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Exponential Stability of Singular Impulsive Systems withTime-varying Delays
Zhiguo Yang and Zhichun Yang
Abstractβ In this article, a model of singular systems involv-ing time-varying delays and impulses is considered. By estab-lishing a singular delay differential inequality with impulsiveinitial condition and using the property of π -matrix, some newsufficient conditions ensuring the global exponential stability ofthe zero solution of singular systems are obtained. The resultscan extend and improve those of the earlier publications. Anexample is given to illustrate the theory.
I. INTRODUCTION
S INGULAR systems (known as semistate systems, dif-ferential algebraic systems, generalized state-space sys-
tems, etc.) have been of interest in the literature since theyhave many important applications in, for example, circuitsystems, robotics, aircraft modelling, social, biological, andmultisector economic systems, dynamics of thermal nuclearreactors singular perturbation systems, and so on. Manyinteresting results in dealing with singular systems have beenreported [1][2]. Furthermore, singular systems with delayshave been extensively studied in the past years due to thefact that delayed singular systems can even accurately depictthe evolutionary processes [3][4][5].
However, besides delay effects, impulsive effects likewiseexist in a wide variety of evolutionary processes in whichstates are changed abruptly at certain moments of time,involving such fields as medicine and biology, economics,mechanics, electronics and telecommunications, etc.. Aspointed out by [6], many sudden and sharp changes occurinstantaneously in singular systems, in the form of impulses.Therefore, it is very important, and indeed necessary, to studysingular impulsive systems.
The stability analysis plays an important role in the studyof singular system theory. Various stability properties ofsingular systems have been investigated in papers [4][5][6].But stability investigation to singular systems with impulsesand delays have not yet been fully developed [7]. Especially,there are few papers dealing with the exponential stability ofsingular impulsive systems with time-varying delays. Thisprompted us to discuss this problem.
Zhiguo Yang is with the College of Mathematics and SoftwareScience, Sichuan Normal University, Chengdu, 610068, China (email:[email protected]). Zhichun Yang is with the Department of Mathe-matics, Chongqing Normal University, Chongqing 400047, China (email:[email protected]).
This work was supported by National Natural Science Foundation ofChina under the grant No. 10926033, 10971147 and 10971240, A ProjectSupported by Scientific Reserch Fund of SiChuan Provincial EducationDepartment (08zb026), Key Research Project of Sichuan Normal University,Natural Science Foundation of Chongqing under Grant CSTC2008BB2364.
More specifically, in this paper, we will obtain some newsufficient conditions ensuring the global exponential stabilityof the zero solution of a singular impulsive system with time-varying delays by establishing a singular delay differentialinequality with impulsive initial condition and using theproperty of π -matrix. The results extend and improve thoseof the earlier publications.
II. MODEL AND PRELIMINARIES
To begin with, we introduce some notations and recallsome basic definitions. Let π© Ξ
= {1, 2, . . . , π}. For π΄,π΅ βπ πΓπ or π΄,π΅ β π π, π΄ β₯ π΅(π΄ β€ π΅,π΄ > π΅,π΄ < π΅)means that each pair of corresponding elements of π΄ and π΅satisfies the inequality ββ₯(β€, >,<)β.πΆ[π,π ] denotes the space of continuous mappings from
the topological space π to the topological space π .ππΆ[π½,π π]
Ξ= {π : π½ β π π β£ π(π‘+) = π(π‘) and π(π‘β)
exists for π‘ β π½ , π(π‘β) = π(π‘) for all but at most countablepoints π β π½}, where π½ β π is an interval, π(π‘+) and π(π‘β)denote the right-hand and left-hand limits of the functionπ(π‘), respectively.
For π₯ = (π₯1, . . . , π₯π)π β π π, π΄ = (πππ)πΓπ β π πΓπ,
π(π‘) = (π1(π‘), . . . , ππ(π‘))π β ππΆ[π½,π π], we define
[π₯]+ = (β£π₯1β£, . . . , β£π₯πβ£)π , [π΄]+ = (β£πππ β£)πΓπ,
[π(π‘)]π = ([π1(π‘)]π , . . . , [ππ(π‘)]π )π , [π(π‘)]+π = [[π(π‘)]+]π ,
where [ππ(π‘)]π = supβπβ€π β€0{ππ(π‘ + π )}, π β π© , π is apositive constant.
In this paper, we consider the following singular impulsivesystem with time-varying delays
πππ·π₯π(π‘) = βπππ₯π(π‘)π·π£π(π‘)
+πβ
π=1
πππππ(π₯π(π‘))π·π’π(π‘)
+
πβπ=1
πππππ(π₯π(π‘β πππ(π‘)))π·π€π(π‘),
π β π© , (1)
with the initial condition
π₯π(π‘0 + π ) = ππ(π ), βπ β€ π β€ 0, π β π© , (2)
where the matrix πΈ = diag{π1, . . . , ππ} β₯ 0 may besingular. The delays functions πππ(π‘) are continuous for π‘ β π and πππ(π‘) β [0, π ], π, π β π© , π is a positive constant.ππ(β ), ππ(β ) β πΆ[π ,π ], π β π© . The initial condition π =(π1(π ), . . . , ππ(π ))
π β ππΆ[[βπ, 0], π π]. π·π₯π(π‘), π·π£π(π‘),
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Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China
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π·π’π(π‘) and π·π€π(π‘) denote the distributional derivatives [8]of the functions π₯π(π‘), π£π(π‘), π’π(π‘) and π€π(π‘) respectively.Without loss of generality, we may assume that
π·π£π(π‘) = 1 +
ββπ=1
πΌπππΏ(π‘β π‘π),
π·π’π(π‘) = 1 +ββπ=1
πΎπππΏ(π‘β π‘π),
π·π€π(π‘) = 1 +
ββπ=1
π½πππΏ(π‘β π‘π), π β π© ,
where πΏ(β ) is the Dirac impulse function. πΌππ, πΎππ and π½ππare constants. The impulsive moments π‘π ( π = 1, 2, . . . )satisfy π‘1 < π‘2 < . . . and limπβ+β π‘π = +β.
Throughout this paper, we assume that for any π βππΆ[[βπ, 0], π π], the system (1) has at least one solutiondenoted by π₯(π‘, π‘0, π) or π₯(π‘). Moreover, we assume thatππ(0) = ππ(0) = 0, π β π© for the stability purpose of thispaper. Then the system (1) admits the zero solution π₯(π‘) β‘ 0.
Definition 1: The zero solution of the singular system(1) is said to be globally exponentially stable if there existconstants πΌ > 0 and π > 0 such that for any solutionπ₯(π‘, π‘0, π) with the initial condition π β ππΆ[[βπ, 0], π π],
β₯π₯(π‘, π‘0, π)β₯ β€ π β₯πβ₯ππβπΌ(π‘βπ‘0), π‘ β₯ π‘0, (3)
where β₯π₯(π‘, π‘0, π)β₯ = max1β€πβ€π{β£π₯π(π‘, π‘0, π)β£}, β₯πβ₯π =max1β€πβ€π{supβπβ€π β€0 β£ππ(π )β£}.
For an π -matrix π [9], we define
Ξ©π (π)Ξ= {π§ β π π β£ ππ§ > 0, π§ > 0}.
Lemma 1: [9] For an π -matrix π, Ξ©π (π) is nonemptyand satisfies,
π1π§1 + π2π§2 β Ξ©π (π),
for any π1, π2 > 0, π§1, π§2 β Ξ©π (π).
III. SINGULAR DELAY DIFFERENTIAL INEQUALITY
For the singular impulsive system (1), we need to estimateevery part on [π‘π, π‘π+1) with its initial function on [π‘πβπ, π‘π]for π = 1, 2, . . . . It is therefore difficult to obtain the estimate(3). To overcome these difficulties we establish the followingsingular delay differential inequality with impulsive initialcondition.
Theorem 1: Let π = (πππ)πΓπ and πππ β₯ 0 for π β= π,π = (πππ)πΓπ β₯ 0 and π = β(π + π) be an π -matrix. For π β (π‘0,+β), let π’(π‘) = (π’1(π‘), . . . , π’π(π‘))
π βπΆ[[π‘0, π), π
π] be a solution of the following delay differen-tial inequality with the initial condition π’(π ) β ππΆ[[π‘0 βπ, π‘0], π
π],
πΎπ·+π’(π‘) β€ β(π‘)(ππ’(π‘) +π[π’(π‘)]π ), π‘ β (π‘0, π), (4)
where πΎ = diag{π1, . . . , ππ} β₯ 0,π·+π’(π‘) is the upper rightderivative of π’(π‘). β(π‘) > 0 satisfies supπ‘β₯π‘0
β« π‘
π‘βπβ(π )ππ β€
π» <β. Then
π’(π‘) β€ π§πβπ
β« π‘π‘0
β(π )ππ , π‘ β [π‘0, π), (5)
provided that the initial condition satisfies
π’(π ) β€ π§πβπ
β« π π‘0
β(π )ππ , π‘0 β π β€ π β€ π‘0, (6)
where π§ = (π§1, . . . , π§π)π β Ξ©π (π) and the positive constant
π satisfies the following inequality
[ππΎ + π +ππππ» ]π§ < 0. (7)
Proof: Since π is an π -matrix, there exists a vector π§ βΞ©π (π) such that
ππ§ > 0 or [π +π]π§ < 0.
By using continuity, we obtain that there must exist a positiveconstant π satisfying the inequality (7), that is,
πβπ=1
[πππ + ππππππ» ]π§π < βππππ§π, π β π© . (8)
We at first shall prove that for any positive constant π
π’π(π‘) β€ (1 + π)π§ππβπ
β« π‘π‘0
β(π )ππ Ξ= π¦π(π‘),
π‘ β [π‘0, π), π β π© . (9)
If inequality (9) is not true, by (6), there must be constantπ‘1 β (π‘0, π) and some integer π β π© such that
π’π(π‘1) = π¦π(π‘1), π·+π’π(π‘1) β₯ π¦β²π(π‘1), (10)
π’π(π‘) β€ π¦π(π‘), π‘ β [π‘0 β π, π‘1], π β π© . (11)
By using (4), (8), (9), (10), (11) and πππ β₯ 0 (π β= π), π β₯0, we obtain that
πππ·+π’π(π‘1) β€ β(π‘1)
πβπ=1
[ππππ’π(π‘1) + πππ [π’π(π‘1)]π ]
β€ β(π‘1)
πβπ=1
[πππ(1 + π)π§ππβπ
β« π‘1π‘0
β(π )ππ
+πππ(1 + π)π§ππβπ
β« π‘1βππ‘0
β(π )ππ ]
β€ β(π‘1)(1 + π)πβπ
β« π‘1π‘0
β(π )ππ
Γπβ
π=1
[πππ + ππππππ» ]π§π
< βππππ§πβ(π‘1)(1 + π)πβπβ« π‘1π‘0
β(π )ππ . (12)
If ππ > 0, the inequality (12) yields that
π·+π’π(π‘1) < π¦β²π(π‘1),
which contradicts the inequality in (10). If ππ = 0, theinequality (12) yields that 0 < 0. It is a contradiction.
Thus (9) holds. Therefore, letting πβ 0, we have
π’π(π‘) β€ π§ππβπ
β« π‘π‘0
β(π )ππ , π‘ β [π‘0, π), π β π© .
The proof is completed.
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IV. GLOBAL EXPONENTIAL STABILIT
In this section, we will obtain the sufficient conditionensuring global exponential stability of the zero solution ofthe singular system (1) by using the property of π -Matrixand employing Theorem 1. Here, we firstly introduce thefollowing assumptions.(π΄1) There exist nonnegative constants οΏ½ΜοΏ½π and π£π such
that continuous functions ππ(β ) and ππ(β ) satisfy
β£ππ(π¦)β£ β€ οΏ½ΜοΏ½π β£π¦β£, β£ππ(π¦)β£ β€ π£π β£π¦β£, π β π© , π¦ β π .
(π΄2) Let π = β(π +π) be an π -matrix, where
π = βπΆ + [π΄]+π, π = [π΅]+π, π΅ = (πππ)πΓπ,
πΆ = diag{π1, . . . , ππ} > 0, π΄ = (πππ)πΓπ,
π = diag{οΏ½ΜοΏ½1, . . . , οΏ½ΜοΏ½π}, π = diag{π£1, . . . , π£π}.
(π΄3) Let ππ = [πΈ+πΆπΌπ]+β [π΄]+π [πΎπ]
+, πβ1π exist and
πβ1π β₯ 0, π = 1, 2, . . . , where
πΈ = diag{π1, . . . , ππ} β₯ 0,
πΌπ = diag{πΌ1π, . . . , πΌππ},
πΎπ = diag{πΎ1π, . . . , πΎππ}.
(π΄4) There exists a positive constant π such that
lnπππ‘π β π‘πβ1
β€ π < π, π = 1, 2, . . . . (13)
Where the positive constant π satisfies
[ππΈ + π +ππππ ]π§β < 0, (14)
for a given π§β β Ξ©π (π) and ππ β₯ 1 satisfies
πππ§β β₯ (πβ1
π πΈ + πβ1π [π΅]+π [π½π]
+πππ )π§β,π = 1, 2, . . . , (15)
where π½π = diag{π½1π, . . . , π½ππ}.Theorem 2: Assume that (π΄1) - (π΄4) hold. Then the zero
solution of the singular system (1) is globally exponentiallystable and the exponential convergence rate is equal to πβπ.
Proof: By the property of the Dirac impulse function πΏ(β ),we see that the derivative π₯β²π(π‘) of π₯π(π‘) exists on (π‘πβ1, π‘π).Thus the system (1) becomes
πππ₯β²π(π‘) = βπππ₯π(π‘) +
πβπ=1
πππππ(π₯π(π‘))
+πβ
π=1
πππππ(π₯π(π‘β πππ(π‘))),
π β π© , π‘ β (π‘πβ1, π‘π). (16)
From (16) and (π΄1), it is easy to obtain
πππ·+β£π₯π(π‘)β£ β€ βππβ£π₯π(π‘)β£+
πβπ=1
β£πππ β£β£ππ(π₯π(π‘))β£
+
πβπ=1
β£πππ β£β£ππ(π₯π(π‘β πππ(π‘)))β£
β€ βππβ£π₯π(π‘)β£+πβ
π=1
β£πππ β£οΏ½ΜοΏ½π β£π₯π(π‘)β£
+
πβπ=1
β£πππ β£π£π β£π₯π(π‘β πππ(π‘))β£,
π β π© , π‘ β (π‘πβ1, π‘π).
That is,
πΈπ·+[π₯(π‘)]+ β€ π [π₯(π‘)]+ +π[π₯(π‘)]+π ,
π‘ β (π‘πβ1, π‘π), π = 1, 2, . . . . (17)
Since π is an π -matrix, from Lemma 1 we may choosea vector π§β = (π§1, . . . , π§π)
π β Ξ©π (π) and π§β β₯ πΌπ =(1, . . . , 1)π β π π such that
ππ§β > 0 or [π +π]π§β < 0.
By using continuity, we obtain that there must exist a positiveconstant π satisfying the inequality (14).
For the initial condition π₯(π‘0 + π ) = π(π ), π β [βπ, 0],where π β ππΆ[[βπ, 0], π π] and π‘0 β π (without loss ofgenerality, we assume π‘0 < π‘1), we can get
[π₯(π‘)]+ β€ π§ββ₯πβ₯ππβπ(π‘βπ‘0), π‘0 β π β€ π‘ β€ π‘0. (18)
Then, all conditions of Theorem 1 are satisfied by (17),(18), Condition (π΄2), β(π‘) β‘ 1 and π» = π . So
[π₯(π‘)]+ β€ π§ββ₯πβ₯ππβπ(π‘βπ‘0), π‘0 β€ π‘ < π‘1. (19)
Suppose that for all π = 1, . . . , π, the inequalities
[π₯(π‘)]+ β€ π0 . . . ππβ1π§ββ₯πβ₯ππβπ(π‘βπ‘0),
π‘πβ1 β€ π‘ < π‘π, (20)
hold, where π0 = 1.On the other hand, (1) implies that
ππ[π₯π(π‘π)β π₯π(π‘π β π)]=
β« π‘π
π‘πβπ
[βπππ₯π(π‘)π·π£π(π )
+
πβπ=1
πππππ(π₯π(π‘))π·ππ’(π )
+
πβπ=1
πππππ(π₯π(π‘β πππ(π‘)))π·π€π(π )], π β π© , (21)
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where π > 0 is sufficiently small, as πβ 0+, which yields
ππ[π₯π(π‘π)β π₯π(π‘βπ )]
= βπππ₯π(π‘π)πΌππ +
πβπ=1
πππππ(π₯π(π‘π))πΎππ
+πβ
π=1
πππππ(π₯π(π‘π β πππ(π‘π)))π½ππ, π β π© . (22)
This, together with (π΄1), yields that
β£(ππ + πππΌππ)π₯π(π‘π)β£
= β£πππ₯π(π‘βπ ) +πβ
π=1
πππππ(π₯π(π‘π))πΎππ
+
πβπ=1
πππππ(π₯π(π‘π β πππ(π‘π)))π½ππβ£
β€ ππβ£π₯(π‘βπ )β£+πβ
π=1
β£πππ β£οΏ½ΜοΏ½π β£π₯π(π‘π)β£β£πΎππβ£
+
πβπ=1
β£πππ β£π£π β£π₯π(π‘π β πππ(π‘π))β£β£π½ππβ£
β€ ππβ£π₯(π‘βπ )β£+πβ
π=1
β£πππ β£οΏ½ΜοΏ½π β£π₯π(π‘π)β£β£πΎππβ£
+πβ
π=1
β£πππ β£π£π [β£π₯π(π‘π)β£]π β£π½ππβ£, π β π© . (23)
That is,
[πΈ + πΆπΌπ]+[π₯(π‘π)]
+
β€ πΈ[π₯(π‘βπ )]+ + [π΄]+π [πΎπ]
+[π₯(π‘π)]+
+[π΅]+π [π½π]+[π₯(π‘π)]
+π . (24)
Then, we have
([πΈ + πΆπΌπ]+ β [π΄]+π [πΎπ]
+)[π₯(π‘π)]+
β€ πΈ[π₯(π‘βπ )]+ + [π΅]+π [π½π]
+[π₯(π‘π)]+π (25)
yielding, together with (π΄3), that
[π₯(π‘π)]+ β€ πβ1
π πΈ[π₯(π‘βπ )]+ + πβ1
π [π΅]+π [π½π]+[π₯(π‘π)]
+π . (26)
Then, from (15), (20) and (26)
[π₯(π‘π)]+ β€ (πβ1
π πΈ + πβ1π [π΅]+π [π½π]
+πππ )
Γπ§β π0 . . . ππβ1β₯πβ₯ππβπ(π‘πβπ‘0)
β€ π§β π0 . . . ππβ1ππβ₯πβ₯ππβπ(π‘πβπ‘0). (27)
This, together with (20) and ππ β₯ 1, π = 1, 2, . . . , leads to
[π₯(π‘)]+ β€ π0 . . . ππβ1πππ§ββ₯πβ₯ππβπ(π‘βπ‘0)
= π0 . . . ππβ1ππβ₯πβ₯ππβπ(π‘πβπ‘0)π§βπβπ(π‘βπ‘π),
π‘ β [π‘π β π, π‘π]. (28)
By Lemma 1 again, the vector
π0 . . . ππβ1ππβ₯πβ₯ππβπ(π‘πβπ‘0)π§β β Ξ©π (π).
Then, all conditions of Theorem 1 are satisfied again by (17),(28), Condition (π΄2), β(π‘) β‘ 1 and π» = π . So
[π₯(π‘)]+ β€ π0 . . . ππβ1ππβ₯πβ₯ππβπ(π‘πβπ‘0)π§βπβπ(π‘βπ‘π)
= π0 . . . ππβ1πππ§ββ₯πβ₯ππβπ(π‘βπ‘0),
π‘π β€ π‘ < π‘π+1. (29)
By the mathematical induction, we can conclude that
[π₯(π‘)]+ β€ π0 . . . ππβ1π§ββ₯πβ₯ππβπ(π‘βπ‘0),
π‘πβ1 β€ π‘ < π‘π, π = 1, 2, . . . . (30)
Noticing that ππ β€ ππ(π‘πβπ‘πβ1) by (13), we can use (30) toconclude that
[π₯(π‘)]+ β€ ππ(π‘1βπ‘0) . . . ππ(π‘πβ1βπ‘πβ2)π§ββ₯πβ₯ππβπ(π‘βπ‘0)
β€ π§ββ₯πβ₯πππ(π‘βπ‘0)πβπ(π‘βπ‘0)
= π§ββ₯πβ₯ππβ(πβπ)(π‘βπ‘0),
π‘ β [π‘πβ1, π‘π), π = 1, 2, . . . .
So,
[π₯(π‘)]+ β€ π§ββ₯πβ₯ππβ(πβπ)(π‘βπ‘0), π‘ β₯ π‘0.
This implies that the conclusion of the theorem holds.Remark 1: Condition (π΄3) must hold when πΌπ > 0 and
πΎπ is sufficiently small. In fact, the positive diagonal matrix[πΈ + πΆπΌπ]
+ is an π -matrix when πΌπ > 0. So, ππ = [πΈ +πΆπΌπ]
+ β [π΄]+π [πΎπ]+ is also an π -matrix for sufficiently
small πΎπ. Then, πβ1π exists and πβ1
π β₯ 0.Remark 2: When πππ(π‘) β‘ π and πΎπ = π½π, some global
exponential stability criteria for the singular impulsive system(1) have been established in [7]. However, from assumption1) in Theorem 2 and Corollary 4.1 of [7], the impulsivecoefficient max1β€π<β, 1β€πβ€π{π½ππ} must be bounded whenmax1β€π<β, 1β€πβ€π{πΌππ} is bounded. Here, Theorem 2 abovedoes not require the boundedness of impulsive coefficient anddrops the additional assumption that π‘π β π‘πβ1 β₯ πΏπ, πΏ > 1in [7].
If πΈ is a π Γ π unit matrix, the system (1) becomes theimpulsive delay differential system without singularity,
π·π₯π(π‘) = βπππ₯π(π‘)π·π£π(π‘)
+πβ
π=1
πππππ(π₯π(π‘))π·π’π(π‘)
+πβ
π=1
πππππ(π₯π(π‘β πππ(π‘)))π·π€π(π‘),
π β π© . (31)
For the system (31), we have the following corollary byTheorem 2.
Corollary 1: Assume that (π΄1) - (π΄4) hold. Then the zerosolution of the system (31) is globally exponentially stableand the exponential convergence rate is equal to πβ π.
Remark 3: When coefficient matrix π΅ β‘ 0, the system(31) has been investigated in [10] where function ππ(β ) isdifferentiable and π β²π(β ) is invertible and bounded, π β π© .And Guan and Chen also have discussed the system (31)
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with πππ(π‘) β‘ π [11]. In [11], ππ(β ) and ππ(β ) must satisfyπ§ππ(π§) β₯ 0 and π§ππ(π§) β₯ 0, π§ β π , π β π© . However, inCorollary 1 above, these restrictive conditions are removed.
V. ILLUSTRATIVE EXAMPLE
The following illustrative example will demonstrate theeffectiveness of our results.
Example 1: Consider the following impulsive singularsystem with time-varying delaysβ§β¨β©π·π¦1(π‘) = β6π¦1(π‘)π·π£1(π‘) + 3β£π¦1(π‘β π11(π‘))β£
Γπ·π€1(π‘)β 2π¦2(π‘β π12(π‘))π·π€2(π‘),0 = β5π¦2(π‘)π·π£2(π‘)β β£π¦1(π‘β π21(π‘))β£π·π€1(π‘)
+3 cos(π¦2(π‘β π22(π‘)))π·π€2(π‘),
(32)
where πππ(π‘) = β£ sin((π+ π)π‘)β£ β€ 1Ξ= π and
π·π£π(π‘) = 1+ββπ=1
πΌπΏ(π‘β π‘π), π·π€π(π‘) = 1+ββπ=1
π½πΏ(π‘β π‘π),
for π, π = 1, 2, where πΏ(β ) is the Dirac impulse function andthe impulsive moments π‘π ( π = 1, 2, . . . ) satisfy: π‘1 = 0.3,π‘1 < π‘2 < . . . and limπβ+β π‘π = +β.
Thus the parameters of Conditions (π΄1) - (π΄4) are asfollows:
πΈ =
(1 00 0
), π =
(1 00 1
),
πΆ =
(6 00 5
), π΄ =
(0 00 0
),
πΌπ = diag{πΌ, πΌ}, π½π = diag{π½, π½},
π΅ =
(3 β2β1 3
), π = [π΅]+π =
(3 21 3
),
π = βπΆ + [π΄]+π = βπΆ =
( β6 00 β5
),
π = β(π +π) =
(3 β2β1 2
),
ππ = [πΈ + πΆπΌπ]+ β [π΄]+π [πΎπ]
+
=
( β£1 + 6πΌβ£ 00 5β£πΌβ£
), (33)
which yield that π is an π -matrix, πβ1π exists when πΌ β= 0
and πΌ β= β 16 . Furthermore,
Ξ©π (π) = {(π§1, π§2)π > 0 β£ 23π§2 < π§1 < 2π§2},
πβ1π =
(1
β£1+6πΌβ£ 0
0 15β£πΌβ£
).
Let π§β = (1, 1)π β Ξ©π (π) and π = 0.1 which satisfies theinequality (ππΈ + π + ππππ )π§β < 0. So ππ can satisfy theinequality
πππ§β β₯ (πβ1
π πΈ + πβ1π [π΅]+π [π½π]
+πππ )π§β
= (πβ1π πΈ + πβ1
π [π΅]+β£π½β£π0.1)π§β,π = 1, 2, . . . , (34)
provided that
ππ β₯ max{1 + 5β£π½β£π0.1β£1 + 6πΌβ£ ,
4β£π½β£π0.15β£πΌβ£ },
π = 1, 2, . . . . (35)
Case 1: Let πΌ = β1, π½ = π0.2π, ππ = π0.5π and π‘π βπ‘πβ1 = 6π, then ππ satisfies the inequality (35) and ππ β₯ 1,
lnπππ‘π β π‘πβ1
=ln π0.5π
6πβ€ 0.084 = π < π,
π = 1, 2, . . . .
Clearly, all conditions of Theorem 2 are satisfied. So the zerosolution of the singular system (32) is globally exponentiallystable and the exponential convergence rate is equal to 0.016.
Case 2: Let πΌ = π0.1, π½ = 1, ππ = 1 and π‘π β π‘πβ1 =ππ β (0, π), then ππ satisfies the inequality (35) and ππ β₯ 1,
lnπππ‘π β π‘πβ1
= 0 = π < π, π = 1, 2, . . . .
So by Theorem 2, the zero solution of the singular system(32) is globally exponentially stable and the exponentialconvergence rate is equal to 0.1.
Remark 4: Even if πππ(π‘) β‘ π , the conditions in Theorem4.1 and Corollary 4.1 of [7] are not satisfied since in Case 1π½ππ = π0.2π β β, π β β, while max1β€π<β, 1β€πβ€2{πΌππ}is bounded, and in Case 2 ππ β (0, π) which does not satisfythe assumption that π‘π β π‘πβ1 β₯ πΏπ, πΏ > 1 in [7].
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