Δ푥 (푡푘)=푥(푡+푘)−푥(푡푘)=퐼푘(푥 (푡푘)) ,
푡=푡푘, 푘=1,2,...,푚,
푥(푡 0 )=휉, 푡=[−휏, 0],
(1)
where the initial value 휉∈PC([−휏, 0]; 푅푛), 푥(푡) =
[푥 1 (푡), 푥 2 (푡),...,푥푛(푡)]푇, 푥푡 is regarded as a PC-valued
stochastic process,푥푡={푥(푡+휃):−휏≤휃≤0)},푓:푅+×푅푛×
PC([−휏, 0]; 푅푛)→푅푛,푔:푅+×PC([−휏, 0]; 푅푛)→푅푛×푚,and
푤(푡)is an푚-dimensional standard Brownian motion defined
on the complete probability space.
Definition 1.Let퐶2,1(푅푛×[푡 0 − 휏, ∞); 푅+)denote the family
of all nonnegative functions푉(푥, 푡)on푅푛×[푡 0 −휏,∞)that
are continuously twice differentiable in푥andoncein푡.Fora
푉∈퐶2,1(푅푛×[푡 0 −휏, ∞); 푅+), one can define the Kolmogorov
operatorL푉as follows:
L푉(푥, 푡)=푉푡(푥, 푡)+푉푥(푥, 푡)푓(푥, 푡)
+
1
2
Tr{푔푇(푥, 푡)푉푥푥푔(푥, 푡)},(2)
where푉푡=휕푉(푥, 푡)/휕푡,푉푥=(휕푉(푥, 푡)/휕푥 1 ,...,휕푉(푥,푡)/휕푥푛),
and푉푥푥=휕^2 푉(푥, 푡)/휕푥^2.
Definition 2.The trivial solution of SIS ( 1 )issaidtobethe푝th
moment exponential stable if there exist positive constants
훼>0and퐾≥1such that
퐸‖푥(푡)‖푝≤퐾푒−훼(푡−푡^0 )
儩儩
儩儩푥 0
儩儩
儩儩
푝
,푡>푡 0 ,푡∈푅+. (3)The following lemmas can be found in [ 11 ].Lemma 3.Let푥, 푦 ≥ 0,푎, 푏 > 1,then
푥푦 ≤
푥푎
푎
+
푦푏
푏
,
1
푎
+
1
푏
=1. (4)
Lemma 4.Let푥, 푦 ≥ 0,푝≥푗≥0,then
푥푝−푗푦푗≤
(푝 − 푗) 푥푝+푗푦푝
푝
. (5)
3. Main Results
In this section, we shall focus on sufficient conditions to
achieve exponential stability of the SIS by employing Razu-
mikhin techniques and Lyapunov functions. Moreover, we
will design the impulsive control for the stabilization of
unstable stochastic systems by using the obtained results.
Theorem 5.If there exist positive constants푝, 푐 1 ,푐 2 ,휆,푑푘>1,
and suppose there exists a function푉such that
(i)푐 1 |푥|푝≤ 푉(푥, 푡) ≤ 푐 2 |푥|푝;(ii)퐸푉(푥, 푡+푘)≤푑푘퐸푉(푥, 푡푘);(iii)퐸L푉(휑, 푡) ≤ 푐퐸푉(휑, 푡),forall푡∈(푡푘−1,푡푘];
(iv) ln푑푘≤ 휆(푡푘−푡푘−1),푘=1,2,....Then the corresponding system( 1 )is the푝th moment exponen-
tial stable.Proof.For any푡∈[푡 1 ,푡 2 ], we can get from the conditions (ii)
and (iii)퐸푉(푡)=퐸푉(푡+ 1 )+∫
푡푡 1푐퐸푉(푥(푠),푠)푑푠
≤푑 1 [퐸푉( 0 )+∫
푡 10푐퐸푉(푥(푠),푠)푑푠]
+∫
푡푡 1푐퐸푉(푥(푠),푠)푑푠 = 푑 1 퐸푉( 0 )
+푑 1 ∫
푡 10푐퐸푉(푥(푠),푠)푑푠 +∫
푡푡 1푐퐸푉(푥(푠),푠)푑푠.
(6)
In general for푡∈[푡푘−1,푡푘], one can find that퐸푉(푡)≤ ∏
0≤푡푘≤푡푑푘퐸푉( (^0) )+∫
푡
0
∏
푠≤푡푘≤푡푑푘푐퐸푉(푥(푠),푠)푑푠.
(7)
From condition (iv), we get∏
푠≤푡푘≤푡푑푘≤푒휆(푡^2 −푡^1 )⋅푒휆(푡^3 −푡^2 )⋅⋅⋅푒휆(푡푘−푡푘−1)
=푒휆(푡푘−푡^1 )=푒휆(푡−푠)⋅푒휆(푡푘−푡)⋅푒휆(푠−푡^1 ).
(8)
For푡∈[푡푘−1,푡푘], 푡 1 ,푡 2 ,...,푡푘be impulsive points in[푠, 푡), 푡 >
푠, 휆 < 0,thenweobtain∏
푠≤푡푘≤푡푑푘≤푒휆(푡−푠)⋅푒휆(푡푘−푡)≤푒휆(푡−푠)⋅푒휆(푡푘−푡푘−1)≤훾푒휆(푡−푠). (9)
By ( 7 )and( 8 ), then we can get퐸푉(푡)≤훾퐸푉( 0 )푒휆푡+∫
푡0훾푒휆(푡−푠)푐퐸푉(푥(푠),푠)푑푠
≤훾sup
−휏≤휃≤0퐸푉(휎)푒휆푡.
(10)
It follows from condition (i), that푐 1 퐸|푥(푡)|푝≤퐸푉(푡)≤훾sup
−휏≤휃≤0퐸푉(휎)푒휆푡≤훾푐 2 퐸儩儩儩儩휉儩儩儩儩푝푒휆푡,
(11)
which implies퐸|푥(푡)|푝≤훾푐 2
푐 1
퐸儩儩儩儩휉儩儩儩儩푝푒휆푡. (12)
System ( 1 )isthe푝th moment exponentially stable. The proof
is complete.