Theorem 6.Assume that
(i)퐸푉푡≤ 휂퐸푉(푥(푡)),퐸푉푥≤휂 1 퐸푉(푥(푡))(푝−1)/푝,퐸푉푥푥≤
휂 2 퐸푉(푥(푡))(푝−2)/푝;
(ii)퐸|푓(푥푡)|푝 ≤ 휂 3 sup−휏≤휃≤0퐸푉(푥푡),
퐸(Tr{푔(푥푡)푇푔(푥푡)})푝/2
≤휂 4 sup−휏≤휃≤0퐸푉(푥(푡 − 휏));
(iii)퐸푉(푥(푡 + 휃)) ≤ 푞퐸푉(푥(푡)), 휃 ∈ (−휏, 0];
and conditions ofTheorem 5hold simultaneously, then
the system is pth exponential stable, where퐸denotes the
expectation.
Proof.Take
L푉=푉푡+푉푥푓(푥푡)+1
2
Tr{푔푇(푥푡)푉푥푥푔(푥푡)} (13)then take the mathematical expectation of both sides of the
Formula ( 12 ), we obtain
퐸L푉=퐸푉푡+퐸(푉푥푓) +1
2
퐸[Tr{푔푇푉푥푥푔}]. (14)UsingLemma 3, from (i) and (ii), we obtain
퐸(푉푥푓(푥,푥푡))≤퐸푉푥⋅퐸푓(푥,푥푡)≤휂 1 퐸(푉)(푝−1)/푝⋅퐸儨儨儨儨푓(푥,푥푡)儨儨儨儨≤
휂 1 (푝 − 1)
푝
퐸푉 +
1
푝
휂 3 ⋅퐸儨儨儨儨푓(푥,푥푡)儨儨儨儨
푝
≤휂 1 (푝 − 1)
푝
퐸푉
+
1
푝
휂 3 ⋅푞⋅sup
−휏≤휃≤0퐸푉(푥(푡)),
퐸[Tr{푔푇(푥푡)푉푥푥푔(푥푡)}]≤퐸푉푥푥⋅퐸[Tr{푔푇(푥푡)푔(푥푡)}] ≤휂 2 퐸(푉(푥))(푝−2)/푝⋅퐸(Tr{푔푇(푥푡)푔(푥푡)}) ≤휂 2 (푝 − 2)
푝
퐸푉(푥)
+
2
푝
퐸[Tr{푔푇푔}]푝/2
≤휂 2 (푝 − 2)
푝
퐸푉(푥)
+
2 휂 4
푝
⋅ sup
−휏≤휃≤0퐸푉(푥(푡−휏))≤
휂 2 (푝 − 2)
푝
퐸푉(푥)
+
2 휂 4
푝
⋅푞⋅sup
−휏≤휃≤0퐸푉(푥(푡)).
(15)
It follows that
퐸L푉≤휂 1 퐸푉(푥(푡))+휂 1 (푝 − 1)
푝
퐸푉(푥(푡))
+
휂 2 (푝 − 2)
푝
퐸푉(푥)+
1
푝
휂 3 ⋅푞⋅sup
−휏≤휃≤0퐸푉(푥(푡))
+
2 휂 4
푝
⋅푞⋅sup
−휏≤휃≤0퐸푉(푥(푡)).
(16)
Consequently, by the above statement, the conditions of
Theorem 5are all satisfied. Then, the conclusion follows from
Theorem 5and the proof is complete.Remark 7.From the above consequence, we know that the
unstable stochastic system푑푥(푡) = 푓(푡, 푥, 푥푡)푑푡 + 푔(푡, 푥푡)
푑푤(푡)can be exponentially stabilized by the impulsive control
푢푘(푥) = 퐼푘(푥), 푡푘,푘∈푁. Moreover, the steps of the impulsive
control design satisfy the conditions ofTheorem 5.Remark 8.Consider a special case of system ( 1 )shownas
follows:푑푥(푡)=[퐴푥+푓 1 (푡, 푥(푡−휏))]푑푡+푔(푡, 푥(푡−휏))푑푤(푡),푡≥0, 푡=푡̸푘,Δ푥 (푡푘)=푥(푡+푘)−푥(푡푘)=퐼푘(푥 (푡푘)) ,푡=푡푘, 푘=1,2,...,푚,푥(푡 0 )=휉, 푡=[−휏, 0],
(17)there exist nonnegative function훼푖(푡), 훽푖(푡)such that(i)|푓 1 (푡, 푥(푡 − 휏))| ≤ Σ푚푖=1훼푖(푡)|푥(푡 − 휏푖)|,(ii)‖푔(푡, 푥(푡 − 휏))‖^2 퐹≤Σ푚푖=1훽푖(푡)|푥(푡 − 휏푖)|^2 ,where휏=max1≤푖≤푚휏푖,휆max(⋅)denotes the largest eigenvalue
of a symmetric matrix. Then we derive the following theorem.Theorem 9.Assume that there exist positive constants휅 1 ,휅 2 ,
휅 3 such that휅 1 +(푝−1)휅 2 +
(푝 − 2) 휅 3
2
>0,
휅 2 +휅 3 >0
(18)
hold, where휅 1 =휆max(퐴),휅 2 =Θ푚푖=1훼푖(푡),휅 3 =(푝−1)
Θ푚푖=1훽푖(푡), and if the conditions of Theorems 5 and 6 are
satisfied, then the trivial solution of system( 17 )is p-moment
exponentially stable.Proof.Let푃=푄푇푄,and푉(푡, 푥) = (푥푇푃푥)푝/2
=|푄푥|푝.Then
by Itoformula,wehaveˆL푉=푉푡+푉푥[퐴푥 + 푓 1 (푥(푡−휏))]+
1
2
Tr{푔푇푉푥푥푔}=푝|푄푥|푝−1[푄퐴푥+푄푓 1 (푡, 푥(푡−휏))]+
푝(푝−1)
2
×|푄푥|푝−2Tr{푔푇(푡, 푥(푡−휏))푃푔(푡, 푥(푡−휏))}.
(19)By condition (ii), we haveTr{푔푇(푡, 푥(푡−휏))푃푔(푡, 푥(푡−휏))}