research article new delay-dependent robust exponential ...new delay-dependent robust exponential...
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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 Β© 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, Cauchyβs inequality, modifiedversion of Jensenβs 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 [1β35]. 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, 18β21, 25β28].
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
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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, Cauchyβs inequality, modifiedversion of Jensenβs 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. π + denotes the set ofall real nonnegative numbers; π π denotes the π-dimensionalspace with the vector norm β β β; βπ₯β denotes the Euclideanvector norm of π₯ β π π; π πΓπ denotes the set of π Γ πreal matrices; π΄π denotes the transpose of the matrix π΄;π΄ is symmetric if π΄ = π΄π; πΌ denotes the identity matrix;π(π΄) denotes the set of all eigenvalues of π΄; πmax(π΄) =max{Re π : π β π(π΄)}; πmin(π΄) = min{Re π : π β π(π΄)};πmax(π΄(πΌ)) = max{πmax(π΄ π) : π = 1, 2, . . . , π}; πmin(π΄(πΌ)) =min{πmin(π΄ π) : π = 1, 2, . . . , π}; πΆ([βπ, 0], π
π) denotes the
space of all continuous vector functions mapping [βπ, 0] intoπ π, where π = max{β, π}, β, π β π +; β represents the elements
below the main diagonal of a symmetric matrix.Consider the system described by the following state
equations of the form:
οΏ½ΜοΏ½ (π‘) = π΄ (πΌ) π₯ (π‘) + π΅ (πΌ) π₯ (π‘ β β (π‘))
+ πΆ (πΌ) οΏ½ΜοΏ½ (π‘ β π (π‘)) + π (π‘, π₯ (π‘))
+ π (π‘, π₯ (π‘ β β (π‘))) + π€ (π‘, οΏ½ΜοΏ½ (π‘ β π (π‘))) , π‘ > 0;
π₯ (π‘ + π‘0) = π (π‘) , οΏ½ΜοΏ½ (π‘ + π‘
0) = π (π‘) , π‘ β [βπ, 0] ,
(1)
where π₯(π‘) β π π is the state variable and π΄(πΌ), π΅(πΌ), πΆ(πΌ) βπ πΓπ are uncertain matrices belonging to the polytope
π΄ (πΌ) =
π
β
π=1
πΌππ΄π, π΅ (πΌ) =
π
β
π=1
πΌππ΅π, πΆ (πΌ) =
π
β
π=1
πΌππΆπ,
π
β
π=1
πΌπ= 1, πΌ
πβ₯ 0, π΄
π, π΅π, πΆπβ π πΓπ
, π = 1, . . . , π.
(2)
β(π‘) and π(π‘) are discrete and neutral time-varying delays,respectively,
0 β€ β (π‘) β€ β, βΜ (π‘) β€ βπ, (3)
0 β€ π (π‘) β€ π, Μπ (π‘) β€ ππ, (4)
where β, π, βπ, and π
πare given positive real constants.
Consider the initial functions π(π‘),π(π‘) β πΆ([βπ, 0], π π)withthe norm βπβ = sup
π‘β[βπ,0]βπ(π‘)β and βπβ = sup
π‘β[βπ,0]βπ(π‘)β.
The uncertainties π(π‘, π₯(π‘)), π(π‘, π₯(π‘ β β(π‘))), and π€(π‘, οΏ½ΜοΏ½(π‘ βπ(π‘))) are the nonlinear perturbations with respect to currentstate π₯(π‘), discrete delayed state π₯(π‘ β β(π‘)), and neutraldelayed state οΏ½ΜοΏ½(π‘ β π(π‘)), respectively, and are bounded inmagnitude:
ππ
(π‘, π₯ (π‘)) π (π‘, π₯ (π‘)) β€ π2
π₯π
(π‘) π₯ (π‘) ,
ππ
(π‘, π₯ (π‘ β β (π‘))) π (π‘, π₯ (π‘ β β (π‘)))
β€ π2
π₯π
(π‘ β β (π‘)) π₯ (π‘ β β (π‘)) ,
π€π
(π‘, οΏ½ΜοΏ½ (π‘ β π (π‘))) π€ (π‘, οΏ½ΜοΏ½ (π‘ β π (π‘)))
β€ πΎ2
οΏ½ΜοΏ½π
(π‘ β π (π‘)) οΏ½ΜοΏ½ (π‘ β π (π‘)) ,
(5)
where π, π, and πΎ are given positive real constants.In order to improve the bound of the discrete time-
varying delayed β(π‘) in system (1), let us decompose theconstant matrix π΅(πΌ) as
π΅ (πΌ) = π΅1(πΌ) + π΅
2(πΌ) , (6)
where π΅1(πΌ) = β
π
π=1πΌππ΅1
π, π΅2(πΌ) = β
π
π=1πΌππ΅2
π, and βπ
π=1πΌπ= 1,
πΌπβ₯ 0 with π΅1
π, π΅2
πβ π πΓπ, π = 1, . . . , π being real constant
matrices. By Leibniz-Newton formula, we have
0 = π₯ (π‘) β π₯ (π‘ β β (π‘)) β β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½ (π ) ππ . (7)
By utilizing the following zero equation, we obtain
0 = πΊπ₯ (π‘) β πΊπ₯ (π‘ β π½β (π‘)) β πΊβ«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½ (π ) ππ , (8)
where π½ is a given positive real constant and πΊ β π πΓπ will bechosen to guarantee the robust exponential stability of system(1). By (6), (7), and (8), system (1) can be represented by theform
οΏ½ΜοΏ½ (π‘) = [π΄ (πΌ) + π΅1(πΌ) + πΊ] π₯ (π‘) + π΅
2(πΌ) π₯ (π‘ β β (π‘))
β πΊπ₯ (π‘ β π½β (π‘)) + π (π‘, π₯ (π‘))
+ π (π‘, π₯ (π‘ β β (π‘))) + π€ (π‘, οΏ½ΜοΏ½ (π‘ β π (π‘)))
+ πΆ (πΌ) οΏ½ΜοΏ½ (π‘ β π (π‘)) β π΅1(πΌ) β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½ (π ) ππ
β πΊβ«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½ (π ) ππ .
(9)
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Journal of Applied Mathematics 3
Definition 1. The system (1) is robustly exponentially stable,if there exist positive real constants π and π such that, foreach π(π‘), π(π‘) β πΆ([βπ, 0], π π), the solution π₯(π‘, π, π) of thesystem (1) satisfies
π₯ (π‘, π, π) β€ πmax {
π ,π
} πβππ‘
, βπ‘ β π +
. (10)
Lemma 2 (Cauchy inequality). For any constant symmetricpositive definite matrix π β π πΓπ and π, π β π π, one has
Β±2ππ
π β€ ππ
ππ + ππ
πβ1
π. (11)
Lemma 3 (see [15]). The following inequality holds for any π βπ π, π β π π,π,π β π πΓπ,π β π πΓπ, and π β π πΓπ:
β2ππ
ππ β€ [π
π]
π
[π π β π
β π][
π
π] , (12)
where [π πβ π
] β₯ 0.
Lemma4. For any constant symmetric positive definitematrixπ β π
πΓπ and β(π‘) which is discrete time-varying delayswith (3), vector function π : [ββ, 0] β π π such that theintegrations concerned are well defined; then
ββ«
0
ββ
ππ
(π ) ππ (π ) ππ β₯ β«
0
ββ(π‘)
ππ
(π ) ππ πβ«
0
ββ(π‘)
π (π ) ππ .
(13)
Proof. From Lemma 2, it is easy to see that
ββ«
0
ββ
ππ
(π ) ππ (π ) ππ = ββ«
0
ββ(π‘)
ππ
(π ) ππ (π ) ππ
+ ββ«
ββ(π‘)
ββ
ππ
(π ) ππ (π ) ππ
β₯ β (π‘) β«
0
ββ(π‘)
ππ
(π ) ππ (π ) ππ
=1
2β¬
0
ββ(π‘)
[ππ
(π ) ππ (π )
+ ππ
(π) ππ (π)] ππ ππ
β₯1
2β¬
0
ββ(π‘)
2ππ
(π ) π1/2π
Γ π1/2
π (π) ππ ππ
= β«
0
ββ(π‘)
ππ
(π ) ππ πβ«
0
ββ(π‘)
π (π ) ππ .
(14)
Lemma 5. Let π₯(π‘) β π π be a vector-valued function with firstorder continuous derivative entries.Then, the following integralinequality holds for any matrices π
πβ π πΓπ
, π = 1, 2, . . . , 5,
and β(π‘) is discrete time-varying delays with (3) and symmetricpositive definite matrix π β π πΓπ:
ββ«
π‘
π‘ββ
οΏ½ΜοΏ½π
(π )ποΏ½ΜοΏ½ (π ) ππ β€ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
Γ [π1+ππ
1βππ
1+π2
β βπ2βππ
2
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
+ β[π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
[π3
π4
β π5
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))] ,
(15)
where
[
[
π π1
π2
β π3
π4
β β π5
]
]
β₯ 0. (16)
Proof. From the Leibniz-Newton formula, one has
0 = π₯ (π‘) β π₯ (π‘ β β (π‘)) β β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½ (π‘) ππ . (17)
Therefore, for any π»1, π»2β π πΓπ, the following equation is
true:
0 = 2 [π₯π
(π‘) β π₯π
(π‘ β β (π‘)) β β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π‘) ππ ]
Γ [π»1π₯ (π‘) + π»
2π₯ (π‘ β β (π‘))]
= 2π₯π
(π‘)π»1π₯ (π‘) + 2π₯
π
(π‘)π»2π₯ (π‘ β β (π‘))
β 2π₯π
(π‘ β β (π‘))π»1π₯ (π‘)
β 2π₯π
(π‘ β β (π‘))π»π
2π₯ (π‘ β β (π‘))
β 2β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π ) ππ π»1π₯ (π‘)
β 2β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π ) ππ π»2π₯ (π‘ β β (π‘))
= [π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
[
[
π»1+ π»π
1βπ»π
1+ π»2
β βπ»2β π»π
2
]
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
β 2β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π ) [π»1
π»2] [
π₯ (π‘)
π₯ (π‘ β β (π‘))] ππ .
(18)
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4 Journal of Applied Mathematics
Using Lemma 3 with π = οΏ½ΜοΏ½(π ), π = [ π₯(π‘)π₯(π‘ββ(π‘))
], π = [π»1
π»2], π = [π
1π2], and π = [π3 π4
β π5], we obtain
β 2β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π ) [π»1
π»2] [
π₯ (π‘)
π₯ (π‘ β β (π‘))] ππ
β€ β«
π‘
π‘ββ(π‘)
[
[
οΏ½ΜοΏ½ (π )
π₯ (π‘)
π₯ (π‘ β β (π‘))
]
]
π
[
[
π π1β π»1
π2β π»2
β π3
π4
β β π5
]
]
Γ [
[
οΏ½ΜοΏ½ (π )
π₯ (π‘)
π₯ (π‘ β β (π‘))
]
]
ππ
= β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π )ποΏ½ΜοΏ½ (π ) ππ
+ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
Γ [π1+ππ
1β π»1β π»π
1βππ
1+π2+ π»π
1β π»2
β βπ2βππ
2+ π»2+ π»π
2
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))] + β (π‘) [
π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
[π3
π4
β π5
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
β€ β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π )ποΏ½ΜοΏ½ (π ) ππ + [π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
Γ ([π1+ππ
1βππ
1+π2
β βπ2βππ
2
]
β [π»1+ π»π
1βπ»π
1+ π»2
β βπ»2β π»π
2
]) [π₯ (π‘)
π₯ (π‘ β β (π‘))]
+ β[π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
[π3
π4
β π5
] [π₯ (π‘)
π₯ (π‘ β β (π‘))] .
(19)
Substituting (19) into (18), we obtain
ββ«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π )ποΏ½ΜοΏ½ (π ) ππ β€ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
Γ [π»1+ π»π
1βπ»π
1+ π»2
β βπ»2β π»π
2
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
+ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
Γ ([
[
π1+ππ
1βππ
1+π2
β βπ2βππ
2
]
]
β [π»1+ π»π
1βπ»π
1+ π»2
β βπ»2β π»π
2
])
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
+ β[π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
[π3
π4
β π5
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
= [π₯(π‘)
π₯(π‘ β β(π‘))]
π
Γ [π1+ππ
1βππ
1+π2
β βπ2βππ
2
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
+ β[π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
[π3
π4
β π5
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))] .
(20)
From (3), it is clear that
ββ«
π‘
π‘ββ
οΏ½ΜοΏ½π
(π )ποΏ½ΜοΏ½ (π ) ππ β€ ββ«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π )ποΏ½ΜοΏ½ (π ) ππ . (21)
From (20) and (21), the integral inequality becomes
ββ«
π‘
π‘ββ
οΏ½ΜοΏ½π
(π )ποΏ½ΜοΏ½ (π ) ππ β€ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
Γ [
[
π1+ππ
1βππ
1+π2
β βπ2βππ
2
]
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))]
+ β[π₯ (π‘)
π₯ (π‘ β β (π‘))]
π
[π3
π4
β π5
]
Γ [π₯ (π‘)
π₯ (π‘ β β (π‘))] .
(22)
The proof of the theorem is complete.
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Journal of Applied Mathematics 5
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:
ππ(πΌ) =
π
β
π=1
πΌπππ
π, π
1(πΌ) =
π
β
π=1
πΌππ1
π,
π π(πΌ) =
π
β
π=1
πΌππ π
π, π
π(πΌ) =
π
β
π=1
πΌπππ
π,
ππ(πΌ) =
π
β
π=1
πΌπππ
π, π
π(πΌ) =
π
β
π=1
πΌπππ
π,
ππ(πΌ) =
π
β
π=1
πΌπππ
π,
π
β
π=1
πΌπ= 1, πΌ
πβ₯ 0,
ππ
π, π1
π, π π
π,ππ
π, ππ
π, ππ
π,ππ
πβ π πΓπ
,
π = 1, 2, . . . , 10, π = 1, 2, . . . , 6, π = 1, 2, . . . , π;
β
π,π
=
[[[[[[[[[[[[[[[[[
[
Ξ£1,1
π,πΞ£1,2
π,πΞ£1,3
π,πΞ£1,4
π,πΞ£1,5
π,πΞ£1,6
π,πΞ£1,7
π,πΞ£1,8
π,πΞ£1,9
π,πΞ£1,10
π,π
β Ξ£2,2
π,πΞ£2,3
π,πΞ£2,4
π,πΞ£2,5
π,πΞ£2,6
π,πΞ£2,7
π,πΞ£2,8
π,πΞ£2,9
π,πΞ£2,10
π,π
β β Ξ£3,3
πΞ£3,4
π,πΞ£3,5
πΞ£3,6
πΞ£3,7
π,πΞ£3,8
πΞ£3,9
πΞ£3,10
π
β β β Ξ£4,4
π,πΞ£4,5
π,πΞ£4,6
π,πΞ£4,7
π,πΞ£4,8
πΞ£4,9
πΞ£4,10
π
β β β β Ξ£5,5
πΞ£5,6
πΞ£5,7
π,πΞ£5,8
πΞ£5,9
πΞ£5,10
π
β β β β β Ξ£6,6
πΞ£6,7
π,πΞ£6,8
πΞ£6,9
πΞ£6,10
π
β β β β β β Ξ£7,7
π,πΞ£7,8
π,πΞ£7,9
π,πΞ£7,10
π,π
β β β β β β β Ξ£8,8
πΞ£8,9
πΞ£8,10
π
β β β β β β β β Ξ£9,9
πΞ£9,10
π
β β β β β β β β β Ξ£10,10
π
]]]]]]]]]]]]]]]]]
]
,
(23)
where
Ξ£1,1
π,π= π΄π
ππ1
π+ π΅1
π
π
π1
π+ π1
ππ΄π+ π1
ππ΅1
π+ π1
π+ π1
π
π
+ π3
π+ π4
π+ βπ
1
π
π
+ βπ1
πβ πβ2πβ
π9
π
β πβ2ππ½β
π10
π+ β2
π3
π+ π½βπ
6
π+ π½βπ
6
π
π
+ π½2
β2
π8
π
+ π1
π
π
+ π1
π+ π1
π
π
+ π1
π+ π΄π
ππ1
π+π1
π
π
π΄π
+ π΅1
π
π
π1
π+π1
π
π
π΅1
π+ π1π2
πΌ + 2ππ1
π,
Ξ£1,2
π,π= π1
ππ΅2
πβ βπ
1
π
π
+ βπ2
π+ β2
π4
π+ πβ2πβ
π9
π
+ βπ 2
πβ π1
π
π
+ π2
π+ π2
π+π1
π
π
π΅2
π
+ π΄π
ππ2
π+ π΅1
π
π
π2
π,
Ξ£1,3
π,π= βπ1
πβ π½βπ
6
π
π
+ π½βπ7
π+ π½β2
π9
π+ π½βπ
5
π
+ πβ2ππ½β
π10
π+ π3
πβ π1
π
π
+ π3
π
+ π΄π
ππ3
π+ π΅1
π
π
π3
π,
Ξ£1,4
π,π= βπ1
ππ΅1
πβ π1
π
π
+ π4
π+ π4
π
βπ1
π
π
π΅1
π+ π΄π
ππ4
π+ π΅1
π
π
π4
π,
Ξ£1,5
π,π= βπ1
π+ π5
πβ π1
π
π
+ π5
π+ π΄π
ππ5
π+ π΅1
π
π
π5
π,
Ξ£1,6
π,π= π6
π+ π6
πβπ1
π
π
+ π΄π
ππ6
π+ π΅1
π
π
π6
π,
Ξ£1,7
π,π= π1
ππΆπ+ π7
π+ π7
π+π1
π
π
πΆπ+ π΄π
ππ7
π+ π΅1
π
π
π7
π,
Ξ£1,8
π,π= π1
π+ π8
π+ π8
π+π1
π
π
+ π΄π
ππ8
π+ π΅1
π
π
π8
π,
Ξ£1,9
π,π= π1
π+ π9
π+ π9
π+π1
π
π
+ π΄π
ππ9
π+ π΅1
π
π
π9
π,
Ξ£1,10
π,π= π1
π+ π10
π+ π10
π+π1
π
π
+ π΄π
ππ10
π+ π΅1
π
π
π10
π,
Ξ£2,2
π,π= β (1 β β
π) πβ2πβ
π3
πβ βπ
2
π
π
β βπ2
π+ β2
π5
π
+ β2
π 1
πβ 2βπ
2
πβ πβ2πβ
π9
π+ π2π2
πΌ β π2
π
π
β π2
π+π2
π
π
π΅2
π+ π΅2
π
π
π2
π,
Ξ£2,3
π,π= βπ
3
πβ π2
π
π
+ π΅2
π
π
π3
π,
Ξ£2,4
π,π= βπ
2
π
π
β π4
πβπ2
π
π
π΅1
π+ π΅2
π
π
π4
π,
Ξ£2,5
π,π= βπ
5
πβ π2
π
π
+ π΅2
π
π
π5
π,
Ξ£2,6
π,π= βπ
6
πβπ2
π
π
+ π΅2
π
π
π6
π,
Ξ£2,7
π,π= βπ
7
π+π2
π
π
πΆπ+ π΅2
π
π
π7
π,
Ξ£2,8
π,π= βπ
8
π+π2
π
π
+ π΅2
π
π
π8
π,
Ξ£2,9
π,π= βπ
9
π
π
βπ2
π
π
+ π΅2
π
π
π9
π,
Ξ£2,10
π,π= βπ
10
π+π2
π
π
+ π΅2
π
π
π10
π,
Ξ£3,3
π= β (1 β π½β
π) πβ2π½πβ
π4
πβ πβ2π½πβ
π10
π+ π½2
β2
π 4
π
β 2π½βπ 5
πβ π½βπ
7
π
π
β π½βπ7
π+ π½2
β2
π10
π
β π3
π
π
β π3
π,
-
6 Journal of Applied Mathematics
Ξ£3,4
π,π= βπ
3
π
π
β π4
πβπ3
π
π
π΅1
π,
Ξ£3,5
π= βπ3
π
π
β π5
π, Ξ£
3,6
π= βπ9
π+π3
π
π
,
Ξ£3,7
π= βπ7
π+π3
π
π
πΆπ, Ξ£
3,8
π= βπ8
π+π3
π
π
,
Ξ£3,9
π= βπ9
π+π3
π
π
, Ξ£3,10
π= βπ10
π+π3
π
π
,
Ξ£4,4
π,π= βπ2πβ
π7
πβ π4
π
π
β π4
π+π4
π
π
π΅1
πβ π΅1
π
π
π4
π,
Ξ£4,5
π,π= βπ
5
πβ π4
π
π
β π΅1
π
π
π5
π,
Ξ£4,6
π,π= βπ
6
πβπ4
π
π
β π΅1
π
π
π6
π,
Ξ£4,7
π,π= βπ
7
π+π4
π
π
πΆπβ π΅1
π
π
π7
π, Ξ£
4,8
π= βπ8
π+π3
π
π
,
Ξ£4,9
π= βπ9
π+π3
π
π
, Ξ£4,10
π= βπ10
π+π3
π
π
,
Ξ£5,5
π= βπ2π½πβ
π8
πβ π5
π
π
β π5
π, Ξ£
5,6
π= βπ6
πβπ5
π
π
,
Ξ£5,7
π,π= βπ7
π+π5
π
π
πΆπ, Ξ£
5,8
π= βπ8
π+π5
π
π
,
Ξ£5,9
π= βπ9
π+π5
π
π
, Ξ£5,10
π= βπ10
π+π5
π
π
,
Ξ£6,6
π= π2
π+ β2
π5
π+ π½2
β2
π6
π+ β2
π7
π+ π½2
β2
π8
π
+ β2
π9
π+ π½2
β2
π10
πβπ6
π
π
βπ6
π,
Ξ£6,7
π,π= π6
π
π
πΆπβπ7
π, Ξ£
6,8
π= π6
π
π
βπ8
π,
Ξ£6,9
π= π6
π
π
βπ9
π, Ξ£
6,10
π= π6
π
π
βπ10
π,
Ξ£7,7
π,π= β (1 β π
π) πβ2ππ
π2
π+ π3πΎ2
πΌ + π7
π
π
πΆπ+ πΆπ
ππ7
π,
Ξ£7,8
π,π= π7
π
π
β πΆπ
ππ8
π, Ξ£
7,9
π,π= π7
π
π
β πΆπ
ππ9
π,
Ξ£7,10
π,π= π7
π
π
β πΆπ
ππ10
π, Ξ£
8,8
π= π8
π
π
+π8
πβ π1πΌ,
Ξ£8,9
π= π8
π
π
+π9
π, Ξ£
8,10
π= π8
π
π
+π10
π,
Ξ£9,9
π= π9
π
π
+π9
πβ π2πΌ, Ξ£
9,10
π= π9
π
π
+π10
π,
Ξ£10,10
π= π10
π
π
+π10
πβ π3πΌ,
ΜΞ£7,7
π= β (1 β ππ) π
2ππ
π2
π+π7
π
π
πΆπ+ πΆπ
ππ7
π,
π1
π= π1
ππΊ.
(24)
Theorem 7. For βπΆπβ + πΎ < 1, π = 1, 2, . . . , π and given
positive real constants β, βπ, π, ππ, π, π, πΎ, and π½, system (1)
is robustly exponentially stable with a decay rate π, if thereexist symmetric positive definite matrices ππ
π, any appropriate
dimensional matrices π ππ, πππ, πππ, πππ, πππ, πΊ, π = 1, 2, . . . 6,
π = 1, 2, . . . , 10, π = 1, 2, . . . , π, and positive real constants
π1, π2, and π
3such that the following symmetric linear matrix
inequalities hold:
[
[
π 1
ππ 2
π
β π 3
π
]
]
> 0, π = 1, 2, . . . , π,
[
[
π 4
ππ 5
π
β π 6
π
]
]
> 0, π = 1, 2, . . . , π,
[
[
πβ2πβ
π3
πβ π 3
ππ1
ππ2
π
β π3
ππ4
π
β β π5
π
]
]
β₯ 0, π = 1, 2, . . . , π,
[
[
πβ2π½πβ
π6
πβ π 6
ππ6
ππ7
π
β π8
ππ9
π
β β π10
π
]
]
β₯ 0, π = 1, 2, . . . , π,
β
π,π
< βπΌ, π = 1, 2, . . . , π,
β
π,π
+β
π,π
<2
(π β 1)πΌ, π = 1, 2, . . . , π β 1,
π = π + 1, π + 2, . . . , π.
(25)
Moreover, the solution π₯(π‘, π, π) satisfies the inequality
π₯ (π‘, π, π) β€ β
πΏ
πmin (π1 (πΌ))max [π
,π
] πβππ‘
,
βπ‘ β π +
,
(26)
where πΏ = πmax(π1(πΌ)) + ππmax(π2(πΌ)) + βπmax(π3(πΌ))+π½βπmax(π4(πΌ)) + β
3πmax(π5(πΌ)+π7(πΌ) + π9(πΌ)) +
(π½β)3
πmax(π6(πΌ)+π8(πΌ) + π10(πΌ)) + β3πmax ([π 1(πΌ) π 2(πΌ)
β π 3(πΌ)])+
(π½β)3
πmax ([π 4(πΌ) π 5(πΌ)
β π 6(πΌ)]).
Proof. Choose a parameter-dependent Lyapunov-Krasovskiifunctional candidate for system (9) as
π (π‘) =
7
β
π=1
ππ(π‘) , (27)
where
π1(π‘) = π₯
π
(π‘) π1(πΌ) π₯ (π‘) ,
π2(π‘) = β«
π‘
π‘βπ(π‘)
π2π(π βπ‘)
οΏ½ΜοΏ½π
(π ) π2(πΌ) οΏ½ΜοΏ½ (π ) ππ ,
π3(π‘) = β«
π‘
π‘ββ(π‘)
π2π(π βπ‘)
π₯π
(π ) π3(πΌ) π₯ (π ) ππ
+ β«
π‘
π‘βπ½β(π‘)
π2π(π βπ‘)
π₯π
(π ) π4(πΌ) π₯ (π ) ππ ,
-
Journal of Applied Mathematics 7
π4(π‘) = ββ«
0
ββ
β«
π‘
π‘+π
π2π(π βπ‘)
οΏ½ΜοΏ½π
(π ) π5(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ
+ π½ββ«
0
βπ½β
β«
π‘
π‘+π
π2π(π βπ‘)
οΏ½ΜοΏ½π
(π ) π6(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ,
π5(π‘) = ββ«
0
ββ
β«
π‘
π‘+π
π2π(π βπ‘)
οΏ½ΜοΏ½π
(π ) π7(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ
+ π½ββ«
0
βπ½β
β«
π‘
π‘+π
π2π(π βπ‘)
οΏ½ΜοΏ½π
(π ) π8(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ,
π6(π‘) = ββ«
0
ββ
β«
π‘
π‘+π
π2π(π βπ‘)
οΏ½ΜοΏ½π
(π ) π9(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ
+ π½ββ«
0
βπ½β
β«
π‘
π‘+π
π2π(π βπ‘)
οΏ½ΜοΏ½π
(π ) π10(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ,
π7(π‘) = ββ«
π‘
ββ
β«
π
πββ(π)
π2π(πβπ‘)
[π₯ (π β β (π))
οΏ½ΜοΏ½ (π )]
π
Γ [π 1(πΌ) π
2(πΌ)
β π 3(πΌ)
]
Γ [π₯ (π β β (π))
οΏ½ΜοΏ½ (π )] ππ ππ
+ π½ββ«
π‘
βπ½β
β«
π
πβπ½β(π)
π2π(πβπ‘)
[π₯ (π β π½β (π))
οΏ½ΜοΏ½ (π )]
π
Γ [π 4(πΌ) π
5(πΌ)
β π 6(πΌ)
]
Γ [π₯ (π β π½β (π))
οΏ½ΜοΏ½ (π )] ππ ππ.
(28)
Calculating the time derivatives of ππ(π‘), π = 1, 2, 3, . . . , 7,
along the trajectory of (9), yields
οΏ½ΜοΏ½1(π‘) = 2π₯
π
(π‘) π1(πΌ) οΏ½ΜοΏ½ (π‘)
= 2π₯π
π1(πΌ) [ (π΄ (πΌ) + π΅
1(πΌ) + πΊ) π₯ (π‘)
+ π΅2(πΌ) π₯ (π‘ β β (π‘)) β πΊπ₯ (π‘ β π½β (π‘))
+ π (π‘, π₯ (π‘)) + π (π‘, π₯ (π‘ β β (π‘)))
+ π€ (π‘, οΏ½ΜοΏ½ (π‘ β π (π‘))) + πΆ (πΌ) οΏ½ΜοΏ½ (π‘ β π (π‘))
β π΅1(πΌ) β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½ (π ) ππ
βπΊβ«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½ (π ) ππ ]
+ 2ππ₯π
(π‘) π1(πΌ) π₯ (π‘) β 2ππ
1(π‘) ,
οΏ½ΜοΏ½2(π‘) = οΏ½ΜοΏ½
π
(π‘) π2(πΌ) οΏ½ΜοΏ½ (π‘)
β (1 β Μπ (π‘)) πβ2ππ(π‘)
οΏ½ΜοΏ½π
(π‘ β π (π‘))
Γ π2(πΌ) οΏ½ΜοΏ½ (π‘ β π (π‘)) β 2ππ
2(π‘)
β€ οΏ½ΜοΏ½π
(π‘) π2(πΌ) οΏ½ΜοΏ½ (π‘) β π
β2ππ
οΏ½ΜοΏ½π
(π‘ β π (π‘))
Γ π2(πΌ) οΏ½ΜοΏ½ (π‘ β π (π‘))
+ πποΏ½ΜοΏ½π
(π‘ β π (π‘)) π2(πΌ) οΏ½ΜοΏ½ (π‘ β π (π‘))
β 2ππ2(π‘) .
(29)
The time derivative of π3(π‘) is
οΏ½ΜοΏ½3(π‘) = π₯
π
(π‘) π3(πΌ) π₯ (π‘) β (1 β βΜ (π‘)) π
β2πβ(π‘)
π₯π
(π‘ β β (π‘))
Γ π3(πΌ) π₯ (π‘ β β (π‘)) + π₯
π
(π‘) π4(πΌ) π₯ (π‘)
β (1 β π½βΜ (π‘)) πβ2ππ½β(π‘)
π₯π
(π‘ β π½β (π‘))
Γ π4(πΌ) π₯ (π‘ β π½β (π‘)) β 2ππ
3(π‘)
β€ π₯π
(π‘) π3(πΌ) π₯ (π‘) β π
β2πβ
π₯π
(π‘ β β (π‘))
Γ π3(πΌ) π₯ (π‘ β β (π‘)) + β
ππβ2πβ
π₯π
(π‘ β β (π‘))
Γ π3(πΌ) π₯ (π‘ β β (π‘)) + π₯
π
(π‘) π4(πΌ) π₯ (π‘)
β πβ2πΌπ½β
π₯π
(π‘ β π½β (π‘)) π4(πΌ) π₯ (π‘ β π½β (π‘))
+ π½βππβ2ππ½β
π₯π
(π‘ β π½β (π‘)) π4(πΌ) π₯ (π‘ β π½β (π‘))
β 2ππ3(π‘) .
(30)
Obviously, for any scalar π β [π‘ β β, π‘], we get πβ2πβ β€πβ2π(π βπ‘)
β€ 1 and for any scalar π β [π‘ β π½β, π‘], we obtainπβ2π½πβ
β€ πβ2π½π(π βπ‘)
β€ 1. Together with Lemma 4, we obtain
οΏ½ΜοΏ½4(π‘) = β
2
οΏ½ΜοΏ½π
(π‘) π5(πΌ) οΏ½ΜοΏ½ (π‘)
β ββ«
0
ββ
π2ππ
οΏ½ΜοΏ½π
(π‘ + π ) π5(πΌ) οΏ½ΜοΏ½ (π‘ + π ) ππ
+ π½2
β2
οΏ½ΜοΏ½π
(π‘) π6(πΌ) οΏ½ΜοΏ½ (π‘)
β π½ββ«
0
βπ½β
π2π½ππ
οΏ½ΜοΏ½π
(π‘ + π ) π6(πΌ) οΏ½ΜοΏ½ (π‘ + π ) ππ
β 2ππ4(π‘)
-
8 Journal of Applied Mathematics
β€ β2
οΏ½ΜοΏ½π
(π‘) π5(πΌ) οΏ½ΜοΏ½ (π‘)
β ββ«
π‘
π‘ββ
π2π(π βπ‘)
οΏ½ΜοΏ½π
(π ) π5(πΌ) οΏ½ΜοΏ½ (π ) ππ
+ π½2
β2
οΏ½ΜοΏ½π
(π‘) π6(πΌ) οΏ½ΜοΏ½ (π‘)
β π½ββ«
π‘
π‘βπ½β
π2π½π(π βπ‘)
οΏ½ΜοΏ½π
(π ) π6(πΌ) οΏ½ΜοΏ½ (π ) ππ
β 2ππ4(π‘)
β€ β2
οΏ½ΜοΏ½π
(π‘) π5(πΌ) οΏ½ΜοΏ½ (π‘)
β βπβ2πβ
β«
π‘
π‘ββ
οΏ½ΜοΏ½π
(π ) π5(πΌ) οΏ½ΜοΏ½ (π ) ππ
+ π½2
β2
οΏ½ΜοΏ½π
(π‘) π6(πΌ) οΏ½ΜοΏ½ (π‘)
β π½βπβ2π½πβ
β«
π‘
π‘βπ½β
οΏ½ΜοΏ½π
(π ) π6(πΌ) οΏ½ΜοΏ½ (π ) ππ
β 2ππ4(π‘) ,
οΏ½ΜοΏ½5(π‘) β€ β
2
οΏ½ΜοΏ½π
(π‘) π7(πΌ) οΏ½ΜοΏ½ (π‘)
β βπβ2πβ
β«
π‘
π‘ββ
οΏ½ΜοΏ½π
(π ) π7(πΌ) οΏ½ΜοΏ½ (π ) ππ
+ π½2
β2
οΏ½ΜοΏ½π
(π‘) π8(πΌ) οΏ½ΜοΏ½ (π‘)
β π½βπβ2π½πβ
β«
π‘
π‘βπ½β
οΏ½ΜοΏ½π
(π ) π8(πΌ) οΏ½ΜοΏ½ (π ) ππ
β 2ππ5(π‘)
β€ β2
οΏ½ΜοΏ½π
(π‘) π7(πΌ) οΏ½ΜοΏ½ (π‘)
β πβ2πβ
β«
π‘
π‘ββ
οΏ½ΜοΏ½π
(π ) ππ π7(πΌ)β«
π‘
π‘ββ
οΏ½ΜοΏ½ (π ) ππ
+ π½2
β2
οΏ½ΜοΏ½π
(π‘) π8(πΌ) οΏ½ΜοΏ½ (π‘)
β πβ2π½πβ
β«
π‘
π‘βπ½β
οΏ½ΜοΏ½π
(π ) ππ π8(πΌ) β«
π‘
π‘βπ½β
οΏ½ΜοΏ½ (π ) ππ
β 2ππ5(π‘)
β€ β2
οΏ½ΜοΏ½π
(π‘) π7(πΌ) οΏ½ΜοΏ½ (π‘)
β πβ2πβ
β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π ) ππ π7(πΌ) β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½ (π ) ππ
+ π½2
β2
οΏ½ΜοΏ½π
(π‘) π8(πΌ) οΏ½ΜοΏ½ (π‘)
β πβ2π½πβ
β«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½π
(π ) ππ π8(πΌ) β«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½ (π ) ππ
β 2πΌπ5(π‘) ,
οΏ½ΜοΏ½6(π‘) β€ β
2
οΏ½ΜοΏ½π
(π‘) π9(πΌ) οΏ½ΜοΏ½ (π‘)
β πβ2πβ
β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π ) ππ π9(πΌ) β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½ (π ) ππ
+ π½2
β2
οΏ½ΜοΏ½π
(π‘) π10(πΌ) οΏ½ΜοΏ½ (π‘)
β πβ2π½πβ
β«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½π
(π ) ππ π10(πΌ)β«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½ (π ) ππ
β 2ππ6(π‘)
= β2
οΏ½ΜοΏ½π
(π‘) π9(πΌ) οΏ½ΜοΏ½ (π‘)
β πβ2πβ
[π₯π
(π‘) β π₯π
(π‘ β β (π‘))]
Γ π9(πΌ) [π₯ (π‘) β π₯ (π‘ β β (π‘))]
+ π½β2
οΏ½ΜοΏ½π
(π‘) π10(πΌ) οΏ½ΜοΏ½ (π‘)
β πβ2π½πβ
[π₯π
(π‘) β π₯π
(π‘ β π½β (π‘))]
Γ π10(πΌ) [π₯ (π‘) β π₯ (π‘ β π½β (π‘))] β 2ππ
6(π‘) .
(31)
Taking the time derivative of π7(π‘), we obtain
οΏ½ΜοΏ½7(π‘) = ββ (π‘) π₯
π
(π‘ β β (π‘)) π 1(πΌ) π₯ (π‘ β β (π‘))
+ 2βπ₯π
(π‘ β β (π‘)) π 2(πΌ) π₯ (π‘)
β 2βπ₯π
(π‘ β β (π‘)) π 2(πΌ) π₯ (π‘ β β (π‘))
+ ββ«
π‘
π‘ββ
οΏ½ΜοΏ½π
(π ) π 3(πΌ) οΏ½ΜοΏ½ (π ) ππ
+ π½2
ββ (π‘) π₯π
(π‘ β π½β (π‘)) π 4(πΌ) π₯ (π‘ β π½β (π‘))
+ 2π½βπ₯π
(π‘ β β (π‘)) π 5(πΌ) π₯ (π‘)
β 2π½βπ₯π
(π‘ β π½β (π‘)) π 5(πΌ) π₯ (π‘ β π½β (π‘))
+ π½ββ«
π‘
π‘βπ½β
οΏ½ΜοΏ½π
(π ) π 6(πΌ) οΏ½ΜοΏ½ (π ) ππ β 2ππ
7(π‘)
β€ β2
π₯π
(π‘ β β (π‘)) π 1(πΌ) π₯ (π‘ β β (π‘))
+ 2βπ₯π
(π‘ β β (π‘)) π 2(πΌ) π₯ (π‘)
β 2βπ₯π
(π‘ β β (π‘)) π 2(πΌ) π₯ (π‘ β β (π‘))
+ ββ«
π‘
π‘ββ
οΏ½ΜοΏ½π
(π ) π 3(πΌ) οΏ½ΜοΏ½ (π ) ππ
+ π½2
β2
π₯π
(π‘ β π½β (π‘)) π 4(πΌ) π₯ (π‘ β π½β (π‘))
+ 2π½βπ₯π
(π‘ β β (π‘)) π 5(πΌ) π₯ (π‘)
β 2π½βπ₯π
(π‘ β π½β (π‘)) π 5(πΌ) π₯ (π‘ β π½β (π‘))
+ π½ββ«
π‘
π‘βπ½β
οΏ½ΜοΏ½π
(π ) π 6(πΌ) οΏ½ΜοΏ½ (π ) ππ β 2ππ
7(π‘) .
(32)
-
Journal of Applied Mathematics 9
By Lemma 5 and the integral term of the right hand sideof οΏ½ΜοΏ½4(π‘) and οΏ½ΜοΏ½
7(π‘), we obtain
β ββ«
π‘
π‘ββ
οΏ½ΜοΏ½π
(π ) [πβ2πβ
π5(πΌ) β π
3(πΌ)] οΏ½ΜοΏ½ (π ) ππ
β π½ββ«
π‘
π‘βπ½β
οΏ½ΜοΏ½π
(π ) [πβ2π½πβ
π6(πΌ) β π
6(πΌ)] οΏ½ΜοΏ½ (π ) ππ
β€ β[π₯ (π‘)
π₯π‘ β β (π‘)]
π
Γ [ππ
1(πΌ) + π
1(πΌ) βπ
π
1(πΌ) + π
2(πΌ)
β βππ
2(πΌ) β π
2(πΌ)
]
Γ [π₯ (π‘)
π₯π‘ β β (π‘)]
+ β2
[π₯ (π‘)
π₯π‘ β β (π‘)]
π
[π3(πΌ) π
4(πΌ)
β π5(πΌ)
]
Γ [π₯ (π‘)
π₯π‘ β β (π‘)] + π½β[
π₯ (π‘)
π₯π‘ β π½β (π‘)]
π
Γ [ππ
6(πΌ) + π
6(πΌ) βπ
π
6(πΌ) + π
7(πΌ)
β βππ
7(πΌ) β π
7(πΌ)
]
Γ [π₯ (π‘)
π₯π‘ β π½β (π‘)]
+ π½2
β2
[π₯ (π‘)
π₯π‘ β π½β (π‘)]
π
[π8(πΌ) π
9(πΌ)
β π10(πΌ)
]
Γ [π₯ (π‘)
π₯π‘ β π½β (π‘)] .
(33)
From the Leibniz-Newton formula, the following equa-tions are true for any parameter-dependent real matricesππ(πΌ), π
π(πΌ), π = 1, 2, . . . , 10 with appropriate dimensions:
2 [π₯π
(π‘)ππ
1(πΌ) + π₯
π
(π‘ β β (π‘))ππ
2(πΌ)
+ π₯π
(π‘ β π½β (π‘))ππ
3(πΌ) + β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π ) ππ ππ
4(πΌ)
+ β«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½π
(π ) ππ ππ
5(πΌ) + οΏ½ΜοΏ½
π
(π‘)ππ
6(πΌ)
+ οΏ½ΜοΏ½π
(π‘ β π (π‘))ππ
7(πΌ) + π
π
(π‘, π₯ (π‘))ππ
8(πΌ)
+ππ
(π‘, π₯ (π‘ β β (π‘)))ππ
9(πΌ) + π€
π
(π‘, οΏ½ΜοΏ½ (π‘ β π (π‘)))ππ
10(πΌ) ]
Γ [π₯ (π‘) β π₯ (π‘ β β (π‘)) β β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½ (π ) ππ ] = 0,
2 [π₯π
(π‘) ππ
1(πΌ) + π₯
π
(π‘ β β (π‘)) ππ
2(πΌ)
+ π₯π
(π‘ β π½β (π‘)) ππ
3(πΌ) + β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π ) ππ ππ
4(πΌ)
+ β«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½π
(π ) ππ ππ
5(πΌ) + οΏ½ΜοΏ½
π
(π‘) ππ
6(πΌ)
+ οΏ½ΜοΏ½π
(π‘ β π (π‘)) ππ
7(πΌ) + π
π
(π‘, π₯ (π‘)) ππ
8(πΌ)
+ ππ
(π‘, π₯ (π‘ β β (π‘))) ππ
9(πΌ) +π€
π
(π‘, οΏ½ΜοΏ½ (π‘ β π (π‘))) ππ
10(πΌ) ]
Γ [π₯ (π‘) β π₯ (π‘ β π½β (π‘)) β β«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½ (π ) ππ ] = 0.
(34)
From the utilization of zero equation, the followingequation is true for any parameter-dependent real matricesππ, π = 1, 2, . . . , 10 with appropriate dimensions:
2 [π₯π
(π‘)ππ
1(πΌ) + π₯
π
(π‘ β β (π‘))ππ
2(πΌ)
+ π₯π
(π‘ β π½β (π‘))ππ
3(πΌ) + β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½π
(π ) ππ ππ
4(πΌ)
+ β«
π‘
π‘βπ½β(π‘)
οΏ½ΜοΏ½π
(π ) ππ ππ
5(πΌ) + οΏ½ΜοΏ½
π
(π‘)ππ
6(πΌ)
+ οΏ½ΜοΏ½π
(π‘ β π (π‘))ππ
7(πΌ) + π
π
(π‘, π₯ (π‘))π8(πΌ)
+ ππ
(π‘, π₯ (π‘ β β (π‘)))ππ
9(πΌ)
+π€π
(π‘, οΏ½ΜοΏ½ (π‘ β π (π‘)))ππ
10(πΌ) ]
Γ [ (π΄ (πΌ) + π΅1(πΌ)) π₯ (π‘) + π΅
2(πΌ) π₯ (π‘ β β (π‘))
+ πΆ (πΌ) οΏ½ΜοΏ½ (π‘ β π (π‘)) + π (π‘, π₯ (π‘)) + π (π‘, π₯ (π‘ β β (π‘)))
+ π€ (π‘, οΏ½ΜοΏ½ (π‘ β π (π‘))) β π΅1(πΌ) β«
π‘
π‘ββ(π‘)
οΏ½ΜοΏ½ (π ) ππ β οΏ½ΜοΏ½ (π‘)]
= 0.
(35)
From (5), we obtain, for any positive real constants π1, π2,
and π3,
0 β€ π1π2
π₯π
(π‘) π₯ (π‘) β π1ππ
(π‘, π₯ (π‘)) π (π‘, π₯ (π‘)) ,
0 β€ π2π2
π₯π
(π‘ β β (π‘)) π₯ (π‘ β β (π‘))
β π2ππ
(π‘, π₯ (π‘ β β (π‘))) π (π‘, π₯ (π‘ β β (π‘))) ,
0 β€ π3πΎ2
οΏ½ΜοΏ½π
(π‘ β π (π‘)) οΏ½ΜοΏ½ (π‘ β π (π‘))
β π3π€π
(π‘, οΏ½ΜοΏ½ (π‘ β π (π‘))) π€ (π‘, οΏ½ΜοΏ½ (π‘ β π (π‘))) .
(36)
-
10 Journal of Applied Mathematics
According to (29)β(36), it is straightforward to see that
οΏ½ΜοΏ½ (π‘) β€ ππ
(π‘)
π
β
π=1
π
β
π=1
πΌππΌπβ
π,π
π (π‘) β 2ππ (π‘) , (37)
where ππ(π‘) = [π₯π(π‘),π₯π(π‘ββ(π‘)),π₯π(π‘βπ½β(π‘)),β«π‘π‘ββ(π‘)
οΏ½ΜοΏ½π(π )ππ ,
β«π‘
π‘βπ½β(π‘)οΏ½ΜοΏ½π(π )ππ , οΏ½ΜοΏ½
π(π‘), οΏ½ΜοΏ½π(π‘ β π(π‘)), ππ(π‘, π₯(π‘)), ππ(π‘, π₯(π‘ β
β(π‘))), π€π(π‘, οΏ½ΜοΏ½(π‘ β π(π‘))] and βπ,πis defined in (23). From the
fact that βππ=1
πΌπ= 1,
π
β
π=1
πΌππ΄π
π
β
π=1
πΌππ΅π=
π
β
π=1
πΌ2
ππ΄ππ΅π+
πβ1
β
π=1
π
β
π=π+1
πΌππΌπ[π΄ππ΅π+ π΄ππ΅π] ,
(π β 1)
π
β
π=1
πΌ2
πβ 2
πβ1
β
π=1
π
β
π=π+1
πΌππΌπ=
πβ1
β
π=1
π
β
π=π+1
[πΌπβ πΌπ]2
β₯ 0.
(38)It is true that if conditions (25) hold, then
οΏ½ΜοΏ½ (π‘) + 2ππ (π‘) β€ 0, βπ‘ β π +
, (39)which gives
π (π‘) β€ π (0) πβ2ππ‘
, βπ‘ β π +
. (40)From (40), it is easy to see that
πmin (π1 (πΌ)) βπ₯ (π‘)β2
β€ π (π‘) β€ π (0) πβ2ππ‘
, (41)
π (0) =
7
β
π=1
ππ(0) , (42)
whereπ1(0) = π₯
π
(0) π1(πΌ) π₯ (0) ,
π2(0) = β«
0
βπ(0)
π2ππ
οΏ½ΜοΏ½π
(π ) π2(πΌ) οΏ½ΜοΏ½ (π ) ππ ,
π3(0) = β«
0
ββ(0)
π2ππ
π₯π
(π ) π3(πΌ) π₯ (π ) ππ
+ β«
0
βπ½β(0)
π2ππ
π₯π
(π ) π4(πΌ) π₯ (π ) ππ ,
π4(0) = ββ«
0
ββ
β«
0
π
π2ππ
οΏ½ΜοΏ½π
(π ) π5(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ
+ π½ββ«
0
βπ½β
β«
0
π
π2ππ
οΏ½ΜοΏ½π
(π ) π6(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ,
π5(0) = ββ«
0
ββ
β«
0
π
π2ππ
οΏ½ΜοΏ½π
(π ) π7(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ
+ π½ββ«
0
βπ½β
β«
0
π
π2ππ
οΏ½ΜοΏ½π
(π ) π8(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ,
π6(0) = ββ«
0
ββ
β«
π‘
π
π2ππ
οΏ½ΜοΏ½π
(π ) π9(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ
+ π½ββ«
0
βπ½β
β«
0
π
π2ππ
οΏ½ΜοΏ½π
(π ) π10(πΌ) οΏ½ΜοΏ½ (π ) ππ ππ,
π7(0) = ββ«
0
ββ
β«
π
πββ(π)
π2ππ
[π₯ (π β β (π))
οΏ½ΜοΏ½ (π )]
π
Γ [π 1(πΌ) π
2(πΌ)
β π 3(πΌ)
]
Γ [π₯ (π β β (π))
οΏ½ΜοΏ½ (π )] ππ ππ
+ π½ββ«
0
βπ½β
β«
π
πβπ½β(π)
π2ππ
[π₯ (π β π½β (π))
οΏ½ΜοΏ½ (π )]
π
Γ [π 4(πΌ) π
5(πΌ)
β π 6(πΌ)
]
Γ [π₯ (π β π½β (π))
οΏ½ΜοΏ½ (π )] ππ ππ.
(43)
From (41), we conclude that
πmin (π1 (πΌ)) βπ₯ (π‘)β2
β€ π (0) πβ2ππ‘
β€ πΏmax [π ,π
]2
πβ2ππ‘
,
(44)
where πΏ = πmax(π1(πΌ)) + ππmax(π2(πΌ))+ βπmax(π3(πΌ)) +π½βπmax(π4(πΌ))+ β
3πmax(π5(πΌ) + π7(πΌ)+π9(πΌ)) + (π½β)
3
πmax(π6(πΌ) +π8(πΌ) +π10(πΌ)) + β3πmax ([
π 1(πΌ) π 2(πΌ)
β π 3(πΌ)]) + (π½β)
3
πmax ([π 4(πΌ) π 5(πΌ)
β π 6(πΌ)]). From (44), this means that the system
(1) is robustly exponentially stable. The proof of the theoremis complete.
Next, we consider the following system:
οΏ½ΜοΏ½ (π‘) = π΄ (πΌ) π₯ (π‘) + π΅ (πΌ) π₯ (π‘ β β (π‘)) + πΆ (πΌ) οΏ½ΜοΏ½ (π‘ β π (π‘))
+ π (π‘, π₯ (π‘)) + π (π‘, π₯ (π‘ β β (π‘))) , π‘ > 0;
π₯ (π‘) = π (π‘) , οΏ½ΜοΏ½ (π‘) = π (π‘) , π‘ β [βπ, 0] .
(45)
We introduce the following notations for later use:
β
π,π
=
[[[[[[[[[[[[[[[[
[
Ξ£1,1
π,πΞ£1,2
π,πΞ£1,3
π,πΞ£1,4
π,πΞ£1,5
π,πΞ£1,6
π,πΞ£1,7
π,πΞ£1,8
π,πΞ£1,9
π,π
β Ξ£2,2
π,πΞ£2,3
π,πΞ£2,4
π,πΞ£2,5
π,πΞ£2,6
π,πΞ£2,7
π,πΞ£2,8
π,πΞ£2,9
π,π
β β Ξ£3,3
πΞ£3,4
π,πΞ£3,5
πΞ£3,6
πΞ£3,7
π,πΞ£3,8
πΞ£3,9
π
β β β Ξ£4,4
π,πΞ£4,5
π,πΞ£4,6
π,πΞ£4,7
π,πΞ£4,8
πΞ£4,9
π
β β β β Ξ£5,5
πΞ£5,6
πΞ£5,7
π,πΞ£5,8
πΞ£5,9
π
β β β β β Ξ£6,6
πΞ£6,7
π,πΞ£6,8
πΞ£6,9
π
β β β β β β Ξ£Μ7,7
π,πΞ£7,8
π,πΞ£7,9
π,π
β β β β β β β Ξ£8,8
πΞ£8,9
π
β β β β β β β β Ξ£9,9
π
]]]]]]]]]]]]]]]]
]
.
(46)
Corollary 8. For βπΆπβ < 1, π = 1, 2, . . . , π and given
positive real constants β, βπ, π, ππ, π, π, and π½, system (45)
is robustly exponentially stable with a decay rate π, if thereexist symmetric positive definite matrices ππ
π, any appropriate
dimensional matrices π ππ, πππ, πππ, πππ, πππ, π = 1, 2, . . . 6,
π = 1, 2, . . . , 10, π = 1, 2, . . . , 9, π = 1, 2, . . . , π, and positive
-
Journal of Applied Mathematics 11
real constants π1, π2such that the following symmetric linear
matrix inequalities hold:
[π 1
ππ 2
π
β π 3
π
] > 0, π = 1, 2, . . . , π,
[π 4
ππ 5
π
β π 6
π
] > 0, π = 1, 2, . . . , π,
[
[
πβ2πβ
π3
πβ π 3
ππ1
ππ2
π
β π3
ππ4
π
β β π5
π
]
]
β₯ 0, π = 1, 2, . . . , π,
[
[
πβ2π½πβ
π6
πβ π 6
ππ6
ππ7
π
β π8
ππ9
π
β β π10
π
]
]
β₯ 0, π = 1, 2, . . . , π,
β
π,π
< βπΌ, π = 1, 2, . . . , π,
β
π,π
+β
π,π
<2
(π β 1)πΌ, π = 1, 2, . . . , π β 1,
π = π + 1, π + 2, . . . , π.
(47)
Moreover, the solution π₯(π‘, π, π) satisfies the inequality
π₯ (π‘, π, π) β€ β
πΏ
πmin (π1 (πΌ))max [π
,π
] πβππ‘
,
βπ‘ β π +
,
(48)
where πΏ = πmax(π1(πΌ)) + ππmax(π2(πΌ)) + βπmax(π3(πΌ)) +π½βπmax(π4(πΌ)) + β
3πmax(π5(πΌ) + π7(πΌ) + π9(πΌ)) +
(π½β)3
πmax(π6(πΌ) + π8(πΌ) + π10(πΌ)) + β3πmax ([
π 1(πΌ) π 2(πΌ)
β π 3(πΌ)]) +
(π½β)3
πmax ([π 4(πΌ) π 5(πΌ)
β π 6(πΌ)]).
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:
β =
[[[[[[[[[[[[[[
[
Ξ£11
Ξ£12
Ξ£13
Ξ£14
Ξ£15
Ξ£16
Ξ£17
Ξ£18
Ξ£19
Ξ£1,10
β Ξ£22
Ξ£23
Ξ£24
Ξ£25
Ξ£26
Ξ£27
Ξ£28
Ξ£29
Ξ£2,10
β β Ξ£33
Ξ£34
Ξ£35
Ξ£36
Ξ£37
Ξ£38
Ξ£39
Ξ£3,10
β β β Ξ£44
Ξ£45
Ξ£46
Ξ£47
Ξ£48
Ξ£49
Ξ£4,10
β β β β Ξ£55
Ξ£56
Ξ£57
Ξ£58
Ξ£59
Ξ£5,10
β β β β β Ξ£66
Ξ£67
Ξ£68
Ξ£69
Ξ£6,10
β β β β β β Ξ£77
Ξ£78
Ξ£79
Ξ£7,10
β β β β β β β Ξ£88
Ξ£89
Ξ£8,10
β β β β β β β β Ξ£99
Ξ£9,10
β β β β β β β β β Ξ£10,10
]]]]]]]]]]]]]]
]
,
(49)
where
Ξ£1,1
= π΄π
π1+ π΅π
1π1+ ππ
1+ π1π΄ + π
1π΅1+ π1+ π3
+ π4+ βπ
π
1+ βπ
1β πβ2πβ
π9β πβ2ππ½β
π10
+ β2
π3+ π½βπ
6+ π½βπ
π
6+ π½2
β2
π8
+ ππ
1+ π1+ ππ
1+ π1+ π΄π
π1+ππ
1π΄
+ π΅π
1π1+ππ
1π΅1+ π1π2
πΌ + 2ππ1,
Ξ£1,2
= π1π΅2β βπ
π
1+ βπ
2+ β2
π4+ πβ2πβ
π9
+ βπ 2β ππ
1+ π2+ π2+ππ
1π΅2
+ π΄π
π2+ π΅π
1π2,
Ξ£1,3
= βπ1β π½βπ
π
6+ π½βπ
7+ π½β2
π9+ π½βπ
5
+ πβ2ππ½β
π10+ π3β ππ
1+ π3
+ π΄π
π3+ π΅π
1π3,
Ξ£1,4
= βπ1π΅1β ππ
1+ π4+ π4
βππ
1π΅1+ π΄π
π4+ π΅π
1π4,
Ξ£1,5
= βπ1+ π5β ππ
1+ π5+ π΄π
π5+ π΅π
1π5,
Ξ£1,6
= π6+ π6βππ
1+ π΄π
π6+ π΅π
1π6,
Ξ£1,7
= π1πΆ + π
7+ π7+ππ
1πΆ + π΄
π
π7+ π΅π
1π7,
Ξ£1,8
= π1+ π8+ π8+ππ
1+ π΄π
π8+ π΅π
1π8,
Ξ£1,9
= π1+ π9+ π9+ππ
1+ π΄π
π9+ π΅π
1π9,
Ξ£1,10
= π1+ π10+ π10+ππ
1+ π΄π
π10+ π΅π
1π10,
Ξ£2,2
= β (1 β βπ) πβ2πβ
π3β βπ
π
2β βπ
2
+ β2
π5+ β2
π 1β 2βπ
2β πβ2πβ
π9
+ π2π2
πΌ β ππ
2β π2+ππ
2π΅2+ π΅π
2π2,
Ξ£2,3
= βπ3β ππ
2+ π΅π
2π3,
Ξ£2,4
= βππ
2β π4βππ
2π΅1+ π΅π
2π4,
Ξ£2,5
= βπ5β ππ
2+ π΅π
2π5,
Ξ£2,6
= βπ6βππ
2+ π΅π
2π6,
Ξ£2,7
= βπ7+ππ
2πΆ + π΅
π
2π7,
Ξ£2,8
= βπ8+ππ
2+ π΅π
2π8,
Ξ£2,9
= βππ
9βππ
2+ π΅π
2π9,
Ξ£2,10
= βπ10+ππ
2+ π΅π
2π10,
-
12 Journal of Applied Mathematics
Ξ£3,3
= β (1 β π½βπ) πβ2π½πβ
π4β πβ2π½πβ
π10+ π½2
β2
π 4
β 2π½βπ 5β π½βπ
π
7β π½βπ
7+ π½2
β2
π10
β ππ
3β π3,
Ξ£3,4
= βππ
3β π4βππ
3π΅1,
Ξ£3,5
= βππ
3β π5,
Ξ£3,6
= βπ9+ππ
3, Ξ£
3,7= βπ7+ππ
3πΆ,
Ξ£3,8
= βπ8+ππ
3, Ξ£
3,9= βπ9+ππ
3,
Ξ£3,10
= βπ10+ππ
3,
Ξ£4,4
= βπ2πβ
π7β ππ
4β π4+ππ
4π΅1β π΅π
1π4,
Ξ£4,5
= βπ5β ππ
4β π΅π
1π5,
Ξ£4,6
= βπ6βππ
4β π΅π
1π6,
Ξ£4,7
= βπ7+ππ
4πΆ β π΅
π
1π7,
Ξ£4,8
= βπ8+ππ
3,
Ξ£4,9
= βπ9+ππ
3, Ξ£
4,10= βπ10+ππ
3,
Ξ£5,5
= βπ2π½πβ
π8β ππ
5β π5,
Ξ£5,6
= βπ6βππ
5,
Ξ£5,7
= βπ7+ππ
5πΆ, Ξ£
5,8= βπ8+ππ
5,
Ξ£5,9
= βπ9+ππ
5,
Ξ£5,10
= βπ10+ππ
5,
Ξ£6,6
= π2+ β2
π5+ π½2
β2
π6+ β2
π7+ π½2
β2
π8
+ β2
π9+ π½2
β2
π10βππ
6βπ6,
Ξ£6,7
= ππ
6πΆ βπ
7, Ξ£
6,8= ππ
6βπ8,
Ξ£6,9
= ππ
6βπ9, Ξ£
6,10= ππ
6βπ10,
Ξ£7,7
= β (1 β ππ) πβ2ππ
π2+ π3πΎ2
πΌ
+ ππ
7πΆ + πΆ
π
π7,
Ξ£7,8
= ππ
7β πΆπ
π8, Ξ£
7,9= ππ
7β πΆπ
π9,
Ξ£7,10
= ππ
7β πΆπ
π10
Ξ£8,8
= ππ
8+π8β π1πΌ,
Ξ£8,9
= ππ
8+π9, Ξ£
8,10= ππ
8+π10,
Ξ£9,9
= ππ
9+π9β π2πΌ, Ξ£
9,10= ππ
9+π10,
Ξ£10,10
= ππ
10+π10β π3πΌ,
Ξ£Μ7,7
= β (1 β ππ) πβ2ππ
π2+ππ
7πΆ + πΆ
π
π7,
π1= π1πΊ.
(50)
If π΄(πΌ) = π΄, π΅(πΌ) = π΅, and πΆ(πΌ) = πΆ, where π΄, π΅, πΆ βπ πΓπ are real constant matrices, then system (1) reduces to the
following system:
οΏ½ΜοΏ½ (π‘) = π΄π₯ (π‘) + π΅π₯ (π‘ β β (π‘)) + πΆοΏ½ΜοΏ½ (π‘ β π (π‘))
+ π (π‘, π₯ (π‘)) + π (π‘, π₯ (π‘ β β (π‘)))
+ π€ (π‘, οΏ½ΜοΏ½ (π‘ β π (π‘))) , π‘ > 0;
π₯ (π‘) = π (π‘) , οΏ½ΜοΏ½ (π‘) = π (π‘) , π‘ β [βπ, 0] .
(51)
Corollary 9. For βπΆβ+πΎ < 1 and given positive real constantsβ, βπ, π, ππ,π,π, πΎ, andπ½, system (51) is exponentially stablewith
a decay rate π, if there exist symmetric positive definitematricesππ, any appropriate dimensional matrices π
π , ππ, ππ, ππ, ππ,
πΊ, π = 1, 2, . . . 6, π = 1, 2, . . . , 10, and positive real constantsπ1, π2, and π
3such that the following symmetric linear matrix
inequalities hold:
[π 1
π 2
β π 3
] > 0,
[π 4
π 5
β π 6
] > 0,
[
[
πβ2πβ
π3β π 3
π1
π2
β π3
π4
β β π5
]
]
β₯ 0,
[
[
πβ2π½πβ
π6β π 6
π6
π7
β π8
π9
β β π10
]
]
β₯ 0,
β < 0.
(52)
Moreover, the solution π₯(π‘, π, π) satisfies the inequality
π₯ (π‘, π, π) β€ β
π·
πmin (π1)max [π
,π
] πβππ‘
, βπ‘ β π +
,
(53)
whereπ· = πmax(π1) + ππmax(π2) + βπmax(π3) + π½βπmax(π4) +β3πmax(π5 + π7 + π9) + (π½β)
3
πmax(π6 + π8 + π10) +
β3πmax ([
π 1 π 2
β π 3]) + (π½β)
3
πmax ([π 4 π 5
β π 6]).
Ifπ΄(πΌ) = π΄,π΅(πΌ) = π΅,πΆ(πΌ) = πΆ, andπ€(π‘, οΏ½ΜοΏ½(π‘βπ(π‘))) = 0,where π΄, π΅, πΆ β π πΓπ are real constant matrices, then system(1) reduces to the following system:
οΏ½ΜοΏ½ (π‘) = π΄π₯ (π‘) + π΅π₯ (π‘ β β (π‘)) + πΆοΏ½ΜοΏ½ (π‘ β π (π‘))
+ π (π‘, π₯ (π‘)) + π (π‘, π₯ (π‘ β β (π‘))) , π‘ > 0;
π₯ (π‘) = π (π‘) , οΏ½ΜοΏ½ (π‘) = π (π‘) , π‘ β [βπ, 0] .
(54)
Corollary 10. For βπΆβ < 1 and given positive real constantsβ, βπ, π, ππ, π, π, and π½, system (54) is exponentially stable
with a decay rate π, if there exist symmetric positive definitematrices π
π, any appropriate dimensional matrices π
π , ππ, ππ,
-
Journal of Applied Mathematics 13
ππ, ππ, and πΊ, where π = 1, 2, . . . 6, π = 1, 2, . . . , 9, and
π = 1, 2, . . . 10, and positive real constants π1, π2such that the
following symmetric linear matrix inequalities hold:
[π 1
π 2
β π 3
] > 0,
[π 4
π 5
β π 6
] > 0,
[
[
πβ2πβ
π3β π 3
π1
π2
β π3
π4
β β π5
]
]
β₯ 0,
[
[
πβ2π½πβ
π6β π 6
π6
π7
β π8
π9
β β π10
]
]
β₯ 0,
β < 0,
(55)
where
β =
[[[[[[[[[[[[[
[
Ξ£11
Ξ£12
Ξ£13
Ξ£14
Ξ£15
Ξ£16
Ξ£17
Ξ£18
Ξ£19
β Ξ£22
Ξ£23
Ξ£24
Ξ£25
Ξ£26
Ξ£27
Ξ£28
Ξ£29
β β Ξ£33
Ξ£34
Ξ£35
Ξ£36
Ξ£37
Ξ£38
Ξ£39
β β β Ξ£44
Ξ£45
Ξ£46
Ξ£47
Ξ£48
Ξ£49
β β β β Ξ£55
Ξ£56
Ξ£57
Ξ£58
Ξ£59
β β β β β Ξ£66
Ξ£67
Ξ£68
Ξ£69
β β β β β β Ξ£Μ77
Ξ£78
Ξ£79
β β β β β β β Ξ£88
Ξ£89
β β β β β β β β Ξ£99
]]]]]]]]]]]]]
]
. (56)
Moreover, the solution π₯(π‘, π, π) satisfies the inequality
π₯ (π‘, π, π) β€ β
π·
πmin (π1)max [π
,π
] πβππ‘
, βπ‘ β π +
,
(57)
whereπ· = πmax(π1) + ππmax(π2) + βπmax(π3) + π½βπmax(π4) +β3πmax(π5 + π7 + π9) + (π½β)
3
πmax(π6 + π8 + π10) +
β3πmax ([
π 1 π 2
β π 3]) + (π½β)
3
πmax ([π 4 π 5
β π 6]).
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
π΄ (πΌ) = πΌ1[β2 0
0 β1] + πΌ2[β2 0
0 β0.8] ,
π΅ (πΌ) = πΌ1[β1 0
β1 β0.8] + πΌ2[β1 0
β1 β1] ,
πΆ (πΌ) = πΌ1[0.1 0
0 0.1] + πΌ2[0.2 0
0 0.1] ,
π (π‘, π₯ (π‘)) = [0.05 sin (π‘) π₯
1(π‘)
0.05 cos (π‘) π₯2(π‘)
] ,
π (π‘, π₯ (π‘ β β (π‘))) = [0.1 cos (π‘)2π₯
1(π‘ β β (π‘))
0.1 sin (π‘)2π₯2(π‘ β β (π‘))
] ,
β (π‘) = 0.9sin2 ( π‘2) ,
π€ (π‘, οΏ½ΜοΏ½ (π‘ β π (π‘))) = [0.1 cos (π‘) οΏ½ΜοΏ½
1(π‘ β π (π‘))
0.1 sin (π‘) οΏ½ΜοΏ½2(π‘ β π (π‘))
] ,
π (π‘) = cos2 ( π‘4) ,
(58)
where π₯(π‘) β π 2. It is easy to see that β = 0.9, βπ= 0.45,
π = 1, ππ= 0.25, π = 0.05, π = 0.1, πΎ = 0.1, and given rate
of convergence π = 0.1. Decompose matrix π΅(πΌ) as follows:π΅(πΌ) = π΅
1(πΌ) + π΅
2(πΌ), where
π΅1(πΌ) = πΌ
1[β0.83 0
β1 β0.63] + πΌ2[β0.83 0
β1 β0.83] ,
π΅2(πΌ) = πΌ
1[β0.17 0
0 β0.17] + πΌ2[β0.17 0
0 β0.17] .
(59)
The numerical solutions π₯1(π‘) and π₯
2(π‘) of system (1) with
(58)-(59) are plotted in Figure 1 where the states π₯1(π‘) and
π₯2(π‘) 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:
π1
1= 103
Γ [3.3091 β0.0455
β0.0455 0.7941] ,
π1
2= 103
Γ [3.3091 β0.0455
β0.0455 0.7941] ,
π2
1= 103
Γ [4.3739 β0.3929
β0.3929 2.7610] ,
π2
2= 103
Γ [4.7896 β0.4618
β0.4618 2.8307] ,
π3
1= 104
Γ [1.5976 0.1384
0.1384 0.3189] ,
π3
2= 104
Γ [1.5976 0.1384
0.1384 0.3189] ,
π4
1= 103
Γ [9.8670 β0.0635
β0.0635 4.5726] ,
-
14 Journal of Applied Mathematics
π4
2= 103
Γ [8.8868 β0.2421
β0.2421 3.2473] ,
π5
1= 103
Γ [1.3009 β0.1372
β0.1372 0.6505] ,
π5
2= 103
Γ [1.0898 β0.0980
β0.0980 0.3986] ,
π6
1= 104
Γ [2.8885 β0.0338
β0.0338 2.6935] ,
π6
2= 104
Γ [2.8526 β0.0449
β0.0449 2.4922] ,
π7
1= 103
Γ [2.6064 β0.3045
β0.3045 2.5259] ,
π7
2= 103
Γ [2.3663 β0.1722
β0.1722 2.5747] ,
π8
1= 104
Γ [1.4801 β0.0277
β0.0277 1.4063] ,
π8
2= 104
Γ [1.4740 β0.0862
β0.0862 1.4207] ,
π9
1= 103
Γ [2.6064 β0.3045
β0.3045 2.5259] ,
π9
2= 103
Γ [2.3663 β0.1722
β0.1722 2.5747] ,
π10
1= 104
Γ [1.4570 β0.0000
β0.0000 1.2074] ,
π10
2= 104
Γ [1.4354 0.0019
0.0019 1.0978] ,
π1
1= 103
Γ [β9.1839 β1.2537
β1.2537 β2.0706] ,
π1
2= 104
Γ [β1.0642 β0.1844
β0.1844 β0.2293] ,
π2
1= 103
Γ [8.8459 1.1503
1.1503 2.0066] ,
π2
2= 104
Γ [1.0288 0.1763
0.1763 0.2238] ,
π3
1= 104
Γ [1.0499 0.1090
0.1090 0.2928] ,
π3
2= 104
Γ [1.1838 0.1766
0.1766 0.2881] ,
π4
1= 103
Γ [β8.8058 β0.9604
β0.9604 β2.6545] ,
π4
2= 104
Γ [β1.0454 β0.1677
β0.1677 β0.2680] ,
π5
1= 104
Γ [1.0303 0.1018
0.1018 0.3190] ,
π5
2= 104
Γ [1.1590 0.1622
0.1622 0.3078] ,
π6
1= [
β34.9655 β18.7513
β18.7513 35.4129] ,
π6
2= [
β39.8158 β24.8476
β24.8476 13.4547] ,
π7
1= [
2.0904 2.7717
2.7717 β7.8109] , π
7
2= [
3.2803 5.2929
5.2929 β4.0693] ,
π8
1= 104
Γ [1.4679 0.0003
0.0003 1.2678] ,
π8
2= 104
Γ [1.4438 β0.0080
β0.0080 1.1830] ,
π9
1= 103
Γ [β0.8905 0.0058
0.0058 β2.9028] ,
π9
2= 103
Γ [β1.1307 β0.0760
β0.0760 β3.7474] ,
π10
1= 104
Γ [1.4684 0.0005
0.0005 1.2673] ,
π10
2= 104
Γ [1.4444 β0.0077
β0.0077 1.1828] ,
π 1
1= 103
Γ [2.9131 0.1748
0.1748 0.7782] ,
π 1
2= 103
Γ [2.3463 0.0353
0.0353 0.5806] ,
π 2
1= 103
Γ [1.1393 0.0531
0.0531 0.1251] ,
π 2
2= [
876.5627 32.5550
32.5550 98.5599] ,
π 3
1= 103
Γ [2.6387 β0.0759
β0.0759 0.5866] ,
π 3
2= 103
Γ [2.0346 β0.0762
β0.0762 0.3589] ,
π 4
1= 103
Γ [5.6600 β0.0415
β0.0415 5.0541] ,
π 4
2= 103
Γ [5.5782 β0.0710
β0.0710 4.6031] ,
π 5
1= 104
Γ [β7.9071 0.0551
0.0551 β3.8199] ,
π 5
2= 104
Γ [β7.3912 0.4576
0.4576 β3.2145] ,
-
Journal of Applied Mathematics 15
π 6
1= 104
Γ [1.4893 β0.0162
β0.0162 1.3778] ,
π 6
2= 104
Γ [1.4703 β0.0223
β0.0223 1.2713] ,
π1
1= 103
Γ [8.7493 3.8534
3.8534 β3.1296] ,
π1
2= 104
Γ [0.6691 0.2686
0.2686 β1.0299] ,
π2
1= 103
Γ [β3.8698 β0.3606
β0.3606 1.4605] ,
π2
2= 103
Γ [β4.5981 β0.9753
β0.9753 1.2772] ,
π3
1= [
β114.8946 β186.6948
β186.6948 β113.8034] ,
π3
2= [
210.0606 β115.9077
β115.9077 β174.4056] ,
π4
1= 104
Γ [β1.1296 β0.0266
β0.0266 0.1446] ,
π4
2= 104
Γ [β1.0062 0.1041
0.1041 0.8782] ,
π5
1= 103
Γ [4.5589 β2.3847
β2.3847 0.8561] ,
π5
2= 103
Γ [5.3887 β2.4363
β2.4363 1.1619] ,
π6
1= 103
Γ [5.9564 β1.9530
β1.9530 1.5688] ,
π6
2= 103
Γ [6.7696 β1.9958
β1.9958 2.0384] ,
π7
1= 103
Γ [5.9602 β0.2891
β0.2891 2.9332] ,
π7
2= 103
Γ [6.3929 β0.0855
β0.0855 3.1668] ,
π8
1= 103
Γ [β9.6508 β0.4780
β0.4780 3.5901] ,
π8
2= 104
Γ [β0.8505 0.0574
0.0574 1.1344] ,
π9
1= 103
Γ [5.5789 β2.0704
β2.0704 1.3737] ,
π9
2= 103
Γ [6.3923 β2.1169
β2.1169 1.8008] ,
π10
1= 103
Γ [β3.6595 3.2032
3.2032 β0.1648] ,
π10
2= 103
Γ [β4.3781 3.2120
3.2120 0.1271] ,
π1
1= 104
Γ [1.5923 β0.7669
β0.7669 1.6350] ,
π1
2= 104
Γ [1.8134 β0.6324
β0.6324 2.3112] ,
π2
1= 104
Γ [1.0371 β0.0001
β0.0001 β0.3635] ,
π2
2= 104
Γ [0.9152 β0.1464
β0.1464 β1.0978] ,
π3
1= 103
Γ [β1.0355 0.1515
0.1515 β0.3626] ,
π3
2= 103
Γ [β1.4990 0.3148
0.3148 β0.4988] ,
π4
1= [
β51.1797 71.7167
71.7167 β390.4844] ,
π4
2= [
β22.7818 β177.9764
β177.9764 237.5644] ,
π5
1= 104
Γ [β1.4569 0.2376
0.2376 β0.4763] ,
π5
2= 104
Γ [β1.5078 0.2277
0.2277 β0.4312] ,
π6
1= 104
Γ [β1.3667 0.2475
0.2475 β0.4459] ,
π6
2= 104
Γ [β1.4169 0.2441
0.2441 β0.4015] ,
π7
1= 103
Γ [5.5038 0.7900
0.7900 β4.7294] ,
π7
2= 104
Γ [0.4206 β0.0170
β0.0170 β1.2249] ,
π8
1= [
244.5684 88.4596
88.4596 481.2263] ,
π8
2= [
272.5078 401.6259
401.6259 β0.2695] ,
π9
1= 104
Γ [β1.3911 0.2447
0.2447 β0.4543] ,
π9
2= 104
Γ [β1.4415 0.2394
0.2394 β0.4095] ,
π10
1= 104
Γ [1.3674 β0.2535
β0.2535 0.3868] ,
π10
2= 104
Γ [1.4277 β0.2479
β0.2479 0.3580] ,
-
16 Journal of Applied Mathematics
π1
1= 104
Γ [1.5106 β0.0468
β0.0468 0.7929] ,
π1
2= 104
Γ [1.4523 β0.0341
β0.0341 0.7069] ,
π2
1= 103
Γ [5.9653 β2.4134
β2.4134 1.7851] ,
π2
2= 103
Γ [6.9091 β2.5666
β2.5666 1.9597] ,
π3
1= [
β289.7085 24.4200
24.4200 β461.3886] ,
π3
2= [
β223.5193 93.6566
93.6566 β234.3346] ,
π4
1= 104
Γ [β1.2746 0.2653
0.2653 β0.3694] ,
π4
2= 104
Γ [β1.3242 0.2636
0.2636 β0.3371] ,
π5
1= 103
Γ [4.8778 β0.0916
β0.0916 3.8521] ,
π5
2= 103
Γ [4.8245 β0.0681
β0.0681 3.9058] ,
π6
1= 103
Γ [7.9418 β0.4415
β0.4415 6.9527] ,
π6
2= 103
Γ [7.9864 β0.4713
β0.4713 7.2384] ,
π7
1= 103
Γ [4.5999 β2.3966
β2.3966 0.1608] ,
π7
2= 103
Γ [5.2537 β2.3858
β2.3858 0.0285] ,
π8
1= 104
Γ [β1.3844 0.2577
0.2577 β0.4094] ,
π8
2= 104
Γ [β1.4426 0.2499
0.2499 β0.3763] ,
π9
1= 103
Γ [7.1086 β0.3474
β0.3474 6.1090] ,
π9
2= 103
Γ [7.1257 β0.3627
β0.3627 6.3346] ,
π10
1= 103
Γ [8.0468 β0.7400
β0.7400 6.5133] ,
π10
2= 103
Γ [8.1365 β0.8020
β0.8020 6.8109] ,
πΊ = [β2.1527 1.4389
6.5441 β11.7139] , π
1= 4.2164 Γ 10
4
,
π2= 5.7269 Γ 10
4
, π3= 5.3163 Γ 10
4
.
(60)
0 1 2 3 4 5 6 7 8Time (s)
β2
β1
0
1
2
3
4
5
6
The s
tate
var
iabl
ex(t)
x1(t)
x2(t)
Figure 1: The simulation solutions π₯1(π‘) and π₯
2(π‘) are presented for
system (1) with (58)-(59) in Example 1, πΌ1= πΌ2= 1/2, and initial
conditions π₯1(π‘) = 2 + 3 cos(π‘), π₯
2(π‘) = 1 + 2 sin(π‘), π‘ β [β1, 0], by
using the Runge-Kutta fourth order method with Matlab.
Table 1: The maximum allowed time delay β for π = 0.1, π = 0.05,π = 0.1, and π½ = 0.1.
βπ= ππ
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]:
οΏ½ΜοΏ½ (π‘) = π΄π₯ (π‘) + π΅π₯ (π‘ β β (π‘)) + πΆοΏ½ΜοΏ½ (π‘ β π (π‘))
+ π (π‘, π₯ (π‘)) + π (π‘, π₯ (π‘ β β (π‘))) ,
(61)
with
π΄ = [β2 0
0 β0.9] , π΅ = [
β1 0
β1 β1] , πΆ = [
0.1 0
0 0.1] .
(62)
Decompose matrix π΅ as follows: π΅ = π΅1+ π΅2, where
π΅1= [
β0.83 0
β1 β0.83] , π΅
2= [
β0.17 0
0 β0.17] , (63)
βπ(π‘, π₯(π‘))β β€ πβπ₯(π‘)β, and βπ(π‘, π₯(π‘ββ(π‘)))β β€ πβπ₯(π‘ββ(π‘))β.The maximum value β for exponential stability of system(61) with (62)-(63) is listed in the comparison in Table 1, fordifferent values of β
πand π
π. In Table 1, we let π = 0.05,
π = 0.1, π½ = 0.1 and β(π‘) = π(π‘). 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]:
οΏ½ΜοΏ½ (π‘) = π΄π₯ (π‘) + π΅π₯ (π‘ β β (π‘)) + πΆοΏ½ΜοΏ½ (π‘ β π (π‘)) + π (π‘, π₯ (π‘))
+ π (π‘, π₯ (π‘ β β (π‘))) + π€ (π‘, οΏ½ΜοΏ½ (π‘ β π (π‘))) ,
(64)
-
Journal of Applied Mathematics 17
Table 2: The maximum allowed time delay β.
βπ= ππ= 0 π = 0.1 π = 0.3 π = 0.5 π = 0.7 π = 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 1.5747βπ= ππ= 0.5 π = 0.1 π = 0.3 π = 0.5 π = 0.7 π = 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
π΄ = [β2 0
0 β2] , π΅ = [
0 0.4
0.4 0] , πΆ = [
0.1 0
0 0.1] .
(65)
Decompose matrix π΅ as follows: π΅ = π΅1+ π΅2, where
π΅1= [
0 0.3
0.2 0] , π΅
2= [
0 0.1
0.2 0] , (66)
βπ(π‘, π₯(π‘))β β€ πβπ₯(π‘)β, βπ(π‘, π₯(π‘ β β(π‘)))β β€ πβπ₯(π‘ β β(π‘))β,and βπ€(π‘, οΏ½ΜοΏ½(π‘ β π(π‘)))β β€ πΎβοΏ½ΜοΏ½(π‘ β π(π‘))β. By Corollary 9 tosystem (64) with (65)-(66), one can obtain the maximumupper bounds of the time delay with different convergencerate π as listed in Table 2. In Table 2, we let β = 0.1, π = 0.1,π = 0.05, πΎ = 0.05, and π½ = 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, Cauchyβs inequality, modified version ofJensenβs 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.
References
[1] K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-DelaySystem, BirkhaΜauser, Berlin, Germany, 2003.
[2] Y. Y. Cao and J. Lam, βComputation of robust stabilitybounds for time-delay systems with nonlinear time-varying
perturbations,β International Journal of Systems Science, vol. 31,pp. 359β365, 2000.
[3] Y. Chen, A. Xue, R. Lu, and S. Zhou, βOn robustly exponentialstability of uncertain neutral systems with time-varying delaysand nonlinear perturbations,βNonlinear Analysis:Theory,Meth-ods & Applications, vol. 68, no. 8, pp. 2464β2470, 2008.
[4] W.-H. Chen and W. X. Zheng, βDelay-dependent robust stabi-lization for uncertain neutral systems with distributed delays,βAutomatica, vol. 43, no. 1, pp. 95β104, 2007.
[5] J. Hale, Theory of Functional Differential Equations, Springer,New York, NY, USA, 2nd edition, 1977.
[6] Y. He, M. Wu, J.-H. She, and G.-P. Liu, βParameter-dependentLyapunov functional for stability of time-delay systems withpolytopic-type uncertainties,β IEEE Transactions on AutomaticControl, vol. 49, no. 5, pp. 828β832, 2004.
[7] Y.He,Q.-G.Wang, C. Lin, andM.Wu, βDelay-range-dependentstability for systems with time-varying delay,β Automatica, vol.43, no. 2, pp. 371β376, 2007.
[8] X. Jiang and Q.-L. Han, βOn π»β
control for linear systemswith interval time-varying delay,βAutomatica, vol. 41, no. 12, pp.2099β2106, 2005.
[9] V. L. Kharitonov and D. Hinrichsen, βExponential estimates fortime delay systems,β Systems & Control Letters, vol. 53, no. 5, pp.395β405, 2004.
[10] O. M. Kwon and J. H. Park, βExponential stability for time-delay systems with interval time-varying delays and nonlinearperturbations,β Journal ofOptimizationTheory andApplications,vol. 139, no. 2, pp. 277β293, 2008.
[11] O. M. Kwon, J. H. Park, and S. M. Lee, βOn robust stabilitycriterion for dynamic systems with time-varying delays andnonlinear perturbations,β Applied Mathematics and Computa-tion, vol. 203, no. 2, pp. 937β942, 2008.
[12] C. Li, J. Sun, andR. Sun, βStability analysis of a class of stochasticdifferential delay equations with nonlinear impulsive effects,βJournal of the Franklin Institute, vol. 347, no. 7, pp. 1186β1198,2010.
[13] C. Li, J. Shi, and J. Sun, βStability of impulsive stochastic differ-ential delay systems and its application to impulsive stochasticneural networks,βNonlinear Analysis: Theory, Methods & Appli-cations, vol. 74, no. 10, pp. 3099β3111, 2011.
[14] Y. Liu, S. M. Lee, O. M. Kwon, and J. H. Park, βDelay-dependentexponential stability criteria for 9 neutral systems with intervaltime-varying delay and nonlinear perturbations,