symmetric linear matrix

inequalities hold:

[𝑅1

𝑅2

∗ 𝑅3

] > 0,

[𝑅4

𝑅5

∗ 𝑅6

] > 0,

[

[

𝑒−2𝑘ℎ

𝑃3− 𝑅3

𝑀1

𝑀2

∗ 𝑀3

𝑀4

∗ ∗ 𝑀5

]

]

≥ 0,

[

[

𝑒−2𝛽𝑘ℎ

𝑃6− 𝑅6

𝑀6

𝑀7

∗ 𝑀8

𝑀9

∗ ∗ 𝑀10

]

]

≥ 0,

∑ < 0.

(52)


𝑥 (𝑡, 𝜙, 𝜓) ≤ √

𝐷

𝜆min (𝑃1)max [𝜙

,𝜓

] 𝑒−𝑘𝑡

, ∀𝑡 ∈ 𝑅+

,

(53)

where𝐷 = 𝜆max(𝑃1) + 𝑟𝜆max(𝑃2) + ℎ𝜆max(𝑃3) + 𝛽ℎ𝜆max(𝑃4) +ℎ3𝜆max(𝑃5 + 𝑃7 + 𝑃9) + (𝛽ℎ)

3

𝜆max(𝑃6 + 𝑃8 + 𝑃10) +

ℎ3𝜆max ([

𝑅1 𝑅2

∗ 𝑅3]) + (𝛽ℎ)

3

𝜆max ([𝑅4 𝑅5

∗ 𝑅6]).

If𝐴(𝛼) = 𝐴,𝐵(𝛼) = 𝐵,𝐶(𝛼) = 𝐶, and𝑤(𝑡, �̇�(𝑡−𝑟(𝑡))) = 0,where 𝐴, 𝐵, 𝐶 ∈ 𝑅𝑛×𝑛 are real constant matrices, then system(1) reduces to the following system:

�̇� (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑥 (𝑡 − ℎ (𝑡)) + 𝐶�̇� (𝑡 − 𝑟 (𝑡))

+ 𝑓 (𝑡, 𝑥 (𝑡)) + 𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡))) , 𝑡 > 0;

𝑥 (𝑡) = 𝜙 (𝑡) , �̇� (𝑡) = 𝜓 (𝑡) , 𝑡 ∈ [−𝑏, 0] .

(54)

Corollary 10. For ‖𝐶‖ < 1 and given positive real constantsℎ, ℎ𝑑, 𝑟, 𝑟𝑑, 𝜂, 𝜌, and 𝛽, system (54) is exponentially stable

with a decay rate 𝑘, if there exist symmetric positive definitematrices 𝑃

𝑗, any appropriate dimensional matrices 𝑅

𝑠, 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, 𝜖2such that the

following symmetric linear matrix inequalities hold:

[𝑅1

𝑅2

∗ 𝑅3

] > 0,

[𝑅4

𝑅5

∗ 𝑅6

] > 0,

[

[

𝑒−2𝑘ℎ

𝑃3− 𝑅3

𝑀1

𝑀2

∗ 𝑀3

𝑀4

∗ ∗ 𝑀5

]

]

≥ 0,

[

[

𝑒−2𝛽𝑘ℎ

𝑃6− 𝑅6

𝑀6

𝑀7

∗ 𝑀8

𝑀9

∗ ∗ 𝑀10

]

]

≥ 0,

∏ < 0,

(55)

where

∏ =

[[[[[[[[[[[[[

[

Σ11

Σ12

Σ13

Σ14

Σ15

Σ16

Σ17

Σ18

Σ19

∗ Σ22

Σ23

Σ24

Σ25

Σ26

Σ27

Σ28

Σ29

∗ ∗ Σ33

Σ34

Σ35

Σ36

Σ37

Σ38

Σ39

∗ ∗ ∗ Σ44

Σ45

Σ46

Σ47

Σ48

Σ49

∗ ∗ ∗ ∗ Σ55

Σ56

Σ57

Σ58

Σ59

∗ ∗ ∗ ∗ ∗ Σ66

Σ67

Σ68

Σ69

∗ ∗ ∗ ∗ ∗ ∗ Σ̃77

Σ78

Σ79

∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ88

Σ89

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Σ99

]]]]]]]]]]]]]

]

. (56)


𝑥 (𝑡, 𝜙, 𝜓) ≤ √

𝐷

𝜆min (𝑃1)max [𝜙

,𝜓

] 𝑒−𝑘𝑡

, ∀𝑡 ∈ 𝑅+

,

(57)

where𝐷 = 𝜆max(𝑃1) + 𝑟𝜆max(𝑃2) + ℎ𝜆max(𝑃3) + 𝛽ℎ𝜆max(𝑃4) +ℎ3𝜆max(𝑃5 + 𝑃7 + 𝑃9) + (𝛽ℎ)

3

𝜆max(𝑃6 + 𝑃8 + 𝑃10) +

ℎ3𝜆max ([

𝑅1 𝑅2

∗ 𝑅3]) + (𝛽ℎ)

3

𝜆max ([𝑅4 𝑅5

∗ 𝑅6]).

4. Numerical Examples

In order to show the effectiveness of the approaches presentedin Section 3, three numerical examples are provided.

Example 1. Consider the robust exponential stability of sys-tem (1) with

𝐴 (𝛼) = 𝛼1[−2 0

0 −1] + 𝛼2[−2 0

0 −0.8] ,

𝐵 (𝛼) = 𝛼1[−1 0

−1 −0.8] + 𝛼2[−1 0

−1 −1] ,

𝐶 (𝛼) = 𝛼1[0.1 0

0 0.1] + 𝛼2[0.2 0

0 0.1] ,

𝑓 (𝑡, 𝑥 (𝑡)) = [0.05 sin (𝑡) 𝑥

1(𝑡)

0.05 cos (𝑡) 𝑥2(𝑡)

] ,

𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡))) = [0.1 cos (𝑡)2𝑥

1(𝑡 − ℎ (𝑡))

0.1 sin (𝑡)2𝑥2(𝑡 − ℎ (𝑡))

] ,

ℎ (𝑡) = 0.9sin2 ( 𝑡2) ,

𝑤 (𝑡, �̇� (𝑡 − 𝑟 (𝑡))) = [0.1 cos (𝑡) �̇�

1(𝑡 − 𝑟 (𝑡))

0.1 sin (𝑡) �̇�2(𝑡 − 𝑟 (𝑡))

] ,

𝑟 (𝑡) = cos2 ( 𝑡4) ,

(58)

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) = 𝐵

1(𝛼) + 𝐵

2(𝛼), where

𝐵1(𝛼) = 𝛼

1[−0.83 0

−1 −0.63] + 𝛼2[−0.83 0

−1 −0.83] ,

𝐵2(𝛼) = 𝛼

1[−0.17 0

0 −0.17] + 𝛼2[−0.17 0

0 −0.17] .

(59)

The numerical solutions 𝑥1(𝑡) and 𝑥

2(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 (withaccuracy 0.01), we use conditions (25) in Theorem 7 forsystem (1) with (58)-(59). The solutions of LMIs verify asfollows:

𝑃1

1= 103

× [3.3091 −0.0455

−0.0455 0.7941] ,

𝑃1

2= 103

× [3.3091 −0.0455

−0.0455 0.7941] ,

𝑃2

1= 103

× [4.3739 −0.3929

−0.3929 2.7610] ,

𝑃2

2= 103

× [4.7896 −0.4618

−0.4618 2.8307] ,

𝑃3

1= 104

× [1.5976 0.1384

0.1384 0.3189] ,

𝑃3

2= 104

× [1.5976 0.1384

0.1384 0.3189] ,

𝑃4

1= 103

× [9.8670 −0.0635

−0.0635 4.5726] ,


𝑃4

2= 103

× [8.8868 −0.2421

−0.2421 3.2473] ,

𝑃5

1= 103

× [1.3009 −0.1372

−0.1372 0.6505] ,

𝑃5

2= 103

× [1.0898 −0.0980

−0.0980 0.3986] ,

𝑃6

1= 104

× [2.8885 −0.0338

−0.0338 2.6935] ,

𝑃6

2= 104

× [2.8526 −0.0449

−0.0449 2.4922] ,

𝑃7

1= 103

× [2.6064 −0.3045

−0.3045 2.5259] ,

𝑃7

2= 103

× [2.3663 −0.1722

−0.1722 2.5747] ,

𝑃8

1= 104

× [1.4801 −0.0277

−0.0277 1.4063] ,

𝑃8

2= 104

× [1.4740 −0.0862

−0.0862 1.4207] ,

𝑃9

1= 103

× [2.6064 −0.3045

−0.3045 2.5259] ,

𝑃9

2= 103

× [2.3663 −0.1722

−0.1722 2.5747] ,

𝑃10

1= 104

× [1.4570 −0.0000

−0.0000 1.2074] ,

𝑃10

2= 104

× [1.4354 0.0019

0.0019 1.0978] ,

𝑀1

1= 103

× [−9.1839 −1.2537

−1.2537 −2.0706] ,

𝑀1

2= 104

× [−1.0642 −0.1844

−0.1844 −0.2293] ,

𝑀2

1= 103

× [8.8459 1.1503

1.1503 2.0066] ,

𝑀2

2= 104

× [1.0288 0.1763

0.1763 0.2238] ,

𝑀3

1= 104

× [1.0499 0.1090

0.1090 0.2928] ,

𝑀3

2= 104

× [1.1838 0.1766

0.1766 0.2881] ,

𝑀4

1= 103

× [−8.8058 −0.9604

−0.9604 −2.6545] ,

𝑀4

2= 104

× [−1.0454 −0.1677

−0.1677 −0.2680] ,

𝑀5

1= 104

× [1.0303 0.1018

0.1018 0.3190] ,

𝑀5

2= 104

× [1.1590 0.1622

0.1622 0.3078] ,

𝑀6

1= [

−34.9655 −18.7513

−18.7513 35.4129] ,

𝑀6

2= [

−39.8158 −24.8476

−24.8476 13.4547] ,

𝑀7

1= [

2.0904 2.7717

2.7717 −7.8109] , 𝑀

7

2= [

3.2803 5.2929

5.2929 −4.0693] ,

𝑀8

1= 104

× [1.4679 0.0003

0.0003 1.2678] ,

𝑀8

2= 104

× [1.4438 −0.0080

−0.0080 1.1830] ,

𝑀9

1= 103

× [−0.8905 0.0058

0.0058 −2.9028] ,

𝑀9

2= 103

× [−1.1307 −0.0760

−0.0760 −3.7474] ,

𝑀10

1= 104

× [1.4684 0.0005

0.0005 1.2673] ,

𝑀10

2= 104

× [1.4444 −0.0077

−0.0077 1.1828] ,

𝑅1

1= 103

× [2.9131 0.1748

0.1748 0.7782] ,

𝑅1

2= 103

× [2.3463 0.0353

0.0353 0.5806] ,

𝑅2

1= 103

× [1.1393 0.0531

0.0531 0.1251] ,

𝑅2

2= [

876.5627 32.5550

32.5550 98.5599] ,

𝑅3

1= 103

× [2.6387 −0.0759

−0.0759 0.5866] ,

𝑅3

2= 103

× [2.0346 −0.0762

−0.0762 0.3589] ,

𝑅4

1= 103

× [5.6600 −0.0415

−0.0415 5.0541] ,

𝑅4

2= 103

× [5.5782 −0.0710

−0.0710 4.6031] ,

𝑅5

1= 104

× [−7.9071 0.0551

0.0551 −3.8199] ,

𝑅5

2= 104

× [−7.3912 0.4576

0.4576 −3.2145] ,


𝑅6

1= 104

× [1.4893 −0.0162

−0.0162 1.3778] ,

𝑅6

2= 104

× [1.4703 −0.0223

−0.0223 1.2713] ,

𝑁1

1= 103

× [8.7493 3.8534

3.8534 −3.1296] ,

𝑁1

2= 104

× [0.6691 0.2686

0.2686 −1.0299] ,

𝑁2

1= 103

× [−3.8698 −0.3606

−0.3606 1.4605] ,

𝑁2

2= 103

× [−4.5981 −0.9753

−0.9753 1.2772] ,

𝑁3

1= [

−114.8946 −186.6948

−186.6948 −113.8034] ,

𝑁3

2= [

210.0606 −115.9077

−115.9077 −174.4056] ,

𝑁4

1= 104

× [−1.1296 −0.0266

−0.0266 0.1446] ,

𝑁4

2= 104

× [−1.0062 0.1041

0.1041 0.8782] ,

𝑁5

1= 103

× [4.5589 −2.3847

−2.3847 0.8561] ,

𝑁5

2= 103

× [5.3887 −2.4363

−2.4363 1.1619] ,

𝑁6

1= 103

× [5.9564 −1.9530

−1.9530 1.5688] ,

𝑁6

2= 103

× [6.7696 −1.9958

−1.9958 2.0384] ,

𝑁7

1= 103

× [5.9602 −0.2891

−0.2891 2.9332] ,

𝑁7

2= 103

× [6.3929 −0.0855

−0.0855 3.1668] ,

𝑁8

1= 103

× [−9.6508 −0.4780

−0.4780 3.5901] ,

𝑁8

2= 104

× [−0.8505 0.0574

0.0574 1.1344] ,

𝑁9

1= 103

× [5.5789 −2.0704

−2.0704 1.3737] ,

𝑁9

2= 103

× [6.3923 −2.1169

−2.1169 1.8008] ,

𝑁10

1= 103

× [−3.6595 3.2032

3.2032 −0.1648] ,

𝑁10

2= 103

× [−4.3781 3.2120

3.2120 0.1271] ,

𝑂1

1= 104

× [1.5923 −0.7669

−0.7669 1.6350] ,

𝑂1

2= 104

× [1.8134 −0.6324

−0.6324 2.3112] ,

𝑂2

1= 104

× [1.0371 −0.0001

−0.0001 −0.3635] ,

𝑂2

2= 104

× [0.9152 −0.1464

−0.1464 −1.0978] ,

𝑂3

1= 103

× [−1.0355 0.1515

0.1515 −0.3626] ,

𝑂3

2= 103

× [−1.4990 0.3148

0.3148 −0.4988] ,

𝑂4

1= [

−51.1797 71.7167

71.7167 −390.4844] ,

𝑂4

2= [

−22.7818 −177.9764

−177.9764 237.5644] ,

𝑂5

1= 104

× [−1.4569 0.2376

0.2376 −0.4763] ,

𝑂5

2= 104

× [−1.5078 0.2277

0.2277 −0.4312] ,

𝑂6

1= 104

× [−1.3667 0.2475

0.2475 −0.4459] ,

𝑂6

2= 104

× [−1.4169 0.2441

0.2441 −0.4015] ,

𝑂7

1= 103

× [5.5038 0.7900

0.7900 −4.7294] ,

𝑂7

2= 104

× [0.4206 −0.0170

−0.0170 −1.2249] ,

𝑂8

1= [

244.5684 88.4596

88.4596 481.2263] ,

𝑂8

2= [

272.5078 401.6259

401.6259 −0.2695] ,

𝑂9

1= 104

× [−1.3911 0.2447

0.2447 −0.4543] ,

𝑂9

2= 104

× [−1.4415 0.2394

0.2394 −0.4095] ,

𝑂10

1= 104

× [1.3674 −0.2535

−0.2535 0.3868] ,

𝑂10

2= 104

× [1.4277 −0.2479

−0.2479 0.3580] ,


𝑊1

1= 104

× [1.5106 −0.0468

−0.0468 0.7929] ,

𝑊1

2= 104

× [1.4523 −0.0341

−0.0341 0.7069] ,

𝑊2

1= 103

× [5.9653 −2.4134

−2.4134 1.7851] ,

𝑊2

2= 103

× [6.9091 −2.5666

−2.5666 1.9597] ,

𝑊3

1= [

−289.7085 24.4200

24.4200 −461.3886] ,

𝑊3

2= [

−223.5193 93.6566

93.6566 −234.3346] ,

𝑊4

1= 104

× [−1.2746 0.2653

0.2653 −0.3694] ,

𝑊4

2= 104

× [−1.3242 0.2636

0.2636 −0.3371] ,

𝑊5

1= 103

× [4.8778 −0.0916

−0.0916 3.8521] ,

𝑊5

2= 103

× [4.8245 −0.0681

−0.0681 3.9058] ,

𝑊6

1= 103

× [7.9418 −0.4415

−0.4415 6.9527] ,

𝑊6

2= 103

× [7.9864 −0.4713

−0.4713 7.2384] ,

𝑊7

1= 103

× [4.5999 −2.3966

−2.3966 0.1608] ,

𝑊7

2= 103

× [5.2537 −2.3858

−2.3858 0.0285] ,

𝑊8

1= 104

× [−1.3844 0.2577

0.2577 −0.4094] ,

𝑊8

2= 104

× [−1.4426 0.2499

0.2499 −0.3763] ,

𝑊9

1= 103

× [7.1086 −0.3474

−0.3474 6.1090] ,

𝑊9

2= 103

× [7.1257 −0.3627

−0.3627 6.3346] ,

𝑊10

1= 103

× [8.0468 −0.7400

−0.7400 6.5133] ,

𝑊10

2= 103

× [8.1365 −0.8020

−0.8020 6.8109] ,

𝐺 = [−2.1527 1.4389

6.5441 −11.7139] , 𝜖

1= 4.2164 × 10

4

,

𝜖2= 5.7269 × 10

4

, 𝜖3= 5.3163 × 10

4

.

(60)

0 1 2 3 4 5 6 7 8Time (s)

−2

−1

0

1

2

3

4

5

6

The s

tate

var

iabl

ex(t)

x1(t)

x2(t)

Figure 1: The simulation solutions 𝑥1(𝑡) and 𝑥

2(𝑡) are presented for

system (1) with (58)-(59) in Example 1, 𝛼1= 𝛼2= 1/2, and initial

conditions 𝑥1(𝑡) = 2 + 3 cos(𝑡), 𝑥

2(𝑡) = 1 + 2 sin(𝑡), 𝑡 ∈ [−1, 0], by

using the Runge-Kutta fourth order method with Matlab.

Table 1: The maximum allowed time delay ℎ for 𝑘 = 0.1, 𝜂 = 0.05,𝜌 = 0.1, and 𝛽 = 0.1.

ℎ𝑑= 𝑟𝑑

0 0.5 0.9Chen et al. (2008) [3] 1.2999 0.9442 0.5471Qiu and Cui (2010) [26] 1.4008 1.0120 0.6438Pinjai and Mukdasai (2011) [25] 1.6237 1.1052 0.6205Corollary 10 6.4417 5.2362 2.5666

Example 2. Consider the following neutral system (54),which is considered in [3, 25, 26]:

�̇� (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑥 (𝑡 − ℎ (𝑡)) + 𝐶�̇� (𝑡 − 𝑟 (𝑡))

+ 𝑓 (𝑡, 𝑥 (𝑡)) + 𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡))) ,

(61)

with

𝐴 = [−2 0

0 −0.9] , 𝐵 = [

−1 0

−1 −1] , 𝐶 = [

0.1 0

0 0.1] .

(62)

Decompose matrix 𝐵 as follows: 𝐵 = 𝐵1+ 𝐵2, where

𝐵1= [

−0.83 0

−1 −0.83] , 𝐵

2= [

−0.17 0

0 −0.17] , (63)

‖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, fordifferent values of ℎ

𝑑and 𝑟

𝑑. In Table 1, we let 𝜂 = 0.05,

𝜌 = 0.1, 𝛽 = 0.1 and ℎ(𝑡) = 𝑟(𝑡). We can see that our resultsin Corollary 8 are much less conservative than in [3, 25, 26].

Example 3. Consider the following neutral system (51), whichis considered in [14, 28]:

�̇� (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑥 (𝑡 − ℎ (𝑡)) + 𝐶�̇� (𝑡 − 𝑟 (𝑡)) + 𝑓 (𝑡, 𝑥 (𝑡))

+ 𝑔 (𝑡, 𝑥 (𝑡 − ℎ (𝑡))) + 𝑤 (𝑡, �̇� (𝑡 − 𝑟 (𝑡))) ,

(64)


Table 2: The maximum allowed time delay ℎ.

ℎ𝑑= 𝑟𝑑= 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 0.3045Liu et al. (2013) [14] 12.2475 3.7460 1.9563 1.1015 0.5957Corollary 9 14.1728 4.7242 2.8345 2.0246 1.5747ℎ𝑑= 𝑟𝑑= 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 0.1027Liu et al. (2013) [14] 10.8211 3.3202 1.7390 0.9662 0.4857Corollary 9 11.5872 3.8624 2.3174 1.6553 1.2874

with

𝐴 = [−2 0

0 −2] , 𝐵 = [

0 0.4

0.4 0] , 𝐶 = [

0.1 0

0 0.1] .

(65)

Decompose matrix 𝐵 as follows: 𝐵 = 𝐵1+ 𝐵2, where

𝐵1= [

0 0.3

0.2 0] , 𝐵

2= [

0 0.1

0.2 0] , (66)

‖𝑓(𝑡, 𝑥(𝑡))‖ ≤ 𝜂‖𝑥(𝑡)‖, ‖𝑔(𝑡, 𝑥(𝑡 − ℎ(𝑡)))‖ ≤ 𝜌‖𝑥(𝑡 − ℎ(𝑡))‖,and ‖𝑤(𝑡, �̇�(𝑡 − 𝑟(𝑡)))‖ ≤ 𝛾‖�̇�(𝑡 − 𝑟(𝑡))‖. By Corollary 9 tosystem (64) with (65)-(66), one can obtain the maximumupper bounds of the time delay with different convergencerate 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 inCorollary 9 give larger delay bounds than the recent resultsin [14, 28].

5. Conclusions

The problem of robust exponential stability for LPD neu-tral systems with mixed time-varying delays and nonlinearuncertainties has been presented. Based on combinationof Leibniz-Newton formula, free-weighting matrices, linearmatrix inequality, Cauchy’s inequality, modified version ofJensen’s inequality, model transformation, and the use of suit-able parameter-dependent Lyapunov-Krasovskii functional,new delay-dependent robust exponential stability criteriaare formulated in terms of LMIs. Numerical examples haveshown significant improvements over some existing results.

Acknowledgments

This work was supported by the Higher Education ResearchPromotion and the National Research University Projectof Thailand, Office of the Higher Education Commission,through the Cluster of Research to Enhance the Qualityof Basic Education, Khon Kaen University, Khon Kaen,Thailand.

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research article new delay-dependent robust exponential ...new delay-dependent robust exponential...

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