New result on exponential stability for neutral stochastic l(5)

324H.Chen,L.Wang/AppliedMathematicsandComputation239(2014)320–325

wherea2ð0;mini¼1;2;...;nfc1;c2;...;cngÞand

()8 9 P>>2 n2>Pn2(>>)>supjwiðhÞj>><61þnj¼1dijðciÀaÞciEh2½À=61þnj¼1dijh;0 20

Mi¼;>0:EsupjwiðhÞjnn2>>iicnh2½Àh;0 >>>>3nj¼1aijÀnj¼1dijþ3cij¼1hij>>:;

Pn

i¼1jxiðtÞj

2

DuetothefactthatjxðtÞj2¼

2

,ityields

nX

EfjxðtÞjg6nM0ieÀat;

i¼1

tP0:

Theproofiscompleted.h

Remark1.Theorem2providestheexponentialstabilitycriteriaforneutralstochasticlinearsystemwiththefasttime-vary-ingdelay.Thus,Theorem2hasawilderapplicationthanonesgivenin[1–6,8–10,12,13,19,21].Besides,byusingtheBorel–CantelliLemma,wecanalsodiscussthealmostsurelyexponentialstabilityforsuchsystem.Owingtothelimitedspace,hereitisomitted.

Remark2.In[7,13,23],theobtainedcriteriaensuringtheexponentialstabilityforstochasticsystemwiththefasttime-varyingdelaycanalsobeobtainedbyutilizingtheRazumikhin-typetheorem,butfromthecomparisonbetweenthederivedconditionsin[7,13,23]andthealgebraicinequalities(7),itiseasilyseenthat(7)areeasilychecked.

Remark3.FromtheproofofTheorem2,Lemma1hastakenactivelypartinobtainingthealgebraicinequalities(7)withoutanyrestrictiveconditionsonthetime-varyingdelayfunctionhðtÞ,whichisextremelydifferentfromthemethodsproposedin[1–6,8–10,12,13,19,21].

3.Anillustrativenumericalexample

Considertheuncertainneutralstochasticsystem:

d½xðtÞÀDxðtÀhðtÞÞ ¼½ÀCxðtÞþAxðtÀhðtÞÞ dtþHxðtÞdBðtÞ;

T

wherexðtÞ¼½xT1ðtÞx2ðtÞ and

T

tP0;

!

ð14Þ

D¼

Thus,

À0:50:0

0:0À0:5

!

;

C¼

4:00:00:0

4:0

!;

A¼

À1:00:0

À0:5À0:5

!;

H¼

0:150:00:0

0:1

:

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