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
: