A Note on Practical Stability of Nonlinear Vibration Systems with Impulsive Effects

8 Pages·1.813 MB·English

by  Qishui Zhong, Hongcai Li, Hui Liu, Juebang Yu.

HindawiPublishingCorporation
JournalofAppliedMathematics
Volume2012,ArticleID340450,7pages
doi:10.1155/2012/340450
Research Article
A Note on Practical Stability of Nonlinear Vibration
Systems with Impulsive Effects
Qishui Zhong,1 Hongcai Li,2 Hui Liu,2 and Juebang Yu3
1InstituteofAstronautics&Aeronautics,UniversityofElectronicScienceandTechnologyofChina,
Chengdu610054,China
2NationalKeyLabofVehicularTransmission,BeijingInstituteofTechnology,Beijing100081,China
3CollegeofElectronicEngineering,UniversityofElectronicScienceandTechnologyofChina,
Chengdu610054,China
CorrespondenceshouldbeaddressedtoQishuiZhong,[email protected]
Received23March2012;Accepted26July2012
AcademicEditor:JunjieWei
Copyrightq2012QishuiZhongetal.ThisisanopenaccessarticledistributedundertheCreative
CommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductionin
anymedium,providedtheoriginalworkisproperlycited.
This paper addresses the issue of vibration characteristics of nonlinear systems with impulsive
effects.ByutilizingaT-Sfuzzymodeltorepresentanonlinearsystem,ageneralstrictpractical
stabilitycriterionisderivedfornonlinearimpulsivesystems.
1. Introduction
In recent years, nonlinear vibration and its control have been widely studied due to
undesirable or harmful behaviors under many circumstances (cid:3)1–6(cid:4). Furthermore, during
working condition, plates usually work with disturbances which can be described with
impulsiveeffects,andimpulsivedifferentialequationsusuallyusedtodealtwiththiskindof
problem.Anumberofpapershavedealwiththetheoryofimpulsivedifferentialequations
(cid:3)7–10(cid:4) and its applications to many kind of systems (cid:3)11, 12(cid:4). Impulsive control has been
demonstratedtobeaneffectiveandattractivecontrolmethodtostabilizelinearandnonlinear
systems(cid:3)11,13(cid:4),especiallytostabilizevariouschaoticvibrationsystems(cid:5)e.g.,see(cid:3)14–16(cid:4)(cid:6).
Furthermore, a highly nonlinear system can usually be represented by the T-S fuzzy
model(cid:3)17(cid:4).TheT-SfuzzymodelisdescribedbyfuzzyIF-THENruleswheretheconsequent
parts represent local linear models for nonlinear systems. Then we can use linear theory to
analyzechaoticsystemsbymeansofT-Sfuzzymodelsatacertaindomain(cid:3)18,19(cid:4).
In the study of the Lyapunov stability, an interesting set of problems deals with
bringing sets close to a certain state, rather than the state x (cid:7) 0 (cid:3)20(cid:4). The desired state of
2 JournalofAppliedMathematics
system may be mathematically unstable and yet the system may oscillate sufficiently near
thisstatethatitsperformanceisacceptable.Manyproblemsfallintothiscategoryincluding
thetravelofaspacevehiclebetweentwopoints,anaircraftoramissilewhichmayoscillate
around a mathematically unstable course yet its performance may be acceptable, and the
problem in a chemical process of keeping the temperature within certain bounds. Such
considerations led to the notion of practical stability which is neither weaker nor stronger
than the Lyapunov stability (cid:3)21, 22(cid:4). In a practical control problem, one aims at controlling
a system into a certain region of interest instead of an exact point. If one wants to control a
systemtoanexactpoint,theexpensemaybeprohibitivelyhighinsomecases.
Inthispaper,westudythepracticalstabilityforthenonlinearimpulsivesystembased
onT-Sfuzzymodel.Afterageneralstrictlypracticalstabilitycriterionisderived,somesimple
andeasilyverifiedsufficientconditionsaregiventostabilizethenonlinearvibrationsystem.
Notation: A > 0 (cid:5)< 0(cid:6) means A is a symmetrical positive (cid:5)negative(cid:6) definite matrix.
R(cid:9), R(cid:9), and N stand for, respectively, the set of all positive real numbers, nonnegative real
numbers, and the set of natural numbers. (cid:2)x(cid:2) denotes the Euclidian norm of vector x, and
λ (cid:5)A(cid:6)andλ (cid:5)A(cid:6)meantheminimalandmaximaleigenvaluesofmatrixA,respectively.
min max
2. Problem Formulation
Considerthefollowingnonlinearsystem
x˙(cid:5)t(cid:6)(cid:7)f(cid:5)x(cid:5)t(cid:6)(cid:6), (cid:5)2.1(cid:6)
where x(cid:5)t(cid:6) ∈ Rn is the state variable, f ∈ C(cid:3)Rn,Rn(cid:4). We can construct the fuzzy model for
(cid:5)2.1(cid:6)asfollows:rule(cid:5)i(cid:6):ifz1(cid:5)t(cid:6)isMi1,...,andzp(cid:5)t(cid:6)isMip,THENx(cid:5)t(cid:6)(cid:7)Aix(cid:5)t(cid:6),i(cid:7)1,2,...,r,
inwhichzi(cid:5)t(cid:6),...,zp(cid:5)t(cid:6)arethepremisevariables,eachMij (cid:5)j (cid:7)1,2,...,p(cid:6)isafuzzyset,and
Ai ∈ Rn×n isaconstantmatrix.Withacenter-averagedefuzzifier,theoverallfuzzysystemis
representedas
(cid:2)r
x˙(cid:5)t(cid:6)(cid:7) hi(cid:5)t(cid:6)Aix(cid:5)t(cid:6), (cid:5)2.2(cid:6)
i(cid:7)1
(cid:3)
w(cid:4)anhpld(cid:7)e1r(cid:3)Meri(cid:7)irl1(cid:5)zhilis(cid:5)(cid:5)tt(cid:6)(cid:6)t(cid:6)h(cid:7),ea1n.ndNuMomteibl(cid:5)etzhrla(cid:5)tto(cid:6)sf(cid:6)yisfsutetzhmzey(cid:5)g2ri.a2md(cid:6)peclaoicnfamltoioecnamlslb,yerhresihp(cid:5)tir(cid:6)pesoe(cid:7)fnztls(cid:5)wyt(cid:6)sit(cid:5)ient(cid:6)m/M(cid:5)(cid:5)2il..1ri(cid:7)O(cid:6)1.fwcoi(cid:5)ut(cid:6)r(cid:6)s,e,whii(cid:5)(cid:5)tt(cid:6)(cid:6)≥(cid:7)0
Disturbances, acting on system (cid:5)2.1(cid:6), are given by a sequence {tk,Ik(cid:5)x(cid:5)tk(cid:6)(cid:6)}, where
0 < t1 < t2 < ··· < tk < ···,tk → ∞ask → ∞,andIk ∈ C(cid:3)Rn,Rn(cid:4)denotestheincremental
changeofthestateattimetk.Thus,thefollowingimpulsivedifferentialequationisobtained
(cid:2)r
x˙(cid:5)t(cid:6)(cid:7) hi(cid:5)t(cid:6)Aix(cid:5)t(cid:6), ∀t≥0, t/(cid:7)tk,
i(cid:7)1 (cid:5)2.3(cid:6)
Δx(cid:5)t(cid:6)(cid:7)I (cid:5)x(cid:5)t(cid:6)(cid:6), t(cid:7)t , k ∈N,
k k
inwhichΔx(cid:5)t(cid:6)(cid:7)x(cid:5)t(cid:9)k(cid:6)−x(cid:5)tk(cid:6),x(cid:5)t(cid:9)k(cid:6)(cid:7)tl→imt(cid:9)x(cid:5)tk(cid:6),k ∈N.
k
JournalofAppliedMathematics 3
AssumethatIk(cid:5)0(cid:6)≡0forallksothatthetrivialsolutionof(cid:5)2.3(cid:6)exists.DenoteS(cid:5)ρ(cid:6)(cid:7)
{x∈Rn :(cid:2)x(cid:2)<ρ}.
We will introduce the following classes of function spaces, definitions, and theorems
forfutureuse:
K (cid:7){a∈C(cid:3)R(cid:9),R(cid:9)(cid:4):aisstrictlyincreasinganda(cid:5)0(cid:6)(cid:7)0}.
CK (cid:7){a∈C(cid:3)R2(cid:9),R(cid:9)(cid:4):a(cid:5)t,u(cid:6)∈Kforeacht∈R(cid:9)}.
Definition2.1. LetV :R(cid:9)×Rn → R(cid:9),thenV issaidtobelongtoclassυ0if:
(cid:5)1(cid:6)V iscontinuousin(cid:5)tk−1,tk(cid:4)×Rnandforeachx∈Rn,k (cid:7)1,2,...,
(cid:5) (cid:6) (cid:5) (cid:6)
(cid:5)t,y(cid:6)l→im(cid:5)t(cid:9),x(cid:6)V t,y lim(cid:7)V t(cid:9)k,x (cid:5)2.4(cid:6)
k
exits;
(cid:5)2(cid:6)V islocallyLipschitzianinx.
Definition2.2. For(cid:5)t,x(cid:6)∈(cid:5)tk−1,tk(cid:4)×Rn,wedefinethederivativesofV(cid:5)t,x(cid:6)as
(cid:7) (cid:5) (cid:6) (cid:8)
D(cid:9)V(cid:5)t,x(cid:6)(cid:7) lim sup1 V t(cid:9)s,x(cid:9)sf(cid:5)t,x(cid:6) −V(cid:5)t,x(cid:6) , (cid:5)2.5(cid:6)
s→0(cid:9) s
(cid:7) (cid:5) (cid:6) (cid:8)
D−V(cid:5)t,x(cid:6)(cid:7) lim inf1 V t(cid:9)s,x(cid:9)sf(cid:5)t,x(cid:6) −V(cid:5)t,x(cid:6) . (cid:5)2.6(cid:6)
s→0− s
Definition2.3. Thetrivialsolutionofthesystem(cid:5)2.1(cid:6)issaidtobe
(cid:5)1(cid:6)practicallystable,ifgiven(cid:5)λ,A(cid:6)with0 < λ < A,onehas(cid:2)x (cid:2) < λimplies(cid:2)x(cid:5)t(cid:6)(cid:2) <
0
A,t≥t0forsomet0 ∈R(cid:9);
(cid:5)2(cid:6)strictlystable,if(cid:5)1(cid:6)holdsandforeveryμ ≤ λthereexistsB < μsuchthat(cid:2)x (cid:2) > μ
0
implies(cid:2)x(cid:5)t(cid:6)(cid:2)>B,t≥t .
0
Theorem2.4(cid:5)see(cid:3)23(cid:4)(cid:6). Supposethat
(cid:5)1(cid:6)0<λ<A<ρ;
(cid:5)2(cid:6)thereexistsV1 : R(cid:9)×S(cid:5)ρ(cid:6) → R(cid:9),V1 ∈ v0,a1 ∈ CK,b1 ∈ Ksuchthata1(cid:5)t0,λ(cid:6) ≤ b1(cid:5)A(cid:6)
forsomet0 ∈R(cid:9),andfor(cid:5)t,x(cid:6)∈R(cid:9)×S(cid:5)ρ(cid:6),
b (cid:5)(cid:2)x(cid:2)(cid:6)≤V (cid:5)t,x(cid:6)≤a (cid:5)t ,(cid:2)x(cid:2)(cid:6),
1 1 1 0
D(cid:9)V1(cid:5)t,x(cid:6)≤0, t/(cid:7)tk, (cid:5)2.7(cid:6)
V (cid:5)t(cid:9),x(cid:9)I (cid:5)x(cid:6)(cid:6)≤V (cid:5)t,x(cid:6), t(cid:7)t ,
1 k 1 k
4 JournalofAppliedMathematics
(cid:5)3(cid:6)thereexistsV2 :R(cid:9)×S(cid:5)ρ(cid:6) → R(cid:9),V2 ∈v0,a2,b2 ∈Ksuchthatfor(cid:5)t,x(cid:6)∈R(cid:9)×S(cid:5)ρ(cid:6),
b (cid:5)(cid:2)x(cid:2)(cid:6)≤V (cid:5)t,x(cid:6)≤a (cid:5)(cid:2)x(cid:2)(cid:6)
2 2 2
D−V2(cid:5)t,x(cid:6)≥0, t/(cid:7)tk, (cid:5)2.8(cid:6)
V (cid:5)t(cid:9),x(cid:9)I (cid:5)x(cid:6)(cid:6)≥V (cid:5)t,x(cid:6), t(cid:7)t .
2 k 2 k
Then,thetrivialsolutionof (cid:5)2.1(cid:6)isstrictlypracticallystable.
3. Main Results
Inthissection,wewillgivesomestrictpracticalstabilitycriteriaofnonlinearvibrationsystem
(cid:5)2.3(cid:6).
Theorem3.1. Thetrivialsolutionofsystem(cid:5)2.3(cid:6)isstrictlypracticallystableifthereexistsamatrix
P >0,α≤0andthefollowingconditionshold:
PAi(cid:9)ATiP ≤αI, i(cid:7)1,...,r, (cid:5)3.1(cid:6)
λ (cid:5)P(cid:6)
(cid:2)x(cid:5)tk(cid:6)(cid:9)Ik(cid:5)x(cid:5)tk(cid:6)(cid:6)(cid:2)2 ≤ λmin(cid:5)P(cid:6)(cid:2)x(cid:5)tk(cid:6)(cid:2)2, k ∈N. (cid:5)3.2(cid:6)
max
Proof. Consider the Lyapunov function V (cid:5)x(cid:5)t(cid:6)(cid:6) (cid:7) (cid:5)1/2(cid:6)xT(cid:5)t(cid:6)Px(cid:5)t(cid:6) and V (cid:5)x(cid:5)t(cid:6)(cid:6) (cid:7)
1 2
exp(cid:5)−xT(cid:5)t(cid:6)Px(cid:5)t(cid:6)(cid:6).Ityieldsthat
1 1
λ (cid:5)P(cid:6)(cid:2)x(cid:5)t(cid:6)(cid:2)2 ≤V (cid:5)x(cid:5)t(cid:6)(cid:6)≤ λ (cid:5)P(cid:6)(cid:2)x(cid:5)t(cid:6)(cid:2)2,
min 1 max
2 (cid:9) (cid:10) 2 (cid:9) (cid:10) (cid:5)3.3(cid:6)
exp −λ (cid:5)P(cid:6)(cid:2)x(cid:5)t(cid:6)(cid:2)2 ≥V (cid:5)x(cid:5)t(cid:6)(cid:6)≥exp −λ (cid:5)P(cid:6)(cid:2)x(cid:5)t(cid:6)(cid:2)2 .
min 2 max
Whent/(cid:7)tk,sincePAi(cid:9)ATiP ≤αI,i(cid:7)1,2,...,r andα≤0,thederivativesofthemare
D(cid:9)V (cid:5)x(cid:5)t(cid:6)(cid:6)(cid:7) 1x˙T(cid:5)t(cid:6)Px(cid:5)t(cid:6)(cid:9)xT(cid:5)t(cid:6)Px˙(cid:5)t(cid:6)
1
2
(cid:2)r (cid:11) (cid:12)
(cid:7) 1 h(cid:5)t(cid:6)xT(cid:5)t(cid:6) PA (cid:9)ATP x(cid:5)t(cid:6)
i i i
2
i(cid:7)1
α
≤ (cid:2)x(cid:5)t(cid:6)(cid:2)2
2
(cid:5) (cid:6)
≤0, t∈(cid:5)tk−1,tk(cid:6), k ∈N, x∈S ρ ,
(cid:5)3.4(cid:6)
(cid:9) (cid:10)(cid:9) (cid:10)
D−V2(cid:5)x(cid:5)t(cid:6)(cid:6)(cid:7)−exp −xT(cid:5)t(cid:6)Px(cid:5)t(cid:6) x˙T(cid:5)t(cid:6)Px(cid:5)t(cid:6)(cid:9)xT(cid:5)t(cid:6)Px˙(cid:5)t(cid:6)
(cid:13) (cid:14)
(cid:9) (cid:10) (cid:2)r (cid:11) (cid:12)
(cid:7)−exp −xT(cid:5)t(cid:6)Px(cid:5)t(cid:6) hi(cid:5)t(cid:6)xT(cid:5)t(cid:6) PAi(cid:9)ATiP x(cid:5)t(cid:6)
i(cid:7)1
JournalofAppliedMathematics 5
(cid:9) (cid:10)
≥−αexp −xT(cid:5)t(cid:6)Px(cid:5)t(cid:6) (cid:2)x(cid:5)t(cid:6)(cid:2)2
(cid:5) (cid:6)
≥0, t∈(cid:5)tk−1,tk(cid:6), k ∈N, x∈S ρ .
(cid:5)3.5(cid:6)
Let a (cid:5)x(cid:6) (cid:7) xT(cid:5)t(cid:6)Px(cid:5)t(cid:6), b (cid:5)x(cid:6) (cid:7) (cid:5)1/4(cid:6)V (cid:5)x(cid:5)t(cid:6)(cid:6), a (cid:5)x(cid:6) (cid:7) 2exp(cid:5)−xT(cid:5)t(cid:6)Px(cid:5)t(cid:6)(cid:6), and b (cid:5)x(cid:6) (cid:7)
1 1 1 2 2
exp(cid:5)−xT(cid:5)t(cid:6)Px(cid:5)t(cid:6)(cid:6),wecaneasilyobtain
b (cid:5)(cid:2)x(cid:2)(cid:6)≤V (cid:5)x(cid:5)t(cid:6)(cid:6)≤a (cid:5)t ,(cid:2)x(cid:2)(cid:6),
1 1 1 0
(cid:5)3.6(cid:6)
b (cid:5)(cid:2)x(cid:2)(cid:6)≤V (cid:5)x(cid:5)t(cid:6)(cid:6)≤a (cid:5)(cid:2)x(cid:2)(cid:6).
2 2 2
Whent(cid:7)tk,accordingto(cid:5)3.2(cid:6)and(cid:5)3.6(cid:6),wehave
V (cid:5)x(cid:5)t(cid:6)(cid:9)I (cid:5)x(cid:5)t(cid:6)(cid:6)(cid:6)(cid:7) 1(cid:5)x(cid:5)t(cid:6)(cid:9)I (cid:5)x(cid:5)t(cid:6)(cid:6)(cid:6)TP(cid:5)x(cid:5)t(cid:6)(cid:9)I (cid:5)x(cid:5)t(cid:6)(cid:6)(cid:6)
1 k k k
2
1
≤ λ (cid:5)P(cid:6)(cid:2)(cid:5)x(cid:5)t(cid:6)(cid:9)I (cid:5)x(cid:5)t(cid:6)(cid:6)(cid:6)(cid:2)2
max k
2 (cid:5)3.7(cid:6)
1
≤ λ (cid:5)P(cid:6)(cid:2)(cid:5)x(cid:5)t(cid:6)(cid:6)(cid:2)2
min
2
≤V (cid:5)x(cid:5)t(cid:6)(cid:6),
1
(cid:9) (cid:10)
V2(cid:5)x(cid:5)t(cid:6)(cid:9)Ik(cid:5)x(cid:5)t(cid:6)(cid:6)(cid:6)(cid:7)exp (cid:5)−(cid:5)x(cid:5)t(cid:6)(cid:9)Ik(cid:5)x(cid:5)t(cid:6)(cid:6)(cid:6)TP(cid:5)x(cid:5)t(cid:6)(cid:9)Ik(cid:5)x(cid:5)t(cid:6)(cid:6)(cid:6)
≥exp(cid:5)−λmax(cid:5)P(cid:6)(cid:2)(cid:5)x(cid:5)t(cid:6)(cid:9)Ik(cid:5)x(cid:5)t(cid:6)(cid:6)(cid:6)(cid:2)(cid:6)2
(cid:9) (cid:10) (cid:5)3.8(cid:6)
≥exp −λ (cid:5)P(cid:6)(cid:2)(cid:5)x(cid:5)t(cid:6)(cid:6)(cid:2)2
min
≥V (cid:5)x(cid:5)t(cid:6)(cid:6).
2
Then,withTheorem2.4,thetrivialsolutionofsystem(cid:5)2.3(cid:6)isstrictlypracticalstable.
Remark 3.2. If system (cid:5)2.3(cid:6) is practically stabilizable, we can design a general linear or
nonlinearimpulsivecontrollaw{tk,Ik(cid:5)x(cid:5)tk(cid:6)(cid:6)}forsystem(cid:5)2.1(cid:6),whichcanmakethesystem
oscillatesufficientlyneartheaimedstateandtheperformanceisconsideredacceptable,that
is, the vibration of system (cid:5)2.1(cid:6) is depressed in the sense of practical stability under the
impulsivecontrol.
If Ik(cid:5)x(cid:5)tk(cid:6)(cid:6) (cid:7) Ckx(cid:5)tk(cid:6),k ∈ N, where each Ck ∈ Rn×n is constant matrix, then system
(cid:5)2.3(cid:6)isrewrittenby
(cid:2)r
x˙(cid:5)t(cid:6)(cid:7) hi(cid:5)t(cid:6)Aix(cid:5)t(cid:6), ∀t≥0, t/(cid:7)tk,
i(cid:7)1 (cid:5)3.9(cid:6)
Δx(cid:5)t(cid:6)(cid:7)C x(cid:5)t(cid:6), t(cid:7)t , k ∈N.
k k
6 JournalofAppliedMathematics
From(cid:5)3.2(cid:6),itiseasilyobtainedthat
λ (cid:5)P(cid:6)
(cid:2)I(cid:9)Ck(cid:2)2 ≤ λmin(cid:5)P(cid:6), k ∈N. (cid:5)3.10(cid:6)
max
Then,wehavethefollowingcorollary.
Corollary3.3. Thetrivialsolutionofsystem(cid:5)3.9(cid:6)isstrictlypracticallystableifthereexistamatrix
P >0,α≤0andtheconditions(cid:5)3.1(cid:6)and(cid:5)3.10(cid:6)hold.
4. Conclusions
In this paper, some strict practical stability criteria have been put forward for nonlinear
impulsive systems based on their T-S models. The reported results are helpful to consider
the vibration characteristics of nonlinear systems with impulsive disturbances and also to
controlthevibrationofnonlinearsystemviatheimpulsivecontrollaw.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under
Grants no. 50905018 and 51075033, the Foundation of Science and Technology on Vehicle
Transmission Laboratory, the Fundamental Research Funds for the Central Universities
under Grant no. ZYGX2009J088, and Sichuan Science & Technology Plan under Grant no.
2011JY0001.
References
(cid:3)1(cid:4) B.Lehman,J.Bentsman,S.VerduynLunel,andE.I.Verriest,“Vibrationalcontrolofnonlineartime
lag systems with bounded delay: averaging theory, stabilizability, and transient behavior,” IEEE
TransactionsonAutomaticControl,vol.39,no.5,pp.898–912,1994.
(cid:3)2(cid:4) A.E.Alshorbagy,M.A.Eltaher,andF.F.Mahmoud,“Freevibrationcharacteristicsofafunctionally
gradedbeambyfiniteelementmethod,”AppliedMathematicalModelling,vol.35,no.1,pp.412–425,
2011.
(cid:3)3(cid:4) B. Farshi and A. Assadi, “Development of a chaotic nonlinear tuned mass damper for optimal
vibrationresponse,”CommunicationsinNonlinearScienceandNumericalSimulation,vol.16,no.11,pp.
4514–4523,2011.
(cid:3)4(cid:4) B.K.Eshmatov,“Nonlinearvibrationsanddynamicstabilityofviscoelasticorthotropicrectangular
plates,”JournalofSoundandVibration,vol.300,no.3–5,pp.709–726,2007.
(cid:3)5(cid:4) X.Zhou,Y.Wu,Y.Li,andZ.Wei,“HopfbifurcationanalysisoftheLiusystem,”Chaos,Solitons&
Fractals,vol.36,no.5,pp.1385–1391,2008.
(cid:3)6(cid:4) X.Zhou,Y.Wu,Y.Li,andH.Xue,“Adaptivecontrolandsynchronizationofanovelhyperchaotic
systemwithuncertainparameters,”AppliedMathematicsandComputation,vol.203,no.1,pp.80–85,
2008.
(cid:3)7(cid:4) V.Lakshmikantham,D.D.Bainov,andP.S.Simeonov,TheoryofImpulsiveDifferentialEquations,Series
inModernAppliedMathematics,WorldScientific,Singapore,1989.
(cid:3)8(cid:4) Q. Zhong, J. Bao, Y. Yu, and X. Liao, “Exponential stabilization for discrete Takagi-Sugeno fuzzy
systemsviaimpulsivecontrol,”Chaos,Solitons&Fractals,vol.41,no.4,pp.2123–2127,2009.
(cid:3)9(cid:4) X.H.Tang,Z.He,andJ.S.Yu,“Stabilitytheoremfordelaydifferentialequationswithimpulses,”
AppliedMathematicsandComputation,vol.131,no.2-3,pp.373–381,2002.
JournalofAppliedMathematics 7
(cid:3)10(cid:4) T. Yang, “Impulsive control,” IEEE Transactions on Automatic Control, vol. 44, no. 5, pp. 1081–1083,
1999.
(cid:3)11(cid:4) Q.Zhong,J.Bao,Y.Yu,andX.Liao,“ImpulsivecontrolforT-Sfuzzymodel-basedchaoticsystems,”
MathematicsandComputersinSimulation,vol.79,no.3,pp.409–415,2008.
(cid:3)12(cid:4) M.DeLaSenandN.Luo,“Anoteonthestabilityoflineartime-delaysystemswithimpulsiveinputs,”
IEEETransactionsonCircuitsandSystems.I,vol.50,no.1,pp.149–152,2003.
(cid:3)13(cid:4) X. Liu and G. Ballinger, “Uniform asymptotic stability of impulsive delay differential equations,”
Computers&MathematicswithApplications,vol.41,no.7-8,pp.903–915,2001.
(cid:3)14(cid:4) X.Liu,“Impulsivestabilizationandcontrolofchaoticsystem,”NonlinearAnalysis:Theory,Methods&
Applications,vol.47,no.2,pp.1081–1092,2001.
(cid:3)15(cid:4) J.Sun,“Impulsivecontrolofanewchaoticsystem,”MathematicsandComputersinSimulation,vol.64,
no.6,pp.669–677,2004.
(cid:3)16(cid:4) Q.S.Zhong,J.F.Bao,Y.B.Yu,andX.F.Liao,“Impulsivecontrolforfractional-orderchaoticsystems,”
ChinesePhysicsLetters,vol.25,no.8,pp.2812–2815,2008.
(cid:3)17(cid:4) T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and
control,”IEEETransactionsonSystems,ManandCybernetics,vol.15,no.1,pp.116–132,1985.
(cid:3)18(cid:4) Y.W.Wang,Z.H.Guan,andH.O.Wang,“ImpulsivesynchronizationforTakagi-Sugenofuzzymodel
anditsapplicationtocontinuouschaoticsystem,”PhysicsLetters,SectionA,vol.339,no.3–5,pp.325–
332,2005.
(cid:3)19(cid:4) Y.-W.Wang,Z.-H.Guan,H.O.Wang,andJ.-W.Xiao,“ImpulsivecontrolforT-Sfuzzysystemand
itsapplicationtochaoticsystems,”InternationalJournalofBifurcationandChaosinAppliedSciencesand
Engineering,vol.16,no.8,pp.2417–2423,2006.
(cid:3)20(cid:4) I.M.Stamova,“VectorLyapunovfunctionsforpracticalstabilityofnonlinearimpulsivefunctional
differentialequations,”JournalofMathematicalAnalysisandApplications,vol.325,no.1,pp.612–623,
2007.
(cid:3)21(cid:4) V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Practical Stability of Nonlinear Systems, World
Scientific,Teaneck,NJ,USA,1990.
(cid:3)22(cid:4) T. Yang, J. A. K. Suykens, and L. O. Chua, “Impulsive control of nonautonomous chaotic systems
using practical stabilization,” International Journal of Bifurcation and Chaos in Applied Sciences and
Engineering,vol.8,no.7,pp.1557–1564,1998.
(cid:3)23(cid:4) S.Li,X.Song,andA.Li,“Strictpracticalstabilityofnonlinearimpulsivesystemsbyemployingtwo
Lyapunov-like functions,” Nonlinear Analysis: Real World Applications, vol. 9, no. 5, pp. 2262–2269,
2008.
Advances in Advances in Journal of Journal of
Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics
Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014
The Scientific
International Journal of
World Journal Differential Equations
Hhtintpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of Advances in
Combinatorics Mathematical Physics
Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014
Journal of Journal of Mathematical Problems Abstract and Discrete Dynamics in
Complex Analysis Mathematics in Engineering Applied Analysis Nature and Society
Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014
International
Journal of
Journal of
Mathematics and Discrete Mathematics
Mathematical
Sciences
Journal of International Journal of Journal of
Function Spaces Stochastic Analysis Optimization
Hhtintpd:a/w/iw Pwuwbl.ihsihnidnagw Ci.ocropmoration Volume 2014 Hhtintpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014 Hhtitnpd:/a/wwiw Pwub.hliisnhdinawg iC.coormporation Volume 2014