Substituting ( 20 )into( 19 ), and using conditions, we obtain
L푉≤푝|푄푥|푝−1[휆max(퐴)|푄푥|+Σ푚푖=1훼푖(푡)儨儨儨儨푄푥 (푡 − 휏푖)儨儨儨儨]+
푝(푝−1)
2
|푄푥|푝−2Σ푚푖=1훽푖(푡)儨儨儨儨푄푥 (푡 − 휏푖)儨儨儨儨
2
.
(21)UsingLemma 4,weget
L푉≤|푄푥|푝[휆
max(퐴)+(푝−1)Σ푚푖=1훼푖(푡)
+
(푝 − 1) (푝 − 2)
2
Σ푚푖=1훽푖(푡)]
+[Σ푚푖=1훼푖(푡)+(푝−1)Σ푚푖=1훽푖(푡)]儨儨儨儨푄푥 (푡 − 휏푖)儨儨儨儨푝
=[휅 1 +(푝−1)휅 2 +
(푝 − 2) 휅 3
2
]푉(푡, 푥)
+[휅 2 +휅 3 ]푉(푡,푥(푡−휏푖)).
(22)
Summinguptheabovestatements,wecanseethatall
the conditions ofTheorem 5and condition (iii) ofTheorem 6
are satisfied. Then the conclusion follows fromTheorem 6
immediately and the proof is completed.
4. Example
In this section, we present an example to demonstrate
our theoretical results. Considering a nonlinear stochastic
impulsive system as follows:
푑푥(푡)=푓(푥(푡))푑푡 + 푔(푥(푡),푥(푡−휏))푑푤(푡),푡≥0, 푡=푡̸푘,Δ푥 (푡푘)=퐼푘(푥 (푡푘)) , 푡 = 푡푘,푘∈푁,(23)
where푓(푥) = 푥(푡),푔(푥, 푥푡) ≤ (1/4)(푥^2 +푥^2 푡),퐼푘= −0.4,휏=2.
Step 1.Calculate the parameters.
Without loss of generality, we choose푐 1 =푐 2 =1,푝=2,
푑푘= 0.37such that they satisfy the conditions of (i) and (ii)
ofTheorem 5.
Step 2.Choose푉(푥, 푡) = 푥^2 ,thenitiseasytocalculatefrom
the Itˆoformulathat
퐸L푉(푥(푡),푥(푡−휏))=2퐸|푥(푡)|^2 +퐸儨儨儨儨푔(푥, 푥푡)儨儨儨儨^2≤
9
4
퐸푉(푥(푡))+
1
4
퐸푉(푥(푡−휏))
(24)
which satisfies condition (iii) ofTheorem 5
퐸L푉(푥(푡),푥(푡−휏))≤푐퐸푉(푥(푡)), (25)where take푞=5,휆 = 0.5,푡푘+1−푡푘= 0.2.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 502468101214161820Timetx(t)Figure 1: Instability of the stochastic delay system ( 23 ) without
impulsive effect.012345678−0.6−0.4−0.200.20.40.60.81Timetx(t)Figure 2: Impulsive Stabilization of the stochastic delay system ( 23 ).Step 3.By calculation, we obtain푐 = 3.5,thenln푑푘= −1.02 < −(푐+휆)(푡푘+1−푡푘)= −0.8. (26)It satisfies condition (iv) ofTheorem 5which means that
the system ( 23 ) is exponentially stable.Figure 1gives the
trajectory of the state of ( 23 ). It is obvious that the system
is not stable without impulsive effect.Figure 1shows that the
solution of the stochastic delayed system ( 23 )isunstable.
Figure 2shows the stability of the delay system with the
impulsive controller.5. Conclusion
In this paper, we have investigated thep-moment stability
and applied the technique of Razumikhin techniques and
Lyapunov functions to impulsive stochastic systems. Some
sufficient conditions about the stability of impulsive stochas-
tic systems in terms of two measures are derived. As a
beneficial supplement in the study of impulsive stochastic
systems with time delay, the concluded criteria are not only
effective but also convenient in practical applications of
specific systems in engineering and physics, etc. We also
provided an illustrative example to show the effectiveness of
our results.