Báo cáo nghiên cứu khoa học: "Về tính ổn định tiệm cận bình ph­ương trung bình của hệ ph­ương trình vi phân ngẫu nhiên có trễ" ppt


dx(t) = (Ax(t) +

r
i=1
A
i
x(t − h
i
))dt + Bx(t)dW (t)
x(u) = ξ(u), ∀u ∈ [−h, 0]; t ≥ 0
. (2.1)
A, A
i
, B ∈ R
d×d
, W (t) h
i
∈ [0, h].
(Ω, , {
t
}
t0
, P )
{
t
}
t0
x
x ∈ R
n

A A A = sup{Ax : x = 1} B
T
B P = (p
ij
)
n×n
λ
max
(P ); λ
min
(P ) P h
C([−h, 0]; R
n
) [−h, 0]
R
n
L
2

t
( [−h, 0]; R
n
) 
t
C([−h, 0]; R
n
)−
ξ = {ξ(u) : −h  u  0}
ξ
2

E
= sup
−hu0
Eξ(u)
2
< ∞,
E(.)
dx(t) = (Ax(t) + A
1
x(t − h))dt + Bx(t)dw(t) t  0 (3.1)
x(u) = ξ(u) −h  u  0, A, A
1
, B ∈ R
n×n
w (Ω, , {
t
}
t0
, P )
ξ ∈ L
2

0
( [ −h, 0 ] ; R
n
) .
(3.1) x(u) = ξ(u)
−h  u  0 x(t, ξ)
(3.1)
ε > 0 r > 0

E|x(t, ξ)|
2
< ε
t > t
0
; ξ ∈ L
2

0
([−h, 0]; R
n
) ξ  r.
(3.1)
E|x(t, ξ)|
2
−→ 0 t −→ ∞. (3.2)

dx(t) = Ax(t)dt + Bx(t)dW (t)
x(t
0
) = I; t  t
0
,
(3.3)
A, B n × n W (t)
(Ω, , {
t
}
t0
, P )

Φ(t, s) (3.3)
(3.3)
K > 0; δ > 0
EΦ(t, t
0
)
2
 Ke
−δ(t−t
0
)
.
Q
A
T
Q + QA + B
T
QB + P = 0. (3.4)
(3.4)
Q = E

∞
0
Φ
T
(t, 0)P Φ(t, 0)dt (3.5)
Φ(t, 0) P
[6] S ∈ R
n×n
P, Q ∈ R

n×n
(P x, x) + 2( Qy, x) − (Sy, y)  ((P + QS
−1
Q
T
)x, x). (3.6)
& [5], p.169
x(t, ξ) x
t
(ξ) = {x(t + θ, ξ) :
−h  θ  0} V : R
+
× C([−h; 0]; R
n
) −→ R
c
1
|ϕ(0)|
2
 V (t, ϕ)  c
2
ϕ
2
∀(t, ϕ) ∈ R
+
× C([−h, 0]; R
n
)
EV (t
2

, x
t
2
(ξ)) − EV (t
1
, x
t
1
(ξ))  −c
3

t
2
t
1
E|x(s, ξ)|
2
ds t
2
> t
1
 0,
c
1
; c
2
; c
3
Q
x

T
Qx 
λ
max
(P )K
δ
x
2
. (4.1)
(3.3)
(Qx, x) = E

∞
0
(P Φ(t, 0)x, Φ(t, 0)x)dt  λ
max
(P )x
2
E

∞
0
Φ(t, 0)
2
dt
 λ
max
(P )x
2
K


∞
0
e
−δt
dt 
λ
max
(P )K
δ
x
2
,
(Qx, x) 
λ
max
(P )K
δ
x
2
.
A
1
 <
δ
2K
. (4.2)
x(t) (3.1)
x
t

(s) = x(t + s), s ∈ [−h; 0] Q (3.4)
P = αI.
V (x
t
, t) = (Qx(t), x(t)) +

0
−h
x(t + τ)
2
dτ.
(4.1)
λ
min
(Q)x(t)
2
 V (x
t
, t)  (
λ
max
(P )K
δ
+ h)x
t

2
. (4.3)
dV (x
t

, t) = {((A
T
Q + QA + B
T
QB)x(t), x(t)) + 2(QA
1
x(t), x(t − h))
+x(t)
2
− x(t − h)
2
}dt + M(t).
M(t) = 2x(t)
T
QBx(t)dW (t).
(3.6) (4.1)
2(QA
1
x(t), x(t − h)) − x(t − h)
2
 Q
2
A
1

2
x(t)
2

α

2
K
2
δ
2
A
1

2
x(t)
2
.
dV (x
t
, t)  (−α + 1 +
α
2
K
2
δ
2
A
1

2
)x(t)
2
dt + M(t).
EM(t) = 0
EV (x

t
, t) − EV (x
0
, 0) 

t
0
(−α + 1 +
α
2
K
2
δ
2
A
1

2
)Ex(s)
2
ds. (4.4)
(α−1)
α
2
α = 2
(α − 1)δ
2
α
2
K

2

δ
2
4K
2
. (4.5)
(4.2) (4.5) α
0
2
A
1

2
<
(α
0
− 1)δ
2
K
2
α
2
0
c
3
= α
0
− 1 −
α

2
0
K
2
δ
2
A
1

2
> 0. (4.6)
(4.3), (4.4) (4.6)
c
1
= λ
min
(Q); c
2
=
λ
max
(P )K
δ
+ h; c
3
= α
0
− 1 −
α
2

0
K
2
δ
2
A
1

2
r

i=1
A
i

2
<
δ
2
4rK
2
. (4.7)
x(t) (2.1) x
t
(s) = x(t + s)
s ∈ [−h; 0] Q (3.3) P = αI.
V (x
t
, t) = (Qx(t), x(t)) +
r


i=1

0
−h
i
x
t
(τ)
2
dτ.
(4.1)
λ
min
(Q)x(t)
2
 V (x
t
, t)  (
λ
max
(P )K
δ
+
r

i−1
h
i
)x

t

2
. (4.8)
dV (x
t
, t) = {((A
T
Q + QA + B
T
QB)x(t), x(t)) + 2
r

i=1
(QA
i
x(t), x(t − h
i
))+
+rx(t)
2
−
r

i=1
x(t − h
i
)
2
}dt + M(t).

M(t) = 2x
T
(t)QBx(t)dW (t).
(3.6) (4.1)
2(QA
i
x(t), x(t − h)) − x(t − h
i
)
2
 Q
2
A
i

2
x(t)
2

α
2
K
2
δ
2
A
i

2
x(t)

2
.
dV (x
t
, t)  (−α + r +
α
2
K
2
δ
2
(
r

i=1
A
i

2
))x(t)
2
.dt + M(t).
EM(t) = 0
EV (x
t
, t) − EV (x
0
, 0) 

t

0
(−α + r +
α
2
K
2
δ
2
(
r

i=1
A
i

2
))Ex(s)
2
ds. (4.9)
(α−r)
α
2
α = 2r
(α − r)δ
2
α
2
K
2


δ
2
4rK
2
. (4.10)
(4.7) (4.10) α
0
r

i=1
A
i

2
<
(α
0
− r)δ
2
α
2
0
K
2
c
3
= α
0
− r −
α

2
0
K
2
δ
2
(
r

i=1
A
i

2
) > 0. (4.11)
(4.8), (4.9) (4.11)
c
1
= λ
min
(Q); c
2
=
λ
max
(P )K
δ
+

r

i=1
h
i
; c
3
= α
0
− r −
α
2
0
K
2
δ
2
(

r
i=1
A
i

2
)