D(r(t)) = [A(r(t))x(t) + F (x(t), r(t), u) + H(r(t))u]dt
+ G(x(t), r(t), u)dw(t),
{r(t)}
t≥0
S = {1, 2, , N} Γ = (γ
ij
)
N×N
(r(t + ∆) = j|r(t) = i) =

γ
ij
∆ + o(∆) i = j
1 + γ
ii
∆ + o(∆) i = j
∆ > 0 γ
ij
≥ 0 i j
γ
ii
= −

i=j
γ
ij
.
w(t) = (w
1
t
, , w






A
k
1
(i)
0 ··· 0
0
0
0 ··· 0 A
k
p
(i)






,
A
k
j
(i)
, 1  j  p R
k
j





b
k
1
(i)
0 0 ··· 0 0
0 b
k
2
(i)
0 0 0
··· ···
0 0 0 ··· b
k
p−1
(i)
0
0 0 0 ··· 0 b
k
p
(i)







, , F
q
) : R
q
×S ×R
p
−→ R
q
G = (G
1
, , G
q
) : R
q
×S ×R
p
−→ R
q×l
F (0, i, 0) = G(0, i, 0) = 0
λ > 0 ∀j, 1  j  q, x ∈ R
q
u ∈ R
p
,
|F
j
(x, i)| + |G
j
(x, i)|  λ|π
j


·
(A(i), H(i)) K(i) ∈ R
p×q
(R)
M(i) = A(i) + H(i)K(i)
P (i)
M
T
(i)P (i)D(i) + D
T
(i)P (i)M(i) = −I.
x ≡ 0 (1)
α β
Ex(t, t
0
, x
0
)
2
 αx
0

2
e
−β(t−t
0
)
, t ≥ t
0

K(i) = α
i
H
T
(i)φ
−1
(i)H(i)K(i)φ(i),
x ∈ R
q
α
−q
i
|x|  |φ(i)x|  α
−1
i
|x|.
V ∈ C
2,1
(R
n
× R
+
× S; R
+
) LV R
n
× R
+
× S
R

∂x
1
, ,
∂V
∂x
n
)
V
xx
= (
∂
2
V
∂x
i
∂x
j
)
n×n
.
[4] V (x, i) ∈ C
2,1
(R
n
×R
+
×S; R
+
)
c

N
j=1
γ
ij
D
T
(j)φ(j)D(j)

(2λα
i
D
√
q + qλ
2
)P φ(i)

(2λα
i
D
√
q + qλ
2
)P φ(i) −I

< 0.
[2] R = R
T
M = M
T
M + NR

T
(i)φ(i)P (i)φ(i)F (x, i, u) + x
T
N

j=1
γ
ij
D
T
(j)φ(j)P (j)φ(j)D(j)x
+ x
T
[(A(i) + H(i)K(i))
T
φ(i)P (i)φ(i)D (i)
+ D
T
(i)φ(i)P (i)φ(i)(A(i) + H(i)K(i))]x.
P  = max{P (i), i ∈ S} D = max{D(i), i ∈ S}
LV (x, i) = 2α
i
x
T
φ(i)D
T
(i)P (i)φ(i)F (x, i, u)+
+ x
T
N

γ
ij
D
T
(j)φ(j)P (j)φ(j)D(j)x
+ P 

2α
i
D|φ(i)x||φ(i)F (x, i, u)| + |φ(i)G(x, i, u)|
2

|φ(i)F (x, i, u )| 
√
qλ|φ(i)x|
|φ(i)G(x, i, u)|
2
 qλ
2
|φ(i)x|
2
.
LV (x, i) 

−α
2
i
+(2α
i
D