Báo cáo nghiên cứu khoa học: "Về tính ổn định mũ bình phương trung bình của một lớp hệ vi phân ngẫu nhiên có bước nhảy Markov" ppsx

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

l
t
)
T
l
(Ω, F, ) |.| A
A
T
A = sup{|Ax| : |x| = 1}
(1.1) A(i), D(i) H(i) R
q×q
, R
q×q
R
q×p
k
1
(i), , k
p
(i)

p
j=1
k
j
(i) =
q
A(i)
A(i) =


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

×k
J
A
k
j
(i)
=






0 1 0 ··· 0 0
0 0 1 ··· 0 0
···
0 0 0 ··· 0 1
0 0 0 0 ··· 0 0






;
H(i)
H(i) =








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)








,
b
k
j
(i)
, 1  j  p R
k
j
b
k
j
(i)
=





0
0
1





D(i) A(i)
F = (F
1

, , 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

(x)|
π
j
R
q
R
j
;
u α ∈ R, α > 1 i ∈ S
φ(i) =






α
−1
i
0 ··· 0
0
0
0 ··· 0 α
−q
i







·
(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

.
[3] x ∈ R
q
u ∈ R
p
|φ(i)F (x, i, u )| 
√
qλ|φ(i)x|
|φ(i)G(x, i, u)| 
√
qλ|φ(i)x|.
[3]
α
i
φ
−1
(i)A(i)φ(i) = A(i); α
i
φ
−1
(i)D(i)φ(i) = D(i)
K(i) ∈ R
p×q
K(i)
H(i)K(i) = α
i
φ
−1
(i)H(i)K(i)φ(i)
K(i)

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

LV (x, t, i) = V
t
(x, t, i) + V
x
(x, t, i)f(x, t, i)
+
1
2
trace[g
T
(x, t, i)V
xx
(x, t, i)g(x, t, i)] +
N

j=1
γ
ij
V (x, t, i),
f(x, t, i) = F (x(t), r(t), u)
g(x, t, i) = G(x(t), r(t), u)dw(t)
V
t
=
∂V
∂t
, V
x
= (
∂V

∂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

1
, c
2
c
3
c
1
|x|
2
 V (x, i)  c
2
|x|
2
LV (x, i) < −c
3
|x|
2
(1)
u = K(i)x (1)
α
i
, i ∈ S

−α
2
i
φ
2
(i) +


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

−1
N
T
< 0

M N
N
T
−R

< 0
R = R
T
i ∈ S K(i) ∈ R
p×q
(R)
A(i) + H(i)K(i) = α
i
φ
−1
(i)M(i)φ(i).
V (x, i) = (D(i)x)
T
φ(i)P (i)φ(i)(D (i)x).
LV (x, i) = [G
T
(x, i, u)φ(i)P (i)φ(i)G(x, i, u)]+
2x
T
D

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


j=1
γ
ij
D
T
(j)φ(j)P (j)φ(j)D(j)x
+ α
2
i
x
T
φ(i)[(M(i)
T
P (i)D(i) + D
T
(i)P (i)M(i)]φ(i)x
+ [G
T
(x, i, u)φ(i)P (i)φ(i)G(x, i, u)]
 −α
2
i
x
T
φ(i)φ(i)x + x
T
N

j=1

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

√
qλ+qλ
2
)P 

|φ(i)x|
2
+x
T

N
j=1
γ
ij
D
T
(j)φ(j)P (j)φ(j)D(j)x.
c > 0
LV (x, i)  −c|x|
2
.
(1)