1

Exponential Stability of Singular Impulsive Systems withTime-varying Delays

Zhiguo Yang and Zhichun Yang

Abstract— 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 𝑀 -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:zhiguoyang@126.com). Zhichun Yang is with the Department of Mathe-matics, Chongqing Normal University, Chongqing 400047, China (email:zhichy@yahoo.com.cn).

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 𝑀 -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 𝒩 Δ

= {1, 2, . . . , 𝑛}. For 𝐴,𝐵 ∈𝑅𝑛×𝑛 or 𝐴,𝐵 ∈ 𝑅𝑛, 𝐴 ≥ 𝐵(𝐴 ≤ 𝐵,𝐴 > 𝐵,𝐴 < 𝐵)means that each pair of corresponding elements of 𝐴 and 𝐵satisfies the inequality “≥(≤, >,<)”.𝐶[𝑋,𝑌 ] denotes the space of continuous mappings from

the topological space 𝑋 to the topological space 𝑌 .𝑃𝐶[𝐽,𝑅𝑛]

Δ= {𝜓 : 𝐽 → 𝑅𝑛 ∣ 𝜓(𝑡+) = 𝜓(𝑡) and 𝜓(𝑡−)

exists for 𝑡 ∈ 𝐽 , 𝜓(𝑡−) = 𝜓(𝑡) for all but at most countablepoints 𝑠 ∈ 𝐽}, where 𝐽 ⊂ 𝑅 is an interval, 𝜓(𝑡+) and 𝜓(𝑡−)denote the right-hand and left-hand limits of the function𝜓(𝑡), respectively.

For 𝑥 = (𝑥1, . . . , 𝑥𝑛)𝑇 ∈ 𝑅𝑛, 𝐴 = (𝑎𝑖𝑗)𝑛×𝑛 ∈ 𝑅𝑛×𝑛,

𝜑(𝑡) = (𝜑1(𝑡), . . . , 𝜑𝑛(𝑡))𝑇 ∈ 𝑃𝐶[𝐽,𝑅𝑛], we define

[𝑥]+ = (∣𝑥1∣, . . . , ∣𝑥𝑛∣)𝑇 , [𝐴]+ = (∣𝑎𝑖𝑗 ∣)𝑛×𝑛,

[𝜑(𝑡)]𝜏 = ([𝜑1(𝑡)]𝜏 , . . . , [𝜑𝑛(𝑡)]𝜏 )𝑇 , [𝜑(𝑡)]+𝜏 = [[𝜑(𝑡)]+]𝜏 ,

where [𝜑𝑖(𝑡)]𝜏 = sup−𝜏≤𝑠≤0{𝜑𝑖(𝑡 + 𝑠)}, 𝑖 ∈ 𝒩 , 𝜏 is apositive constant.

In this paper, we consider the following singular impulsivesystem with time-varying delays

𝑒𝑖𝐷𝑥𝑖(𝑡) = −𝑐𝑖𝑥𝑖(𝑡)𝐷𝑣𝑖(𝑡)

+𝑛∑

𝑗=1

𝑎𝑖𝑗𝑓𝑗(𝑥𝑗(𝑡))𝐷𝑢𝑗(𝑡)

+

𝑛∑𝑗=1

𝑏𝑖𝑗𝑔𝑗(𝑥𝑗(𝑡− 𝜏𝑖𝑗(𝑡)))𝐷𝑤𝑗(𝑡),

𝑖 ∈ 𝒩 , (1)

with the initial condition

𝑥𝑖(𝑡0 + 𝑠) = 𝜙𝑖(𝑠), −𝜏 ≤ 𝑠 ≤ 0, 𝑖 ∈ 𝒩 , (2)

where the matrix 𝐸 = diag{𝑒1, . . . , 𝑒𝑛} ≥ 0 may besingular. The delays functions 𝜏𝑖𝑗(𝑡) are continuous for 𝑡 ∈ 𝑅and 𝜏𝑖𝑗(𝑡) ∈ [0, 𝜏 ], 𝑖, 𝑗 ∈ 𝒩 , 𝜏 is a positive constant.𝑓𝑗(⋅), 𝑔𝑗(⋅) ∈ 𝐶[𝑅,𝑅], 𝑗 ∈ 𝒩 . The initial condition 𝜙 =(𝜙1(𝑠), . . . , 𝜙𝑛(𝑠))

𝑇 ∈ 𝑃𝐶[[−𝜏, 0], 𝑅𝑛]. 𝐷𝑥𝑖(𝑡), 𝐷𝑣𝑖(𝑡),

630

Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China

978-1-4244-6337-4/10/$26.00 @2010 IEEE

2

𝐷𝑢𝑗(𝑡) and 𝐷𝑤𝑗(𝑡) denote the distributional derivatives [8]of the functions 𝑥𝑖(𝑡), 𝑣𝑖(𝑡), 𝑢𝑗(𝑡) and 𝑤𝑗(𝑡) respectively.Without loss of generality, we may assume that

𝐷𝑣𝑖(𝑡) = 1 +

∞∑𝑘=1

𝛼𝑖𝑘𝛿(𝑡− 𝑡𝑘),

𝐷𝑢𝑖(𝑡) = 1 +∞∑𝑘=1

𝛾𝑖𝑘𝛿(𝑡− 𝑡𝑘),

𝐷𝑤𝑖(𝑡) = 1 +

∞∑𝑘=1

𝛽𝑖𝑘𝛿(𝑡− 𝑡𝑘), 𝑖 ∈ 𝒩 ,

where 𝛿(⋅) is the Dirac impulse function. 𝛼𝑖𝑘, 𝛾𝑖𝑘 and 𝛽𝑖𝑘are constants. The impulsive moments 𝑡𝑘 ( 𝑘 = 1, 2, . . . )satisfy 𝑡1 < 𝑡2 < . . . and lim𝑘→+∞ 𝑡𝑘 = +∞.

Throughout this paper, we assume that for any 𝜙 ∈𝑃𝐶[[−𝜏, 0], 𝑅𝑛], the system (1) has at least one solutiondenoted by 𝑥(𝑡, 𝑡0, 𝜙) or 𝑥(𝑡). Moreover, we assume that𝑓𝑗(0) = 𝑔𝑗(0) = 0, 𝑗 ∈ 𝒩 for the stability purpose of thispaper. Then the system (1) admits the zero solution 𝑥(𝑡) ≡ 0.

Definition 1: The zero solution of the singular system(1) is said to be globally exponentially stable if there existconstants 𝛼 > 0 and 𝜅 > 0 such that for any solution𝑥(𝑡, 𝑡0, 𝜙) with the initial condition 𝜙 ∈ 𝑃𝐶[[−𝜏, 0], 𝑅𝑛],

∥𝑥(𝑡, 𝑡0, 𝜙)∥ ≤ 𝜅∥𝜙∥𝜏𝑒−𝛼(𝑡−𝑡0), 𝑡 ≥ 𝑡0, (3)

where ∥𝑥(𝑡, 𝑡0, 𝜙)∥ = max1≤𝑖≤𝑛{∣𝑥𝑖(𝑡, 𝑡0, 𝜙)∣}, ∥𝜑∥𝜏 =max1≤𝑖≤𝑛{sup−𝜏≤𝑠≤0 ∣𝜑𝑖(𝑠)∣}.

For an 𝑀 -matrix 𝑆 [9], we define

Ω𝑀 (𝑆)Δ= {𝑧 ∈ 𝑅𝑛 ∣ 𝑆𝑧 > 0, 𝑧 > 0}.

Lemma 1: [9] For an 𝑀 -matrix 𝑆, Ω𝑀 (𝑆) is nonemptyand satisfies,

𝑘1𝑧1 + 𝑘2𝑧2 ∈ Ω𝑀 (𝑆),

for any 𝑘1, 𝑘2 > 0, 𝑧1, 𝑧2 ∈ Ω𝑀 (𝑆).

III. SINGULAR DELAY DIFFERENTIAL INEQUALITY

For the singular impulsive system (1), we need to estimateevery part on [𝑡𝑘, 𝑡𝑘+1) with its initial function on [𝑡𝑘−𝜏, 𝑡𝑘]for 𝑘 = 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 𝑃 = (𝑝𝑖𝑗)𝑛×𝑛 and 𝑝𝑖𝑗 ≥ 0 for 𝑖 ∕= 𝑗,𝑄 = (𝑞𝑖𝑗)𝑛×𝑛 ≥ 0 and 𝑆 = −(𝑃 + 𝑄) be an 𝑀 -matrix. For 𝑏 ∈ (𝑡0,+∞), let 𝑢(𝑡) = (𝑢1(𝑡), . . . , 𝑢𝑛(𝑡))

𝑇 ∈𝐶[[𝑡0, 𝑏), 𝑅

𝑛] be a solution of the following delay differen-tial inequality with the initial condition 𝑢(𝑠) ∈ 𝑃𝐶[[𝑡0 −𝜏, 𝑡0], 𝑅

𝑛],

𝐾𝐷+𝑢(𝑡) ≤ ℎ(𝑡)(𝑃𝑢(𝑡) +𝑄[𝑢(𝑡)]𝜏 ), 𝑡 ∈ (𝑡0, 𝑏), (4)

where 𝐾 = diag{𝑘1, . . . , 𝑘𝑛} ≥ 0,𝐷+𝑢(𝑡) is the upper rightderivative of 𝑢(𝑡). ℎ(𝑡) > 0 satisfies sup𝑡≥𝑡0

∫ 𝑡

𝑡−𝜏ℎ(𝑠)𝑑𝑠 ≤

𝐻 <∞. Then

𝑢(𝑡) ≤ 𝑧𝑒−𝜆

∫ 𝑡𝑡0

ℎ(𝑠)𝑑𝑠, 𝑡 ∈ [𝑡0, 𝑏), (5)

provided that the initial condition satisfies

𝑢(𝑠) ≤ 𝑧𝑒−𝜆

∫ 𝑠𝑡0

ℎ(𝑠)𝑑𝑠, 𝑡0 − 𝜏 ≤ 𝑠 ≤ 𝑡0, (6)

where 𝑧 = (𝑧1, . . . , 𝑧𝑛)𝑇 ∈ Ω𝑀 (𝑆) and the positive constant

𝜆 satisfies the following inequality

[𝜆𝐾 + 𝑃 +𝑄𝑒𝜆𝐻 ]𝑧 < 0. (7)

Proof: Since 𝑆 is an 𝑀 -matrix, there exists a vector 𝑧 ∈Ω𝑀 (𝑆) such that

𝑆𝑧 > 0 or [𝑃 +𝑄]𝑧 < 0.

By using continuity, we obtain that there must exist a positiveconstant 𝜆 satisfying the inequality (7), that is,

𝑛∑𝑗=1

[𝑝𝑖𝑗 + 𝑞𝑖𝑗𝑒𝜆𝐻 ]𝑧𝑗 < −𝜆𝑘𝑖𝑧𝑖, 𝑖 ∈ 𝒩 . (8)

We at first shall prove that for any positive constant 𝜖

𝑢𝑖(𝑡) ≤ (1 + 𝜖)𝑧𝑖𝑒−𝜆

∫ 𝑡𝑡0

ℎ(𝑠)𝑑𝑠 Δ= 𝑦𝑖(𝑡),

𝑡 ∈ [𝑡0, 𝑏), 𝑖 ∈ 𝒩 . (9)

If inequality (9) is not true, by (6), there must be constant𝑡1 ∈ (𝑡0, 𝑏) and some integer 𝑚 ∈ 𝒩 such that

𝑢𝑚(𝑡1) = 𝑦𝑚(𝑡1), 𝐷+𝑢𝑚(𝑡1) ≥ 𝑦′𝑚(𝑡1), (10)

𝑢𝑖(𝑡) ≤ 𝑦𝑖(𝑡), 𝑡 ∈ [𝑡0 − 𝜏, 𝑡1], 𝑖 ∈ 𝒩 . (11)

By using (4), (8), (9), (10), (11) and 𝑝𝑖𝑗 ≥ 0 (𝑖 ∕= 𝑗), 𝑄 ≥0, we obtain that

𝑘𝑚𝐷+𝑢𝑚(𝑡1) ≤ ℎ(𝑡1)

𝑛∑𝑗=1

[𝑝𝑚𝑗𝑢𝑗(𝑡1) + 𝑞𝑚𝑗 [𝑢𝑗(𝑡1)]𝜏 ]

≤ ℎ(𝑡1)

𝑛∑𝑗=1

[𝑝𝑚𝑗(1 + 𝜖)𝑧𝑗𝑒−𝜆

∫ 𝑡1𝑡0

ℎ(𝑠)𝑑𝑠

+𝑞𝑚𝑗(1 + 𝜖)𝑧𝑗𝑒−𝜆

∫ 𝑡1−𝜏𝑡0

ℎ(𝑠)𝑑𝑠]

≤ ℎ(𝑡1)(1 + 𝜖)𝑒−𝜆

∫ 𝑡1𝑡0

ℎ(𝑠)𝑑𝑠

×𝑛∑

𝑗=1

[𝑝𝑚𝑗 + 𝑞𝑚𝑗𝑒𝜆𝐻 ]𝑧𝑗

< −𝜆𝑘𝑚𝑧𝑚ℎ(𝑡1)(1 + 𝜖)𝑒−𝜆∫ 𝑡1𝑡0

ℎ(𝑠)𝑑𝑠. (12)

If 𝑘𝑚 > 0, the inequality (12) yields that

𝐷+𝑢𝑚(𝑡1) < 𝑦′𝑚(𝑡1),

which contradicts the inequality in (10). If 𝑘𝑚 = 0, theinequality (12) yields that 0 < 0. It is a contradiction.

Thus (9) holds. Therefore, letting 𝜖→ 0, we have

𝑢𝑖(𝑡) ≤ 𝑧𝑖𝑒−𝜆

∫ 𝑡𝑡0

ℎ(𝑠)𝑑𝑠, 𝑡 ∈ [𝑡0, 𝑏), 𝑖 ∈ 𝒩 .

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 𝑀 -Matrixand employing Theorem 1. Here, we firstly introduce thefollowing assumptions.(𝐴1) There exist nonnegative constants �̄�𝑗 and 𝑣𝑗 such

that continuous functions 𝑓𝑗(⋅) and 𝑔𝑗(⋅) satisfy

∣𝑓𝑗(𝑦)∣ ≤ �̄�𝑗 ∣𝑦∣, ∣𝑔𝑗(𝑦)∣ ≤ 𝑣𝑗 ∣𝑦∣, 𝑗 ∈ 𝒩 , 𝑦 ∈ 𝑅.

(𝐴2) Let 𝑆 = −(𝑃 +𝑄) be an 𝑀 -matrix, where

𝑃 = −𝐶 + [𝐴]+𝑈, 𝑄 = [𝐵]+𝑉, 𝐵 = (𝑏𝑖𝑗)𝑛×𝑛,

𝐶 = diag{𝑐1, . . . , 𝑐𝑛} > 0, 𝐴 = (𝑎𝑖𝑗)𝑛×𝑛,

𝑈 = diag{�̄�1, . . . , �̄�𝑛}, 𝑉 = diag{𝑣1, . . . , 𝑣𝑛}.

(𝐴3) Let 𝑆𝑘 = [𝐸+𝐶𝛼𝑘]+− [𝐴]+𝑈 [𝛾𝑘]

+, 𝑆−1𝑘 exist and

𝑆−1𝑘 ≥ 0, 𝑘 = 1, 2, . . . , where

𝐸 = diag{𝑒1, . . . , 𝑒𝑛} ≥ 0,

𝛼𝑘 = diag{𝛼1𝑘, . . . , 𝛼𝑛𝑘},

𝛾𝑘 = diag{𝛾1𝑘, . . . , 𝛾𝑛𝑘}.

(𝐴4) There exists a positive constant 𝜎 such that

ln𝜎𝑘𝑡𝑘 − 𝑡𝑘−1

≤ 𝜎 < 𝜆, 𝑘 = 1, 2, . . . . (13)

Where the positive constant 𝜆 satisfies

[𝜆𝐸 + 𝑃 +𝑄𝑒𝜆𝜏 ]𝑧∗ < 0, (14)

for a given 𝑧∗ ∈ Ω𝑀 (𝑆) and 𝜎𝑘 ≥ 1 satisfies

𝜎𝑘𝑧∗ ≥ (𝑆−1

𝑘 𝐸 + 𝑆−1𝑘 [𝐵]+𝑉 [𝛽𝑘]

+𝑒𝜆𝜏 )𝑧∗,𝑘 = 1, 2, . . . , (15)

where 𝛽𝑘 = diag{𝛽1𝑘, . . . , 𝛽𝑛𝑘}.Theorem 2: Assume that (𝐴1) - (𝐴4) hold. Then the zero

solution of the singular system (1) is globally exponentiallystable and the exponential convergence rate is equal to 𝜆−𝜎.

Proof: By the property of the Dirac impulse function 𝛿(⋅),we see that the derivative 𝑥′𝑖(𝑡) of 𝑥𝑖(𝑡) exists on (𝑡𝑘−1, 𝑡𝑘).Thus the system (1) becomes

𝑒𝑖𝑥′𝑖(𝑡) = −𝑐𝑖𝑥𝑖(𝑡) +

𝑛∑𝑗=1

𝑎𝑖𝑗𝑓𝑗(𝑥𝑗(𝑡))

+𝑛∑

𝑗=1

𝑏𝑖𝑗𝑔𝑗(𝑥𝑗(𝑡− 𝜏𝑖𝑗(𝑡))),

𝑖 ∈ 𝒩 , 𝑡 ∈ (𝑡𝑘−1, 𝑡𝑘). (16)

From (16) and (𝐴1), it is easy to obtain

𝑒𝑖𝐷+∣𝑥𝑖(𝑡)∣ ≤ −𝑐𝑖∣𝑥𝑖(𝑡)∣+

𝑛∑𝑗=1

∣𝑎𝑖𝑗 ∣∣𝑓𝑗(𝑥𝑗(𝑡))∣

+

𝑛∑𝑗=1

∣𝑏𝑖𝑗 ∣∣𝑔𝑗(𝑥𝑗(𝑡− 𝜏𝑖𝑗(𝑡)))∣

≤ −𝑐𝑖∣𝑥𝑖(𝑡)∣+𝑛∑

𝑗=1

∣𝑎𝑖𝑗 ∣�̄�𝑗 ∣𝑥𝑗(𝑡)∣

+

𝑛∑𝑗=1

∣𝑏𝑖𝑗 ∣𝑣𝑗 ∣𝑥𝑗(𝑡− 𝜏𝑖𝑗(𝑡))∣,

𝑖 ∈ 𝒩 , 𝑡 ∈ (𝑡𝑘−1, 𝑡𝑘).

That is,

𝐸𝐷+[𝑥(𝑡)]+ ≤ 𝑃 [𝑥(𝑡)]+ +𝑄[𝑥(𝑡)]+𝜏 ,

𝑡 ∈ (𝑡𝑘−1, 𝑡𝑘), 𝑘 = 1, 2, . . . . (17)

Since 𝑆 is an 𝑀 -matrix, from Lemma 1 we may choosea vector 𝑧∗ = (𝑧1, . . . , 𝑧𝑛)

𝑇 ∈ Ω𝑀 (𝑆) and 𝑧∗ ≥ 𝐼𝑛 =(1, . . . , 1)𝑇 ∈ 𝑅𝑛 such that

𝑆𝑧∗ > 0 or [𝑃 +𝑄]𝑧∗ < 0.

By using continuity, we obtain that there must exist a positiveconstant 𝜆 satisfying the inequality (14).

For the initial condition 𝑥(𝑡0 + 𝑠) = 𝜙(𝑠), 𝑠 ∈ [−𝜏, 0],where 𝜙 ∈ 𝑃𝐶[[−𝜏, 0], 𝑅𝑛] and 𝑡0 ∈ 𝑅 (without loss ofgenerality, we assume 𝑡0 < 𝑡1), we can get

[𝑥(𝑡)]+ ≤ 𝑧∗∥𝜙∥𝜏𝑒−𝜆(𝑡−𝑡0), 𝑡0 − 𝜏 ≤ 𝑡 ≤ 𝑡0. (18)

Then, all conditions of Theorem 1 are satisfied by (17),(18), Condition (𝐴2), ℎ(𝑡) ≡ 1 and 𝐻 = 𝜏 . So

[𝑥(𝑡)]+ ≤ 𝑧∗∥𝜙∥𝜏𝑒−𝜆(𝑡−𝑡0), 𝑡0 ≤ 𝑡 < 𝑡1. (19)

Suppose that for all 𝑚 = 1, . . . , 𝑘, the inequalities

[𝑥(𝑡)]+ ≤ 𝜎0 . . . 𝜎𝑚−1𝑧∗∥𝜙∥𝜏𝑒−𝜆(𝑡−𝑡0),

𝑡𝑚−1 ≤ 𝑡 < 𝑡𝑚, (20)

hold, where 𝜎0 = 1.On the other hand, (1) implies that

𝑒𝑖[𝑥𝑖(𝑡𝑘)− 𝑥𝑖(𝑡𝑘 − 𝜖)]=

∫ 𝑡𝑘

𝑡𝑘−𝜖

[−𝑐𝑖𝑥𝑖(𝑡)𝐷𝑣𝑖(𝑠)

+

𝑛∑𝑗=1

𝑎𝑖𝑗𝑓𝑗(𝑥𝑗(𝑡))𝐷𝑗𝑢(𝑠)

+

𝑛∑𝑗=1

𝑏𝑖𝑗𝑔𝑗(𝑥𝑗(𝑡− 𝜏𝑖𝑗(𝑡)))𝐷𝑤𝑗(𝑠)], 𝑖 ∈ 𝒩 , (21)

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4

where 𝜖 > 0 is sufficiently small, as 𝜖→ 0+, which yields

𝑒𝑖[𝑥𝑖(𝑡𝑘)− 𝑥𝑖(𝑡−𝑘 )]

= −𝑐𝑖𝑥𝑖(𝑡𝑘)𝛼𝑖𝑘 +

𝑛∑𝑗=1

𝑎𝑖𝑗𝑓𝑗(𝑥𝑗(𝑡𝑘))𝛾𝑗𝑘

+𝑛∑

𝑗=1

𝑏𝑖𝑗𝑔𝑗(𝑥𝑗(𝑡𝑘 − 𝜏𝑖𝑗(𝑡𝑘)))𝛽𝑗𝑘, 𝑖 ∈ 𝒩 . (22)

This, together with (𝐴1), yields that

∣(𝑒𝑖 + 𝑐𝑖𝛼𝑖𝑘)𝑥𝑖(𝑡𝑘)∣

= ∣𝑒𝑖𝑥𝑖(𝑡−𝑘 ) +𝑛∑

𝑗=1

𝑎𝑖𝑗𝑓𝑗(𝑥𝑗(𝑡𝑘))𝛾𝑗𝑘

+

𝑛∑𝑗=1

𝑏𝑖𝑗𝑔𝑗(𝑥𝑗(𝑡𝑘 − 𝜏𝑖𝑗(𝑡𝑘)))𝛽𝑗𝑘∣

≤ 𝑒𝑖∣𝑥(𝑡−𝑘 )∣+𝑛∑

𝑗=1

∣𝑎𝑖𝑗 ∣�̄�𝑗 ∣𝑥𝑗(𝑡𝑘)∣∣𝛾𝑗𝑘∣

+

𝑛∑𝑗=1

∣𝑏𝑖𝑗 ∣𝑣𝑗 ∣𝑥𝑗(𝑡𝑘 − 𝜏𝑖𝑗(𝑡𝑘))∣∣𝛽𝑗𝑘∣

≤ 𝑒𝑖∣𝑥(𝑡−𝑘 )∣+𝑛∑

𝑗=1

∣𝑎𝑖𝑗 ∣�̄�𝑗 ∣𝑥𝑗(𝑡𝑘)∣∣𝛾𝑗𝑘∣

+𝑛∑

𝑗=1

∣𝑏𝑖𝑗 ∣𝑣𝑗 [∣𝑥𝑗(𝑡𝑘)∣]𝜏 ∣𝛽𝑗𝑘∣, 𝑖 ∈ 𝒩 . (23)

That is,

[𝐸 + 𝐶𝛼𝑘]+[𝑥(𝑡𝑘)]

+

≤ 𝐸[𝑥(𝑡−𝑘 )]+ + [𝐴]+𝑈 [𝛾𝑘]

+[𝑥(𝑡𝑘)]+

+[𝐵]+𝑉 [𝛽𝑘]+[𝑥(𝑡𝑘)]

+𝜏 . (24)

Then, we have

([𝐸 + 𝐶𝛼𝑘]+ − [𝐴]+𝑈 [𝛾𝑘]

+)[𝑥(𝑡𝑘)]+

≤ 𝐸[𝑥(𝑡−𝑘 )]+ + [𝐵]+𝑉 [𝛽𝑘]

+[𝑥(𝑡𝑘)]+𝜏 (25)

yielding, together with (𝐴3), that

[𝑥(𝑡𝑘)]+ ≤ 𝑆−1

𝑘 𝐸[𝑥(𝑡−𝑘 )]+ + 𝑆−1

𝑘 [𝐵]+𝑉 [𝛽𝑘]+[𝑥(𝑡𝑘)]

+𝜏 . (26)

Then, from (15), (20) and (26)

[𝑥(𝑡𝑘)]+ ≤ (𝑆−1

𝑘 𝐸 + 𝑆−1𝑘 [𝐵]+𝑉 [𝛽𝑘]

+𝑒𝜆𝜏 )

×𝑧∗ 𝜎0 . . . 𝜎𝑘−1∥𝜙∥𝜏𝑒−𝜆(𝑡𝑘−𝑡0)

≤ 𝑧∗ 𝜎0 . . . 𝜎𝑘−1𝜎𝑘∥𝜙∥𝜏𝑒−𝜆(𝑡𝑘−𝑡0). (27)

This, together with (20) and 𝜎𝑖 ≥ 1, 𝑖 = 1, 2, . . . , leads to

[𝑥(𝑡)]+ ≤ 𝜎0 . . . 𝜎𝑘−1𝜎𝑘𝑧∗∥𝜙∥𝜏𝑒−𝜆(𝑡−𝑡0)

= 𝜎0 . . . 𝜎𝑘−1𝜎𝑘∥𝜙∥𝜏𝑒−𝜆(𝑡𝑘−𝑡0)𝑧∗𝑒−𝜆(𝑡−𝑡𝑘),

𝑡 ∈ [𝑡𝑘 − 𝜏, 𝑡𝑘]. (28)

By Lemma 1 again, the vector

𝜎0 . . . 𝜎𝑘−1𝜎𝑘∥𝜙∥𝜏𝑒−𝜆(𝑡𝑘−𝑡0)𝑧∗ ∈ Ω𝑀 (𝑆).

Then, all conditions of Theorem 1 are satisfied again by (17),(28), Condition (𝐴2), ℎ(𝑡) ≡ 1 and 𝐻 = 𝜏 . So

[𝑥(𝑡)]+ ≤ 𝜎0 . . . 𝜎𝑘−1𝜎𝑘∥𝜙∥𝜏𝑒−𝜆(𝑡𝑘−𝑡0)𝑧∗𝑒−𝜆(𝑡−𝑡𝑘)

= 𝜎0 . . . 𝜎𝑘−1𝜎𝑘𝑧∗∥𝜙∥𝜏𝑒−𝜆(𝑡−𝑡0),

𝑡𝑘 ≤ 𝑡 < 𝑡𝑘+1. (29)

By the mathematical induction, we can conclude that

[𝑥(𝑡)]+ ≤ 𝜎0 . . . 𝜎𝑘−1𝑧∗∥𝜙∥𝜏𝑒−𝜆(𝑡−𝑡0),

𝑡𝑘−1 ≤ 𝑡 < 𝑡𝑘, 𝑘 = 1, 2, . . . . (30)

Noticing that 𝜎𝑘 ≤ 𝑒𝜎(𝑡𝑘−𝑡𝑘−1) by (13), we can use (30) toconclude that

[𝑥(𝑡)]+ ≤ 𝑒𝜎(𝑡1−𝑡0) . . . 𝑒𝜎(𝑡𝑘−1−𝑡𝑘−2)𝑧∗∥𝜙∥𝜏𝑒−𝜎(𝑡−𝑡0)

≤ 𝑧∗∥𝜙∥𝜏𝑒𝜎(𝑡−𝑡0)𝑒−𝜆(𝑡−𝑡0)

= 𝑧∗∥𝜙∥𝜏𝑒−(𝜆−𝜎)(𝑡−𝑡0),

𝑡 ∈ [𝑡𝑘−1, 𝑡𝑘), 𝑘 = 1, 2, . . . .

So,

[𝑥(𝑡)]+ ≤ 𝑧∗∥𝜙∥𝜏𝑒−(𝜆−𝜎)(𝑡−𝑡0), 𝑡 ≥ 𝑡0.

This implies that the conclusion of the theorem holds.Remark 1: Condition (𝐴3) must hold when 𝛼𝑘 > 0 and

𝛾𝑘 is sufficiently small. In fact, the positive diagonal matrix[𝐸 + 𝐶𝛼𝑘]

+ is an 𝑀 -matrix when 𝛼𝑘 > 0. So, 𝑆𝑘 = [𝐸 +𝐶𝛼𝑘]

+ − [𝐴]+𝑈 [𝛾𝑘]+ is also an 𝑀 -matrix for sufficiently

small 𝛾𝑘. Then, 𝑆−1𝑘 exists and 𝑆−1

𝑘 ≥ 0.Remark 2: When 𝜏𝑖𝑗(𝑡) ≡ 𝜏 and 𝛾𝑘 = 𝛽𝑘, 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 max1≤𝑘<∞, 1≤𝑖≤𝑛{𝛽𝑖𝑘} must be bounded whenmax1≤𝑘<∞, 1≤𝑖≤𝑛{𝛼𝑖𝑘} is bounded. Here, Theorem 2 abovedoes not require the boundedness of impulsive coefficient anddrops the additional assumption that 𝑡𝑘 − 𝑡𝑘−1 ≥ 𝛿𝜏, 𝛿 > 1in [7].

If 𝐸 is a 𝑛 × 𝑛 unit matrix, the system (1) becomes theimpulsive delay differential system without singularity,

𝐷𝑥𝑖(𝑡) = −𝑐𝑖𝑥𝑖(𝑡)𝐷𝑣𝑖(𝑡)

+𝑛∑

𝑗=1

𝑎𝑖𝑗𝑓𝑗(𝑥𝑗(𝑡))𝐷𝑢𝑗(𝑡)

+𝑛∑

𝑗=1

𝑏𝑖𝑗𝑔𝑗(𝑥𝑗(𝑡− 𝜏𝑖𝑗(𝑡)))𝐷𝑤𝑗(𝑡),

𝑖 ∈ 𝒩 . (31)

For the system (31), we have the following corollary byTheorem 2.

Corollary 1: Assume that (𝐴1) - (𝐴4) hold. Then the zerosolution of the system (31) is globally exponentially stableand the exponential convergence rate is equal to 𝜆− 𝜎.

Remark 3: When coefficient matrix 𝐵 ≡ 0, the system(31) has been investigated in [10] where function 𝑓𝑗(⋅) isdifferentiable and 𝑓 ′𝑗(⋅) is invertible and bounded, 𝑗 ∈ 𝒩 .And Guan and Chen also have discussed the system (31)

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5

with 𝜏𝑖𝑗(𝑡) ≡ 𝜏 [11]. In [11], 𝑓𝑗(⋅) and 𝑔𝑗(⋅) must satisfy𝑧𝑓𝑗(𝑧) ≥ 0 and 𝑧𝑔𝑗(𝑧) ≥ 0, 𝑧 ∈ 𝑅, 𝑗 ∈ 𝒩 . 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 delays⎧⎨⎩𝐷𝑦1(𝑡) = −6𝑦1(𝑡)𝐷𝑣1(𝑡) + 3∣𝑦1(𝑡− 𝜏11(𝑡))∣

×𝐷𝑤1(𝑡)− 2𝑦2(𝑡− 𝜏12(𝑡))𝐷𝑤2(𝑡),0 = −5𝑦2(𝑡)𝐷𝑣2(𝑡)− ∣𝑦1(𝑡− 𝜏21(𝑡))∣𝐷𝑤1(𝑡)

+3 cos(𝑦2(𝑡− 𝜏22(𝑡)))𝐷𝑤2(𝑡),

(32)

where 𝜏𝑖𝑗(𝑡) = ∣ sin((𝑖+ 𝑗)𝑡)∣ ≤ 1Δ= 𝜏 and

𝐷𝑣𝑖(𝑡) = 1+∞∑𝑘=1

𝛼𝛿(𝑡− 𝑡𝑘), 𝐷𝑤𝑖(𝑡) = 1+∞∑𝑘=1

𝛽𝛿(𝑡− 𝑡𝑘),

for 𝑖, 𝑗 = 1, 2, where 𝛿(⋅) is the Dirac impulse function andthe impulsive moments 𝑡𝑘 ( 𝑘 = 1, 2, . . . ) satisfy: 𝑡1 = 0.3,𝑡1 < 𝑡2 < . . . and lim𝑘→+∞ 𝑡𝑘 = +∞.

Thus the parameters of Conditions (𝐴1) - (𝐴4) are asfollows:

𝐸 =

(1 00 0

), 𝑉 =

(1 00 1

),

𝐶 =

(6 00 5

), 𝐴 =

(0 00 0

),

𝛼𝑘 = diag{𝛼, 𝛼}, 𝛽𝑘 = diag{𝛽, 𝛽},

𝐵 =

(3 −2−1 3

), 𝑄 = [𝐵]+𝑉 =

(3 21 3

),

𝑃 = −𝐶 + [𝐴]+𝑈 = −𝐶 =

( −6 00 −5

),

𝑆 = −(𝑃 +𝑄) =

(3 −2−1 2

),

𝑆𝑘 = [𝐸 + 𝐶𝛼𝑘]+ − [𝐴]+𝑈 [𝛾𝑘]

+

=

( ∣1 + 6𝛼∣ 00 5∣𝛼∣

), (33)

which yield that 𝑆 is an 𝑀 -matrix, 𝑆−1𝑘 exists when 𝛼 ∕= 0

and 𝛼 ∕= − 16 . Furthermore,

Ω𝑀 (𝑆) = {(𝑧1, 𝑧2)𝑇 > 0 ∣ 23𝑧2 < 𝑧1 < 2𝑧2},

𝑆−1𝑘 =

(1

∣1+6𝛼∣ 0

0 15∣𝛼∣

).

Let 𝑧∗ = (1, 1)𝑇 ∈ Ω𝑀 (𝑆) and 𝜆 = 0.1 which satisfies theinequality (𝜆𝐸 + 𝑃 + 𝑄𝑒𝜆𝜏 )𝑧∗ < 0. So 𝜎𝑘 can satisfy theinequality

𝜎𝑘𝑧∗ ≥ (𝑆−1

𝑘 𝐸 + 𝑆−1𝑘 [𝐵]+𝑉 [𝛽𝑘]

+𝑒𝜆𝜏 )𝑧∗

= (𝑆−1𝑘 𝐸 + 𝑆−1

𝑘 [𝐵]+∣𝛽∣𝑒0.1)𝑧∗,𝑘 = 1, 2, . . . , (34)

provided that

𝜎𝑘 ≥ max{1 + 5∣𝛽∣𝑒0.1∣1 + 6𝛼∣ ,

4∣𝛽∣𝑒0.15∣𝛼∣ },

𝑘 = 1, 2, . . . . (35)

Case 1: Let 𝛼 = −1, 𝛽 = 𝑒0.2𝑘, 𝜎𝑘 = 𝑒0.5𝑘 and 𝑡𝑘 −𝑡𝑘−1 = 6𝑘, then 𝜎𝑘 satisfies the inequality (35) and 𝜎𝑘 ≥ 1,

ln𝜎𝑘𝑡𝑘 − 𝑡𝑘−1

=ln 𝑒0.5𝑘

6𝑘≤ 0.084 = 𝜎 < 𝜆,

𝑘 = 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 𝛼 = 𝑒0.1, 𝛽 = 1, 𝜎𝑘 = 1 and 𝑡𝑘 − 𝑡𝑘−1 =𝜇𝑘 ∈ (0, 𝜏), then 𝜎𝑘 satisfies the inequality (35) and 𝜎𝑘 ≥ 1,

ln𝜎𝑘𝑡𝑘 − 𝑡𝑘−1

= 0 = 𝜎 < 𝜆, 𝑘 = 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 𝜏𝑖𝑗(𝑡) ≡ 𝜏 , the conditions in Theorem4.1 and Corollary 4.1 of [7] are not satisfied since in Case 1𝛽𝑖𝑘 = 𝑒0.2𝑘 → ∞, 𝑘 → ∞, while max1≤𝑘<∞, 1≤𝑖≤2{𝛼𝑖𝑘}is bounded, and in Case 2 𝜇𝑘 ∈ (0, 𝜏) which does not satisfythe assumption that 𝑡𝑘 − 𝑡𝑘−1 ≥ 𝛿𝜏, 𝛿 > 1 in [7].

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[6] Z. Guan, J. Yao and D.J. Hill, “Robust 𝐻∞ control of singularimpulsive systems with uncertain perturbations”, IEEE Transactions onCircuits and Systems - II: Express Briefs, vol. 52, pp. 293-298, 2005.

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[8] S.G. Deo and S.G. Pandit, Differential Systems Involving Impulses.Springer-Verlag, New York, US, 1982.

[9] Z. Yang, D. Xu and L. Xiang, “Exponential p-stability of impulsivestochastic differential equations with delays”, Phys Lett A , vol. 359,pp. 129-137, 2006.

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