Tính ổn định hậu chắc chắn của một dạng phương trình vi phân nhiễu ngẫu nhiên có trễ

dx(t) = f(x(t), t)dt + F (ˆx(t), t)dt + G(ˆx(t), t)dW (t)
ˆx(t) = {x(t + s) : −τ  s  0}
dx(t) = f(x(t), x(t − τ), t)dt.
dx(t) = f(x(t), x(t − τ), t)dt + σ(t)dW (t).
(Ω, F, P) {F
t
}
W = {W
t
}
t0
{F
t
}
t0
.
0  a  b < ∞. M
2
([a, b]; R)
f = {f(t)}
t0
(F
t
)
f
2
a,b
= E


b
a
|f(t)|
2
dt < ∞.
f f M
2
([a, b]; R) f −f
2
a,b
= 0
f f f = f.
 · 
a,b
M
2
([a, b]; R)
f ∈ M
2
([a, b]; R)

f ∈ M
2
([a, b]; R) f
f(t) = lim sup
h↓0
1
h


t
t−h

f(s)ds.
f f = f.
f ∈ M
2
([a, b]; R)
g = {g(t)}
atb
[a, b] a = t
0
< t
1
< . . . <
t
k
= b ξ
i
, 0  i  k − 1 ξ
i
F
t
i
g(t) = ξ
0
I
[t
0
,t

1
]
(t) +
k−1

i=1
ξ
i
I
(t
i
,t
i+1
]
(t).
M
0
([a, b]; R).
M
0
([a, b]; R) ⊂ M
2
([a, b]; R)
f ∈ M
2
([a, b]; R) {g
n
}
lim
n→+∞

E

b
a
|f(t) − g
n
(t)|
2
dt = 0.
g
M
0
([a, b]; R). g
{W
t
}

b
a
g(t)dW
t
=
k−1

i=0
ξ
i
(W
t
i+1

− W
t
i
).
f ∈ M
2
([a, b]; R)
{g
n
}
lim
n→∞
E

b
a
|f(t) − g
n
(t)|
2
dt = 0.
E





b
a
g

n
(t)dW
t
−

b
a
g
m
(t)dW
t




2
= E





b
a
[g
n
(t) − g
m
(t)]dW
t





2
= E

b
a
|g
n
(t) − g
m
(t)|
2
dt → 0 m, n → ∞.


b
a
g
n
(t)dW
t

L
2
(Ω; R)
f ∈ M
2

([a, b]; R)
f {W
t
} [a, b]

b
a
f(t)dW
t

b
a
f(t)dW
t
= L
2
− lim
n→∞

b
a
g
n
(t)dW
t
,
{g
n
} M
0

([a, b]; R)
lim
n→∞
E

b
a
|f(t) − g
n
(t)|
2
dt = 0.
{g
n
}
f, g ∈ M
2
([a, b]; R) α, β

b
a
f(t)dW
t
F
b
E

b
a
f(t)dW

t
= 0;
E




b
a
f(t)dW
t



2
= E

b
a
|f(t)|
2
dt;

b
a
[αf(t) + βg(t)]dW
t
= α

b

a
f(t)dW
t
+ β

b
a
g(t)dW
t
.
T > 0 f ∈ M
2
([0, T ]; R). 0  a < b  T, {f(t)}
atb
∈
M
2
([a, b]; R)

b
a
f(t)dW
t

b
a
f(t)dW
t
+


c
b
f(t)dW
t
=

c
a
f(t)dW
t
,
0  a < b < c  T.
f ∈ M
2
([a, b]; R) t ∈ [a, b]
I(a) = 0; I(t) =

t
a
f(τ)dW
τ
∀ t ∈ (a, b],
f
W
t

t
a
f(τ)dW
τ

f ∈ M
2
([0, T ]; R) {I(t)}
0tT
{F
t
}.
E

sup
0tT





t
0
f(s)dW
s




2

 4E

T
0

|f(s)|
2
ds.
x(t)
t  0
x(t) = x(0) +

t
0
f(s)ds +

t
0
g(s)dB
s
,
f ∈ L
1
(R
+
; R) g ∈ L
2
(R
+
; R). x(t)
dx(t)
dx(t) = f(t)dt + g(t)dB
t
C
2,1

(R
d
× R
+
; R) V (x, t)
R
d
× R
+
x t
V ∈ C
2,1
(R
d
× R
+
; R)
V
t
=
∂V
∂t
, V
x
=

∂V
∂x
1
, . . . ,

∂V
∂x
d

V
xx
=

∂
2
V
∂x
i
∂x
j

d×d
=





∂
2
V
∂x
1
∂x
1

· · ·
∂
2
V
∂x
1
∂x
d
∂
2
V
∂x
d
∂x
1
· · ·
∂
2
V
∂x
d
∂x
d





V ∈ C
2,1

(R × R
+
; R) V
x
=
∂V
∂x
; V
xx
=
∂
2
V
∂x
2
{x(t)}
t0
dx(t) = f(t)dt + g(t)dW
t
,
f ∈ L
1
(R
+
; R) g ∈ L
2
(R
+
; R). V ∈ C
2,1

(R × R
+
; R)
V (x(t), t)
dV (x(t), t) =

V
t
(x(t), t) + V
x
(x(t), t)f(t) +
1
2
V
xx
(x(t), t)g
2
(t)

dt
+V
x
(x(t), t)g(t)dW
t
{x(t)}
t0
dx(t) = f(t)dt + g(t)dW
t
,
f ∈ L

1
(R
+
; R
d
) g ∈ L
2
(R
+
; R
d×m
). V ∈ C
2,1
(R
d
× R
+
; R)
V (x(t), t)
dV (x(t), t) =

V
t
(x(t), t)+V
x
(x(t), t)f(t)+
1
2
trace


g
T
(t)V
xx
(x(t), t)g(t)

dt
+ V
x
(x(t), t)g(t)dW
t
(Ω, F, {F
t
}
t0
, P) {F
t
}
t0
σ
W (t) = (W
1
(t), W
2
(t), . . . , W
m
(t))
T
m
(Ω, F, {F

t
}
t0
, P) τ > 0 C([−τ, 0]; R
n
)
ϕ [−τ; 0] R
n
ϕ = sup
−τθ0
|ϕ(θ)| | · |
R
n
. A A
T
A.
A A |A| =

trace(A
T
A)
A A = sup{|Ax| : |x| = 1}.
A λ
max
λ
min
A. C
b
F
0

([−τ; 0]; R
n
)
C([−τ; 0]; R
n
) F
0
{x(t)}
t−τ
R
n
x
t
= {x(t + θ) : −τ  θ  0} t  0
x
t
C([−τ; 0]; R
n
)
dx(t) = f(x(t), x(t − τ), t)dt + g(x(t), x(t − τ)t)dW (t), t
0
 t  T
x
t
0
= ξ = {ξ(θ) : −τ  θ  0} F
t
0
−
C([−τ, 0]; R

n
) Eξ
2
< ∞.
x(t) = x
0
+

t
t
0
f(x(s), x(s−τ), s)ds+

t
t
0
g(x(s), x(s−τ ), s)dW (s).
R
d
{x(t)}
t∈[t
0
,T ]
{x(t)} (F
t
)
f(x(·), x
(·)
, ·) ∈ L
1

([t
0
, T ]; R
d
) g(x(·), x
(·)
, ·) ∈ L
2
([t
0
, T ]; R
d×m
),
t ∈ [t
0
, T ]
[t
0
, T ]
x(t)
x(t)
P {x(t) = x(t) ∀ t ∈ [t
0
, T ]} = 1.
K K
x, y ∈ R
d
t ∈ [t
0
, T ]

|f(x, yt) − f(x, y, t)|
2
∨ |g(x, y, t) − g(x, y, t)|
2
 K(|x − y|
2
+ |x − y|
2
)
(x, t) ∈ R
d
× [t
0
, T ]
|f(x, y, t)|
2
∨ |g(x, y, t)|
2
 K(1 + |x|
2
+ |y|
2
).
x(t)
x(·) ∈ M
2
([t
0
, T ]; R
d

).
x(t)
E

sup
t
0
tT
|x(t)|
2

 (1 + 3E|x
0
|
2
)e
3K(T −t
0
)(T −t
0
+4)
.
x(·) ∈ M
2
([t
0
, T ]; R
d
).
W (t) = (W

1
(t), · · · , W
m
(t))
T
, t  0
(Ω, F, P)
{F
t
}
t0
p  2. g ∈ M
2
([0, T ]; R
d×m
)
E

T
0
|g(s)|
p
ds < ∞.
E





T

0
g(s)dW (s)




p


p(p − 1)
2

p
2
T
p−2
2
E

T
0
|g(s)|
p
ds.
p = 2
E sup
0tT






t
0
g(s)dW (s)




p


p
3
2(p − 1)

p
2
T
p−2
2
E

T
0
|g(s)|
p
ds
g ∈ L
2

(R
+
; R
d×m
). t  0,
x(t) =

t
0
g(s)dW (s) A(t) =

t
0
|g(s)|
2
ds.
p > 0 c
p
, C
p
c
p
E|A(t)|
p
2
 E

sup
0st
|x(s)|

p

 C
p
E|A(t)|
p
2
t  0.
c
p
=

p
2

p
, C
p
=

32
p

p
2
0 < p < 2;
c
p
= 1, C
p

= 4 p = 2;
c
p
= (2p)
−
p
2
, C
p
=

p
p+1
2(p − 1)
p−1

p
2
p > 2;
g = (g
1
, · · · , g
m
) ∈ L
2
(R
+
; R
1×m
) T, α, β

P

sup
0tT


t
0
g(s)dW (s) −
α
2

t
0
|g(s)|
2
ds

> β

 e
−αβ
.
dx(t) = f(x(t), x(t − τ), t)dt + σ(t)dW (t) t  0,
x(t) = ξ(t), −τ  t  0
f : R
d
× R
d
× R

+
→ R
d
σ : R
+
→ R
d×m
W (t) m ξ ∈ C([−τ, 0]; R
d
), ξ(t)
F
0
x(t, ξ)
x(t, ξ)
γ > 0
lim sup
t→∞
1
t
ln(|x(t, ξ)|)  −γ .
c
1
, c
2
, c
3
(c
1
> c
2

)
x, y ∈ R
d
t  0,
2x
T
f(x, y, t)  −c
1
|x|
2
+ c
2
|y|
2
trace(σ(t), σ(t)
T
)  c
3
e
−c
1
t
.
λ > 0
lim
t→∞
sup
1
t
log|x(t, ξ)|  −λ, h.c.c

ξ
ξ x(t, ξ) = x(t). c
1
> c
2
λ ∈ (0, c
1
)
c
2
e
λτ
c
1
− λ
< 1.
ω ∈ Ω k
0
= k
0
(ω)
C = C(ω) k  k
0

T
0
e
λt
|x(t)|
2

dt  Ck 0  T  k.
e
c
1
t
|x(t)|
2
= |x(0)|
2
+ M(t) +

t
0
e
c
1
s
(c
1
|x(s)|
2
+ 2x(s)
T
f(x(s), x(s − τ), s) + trace(σ(s)σ(s)
T
)ds
 |ξ(0)|
2
+ M(t) + c
3

t + c
2

t
0
e
c
1
s
|x(s − τ)|
2
ds
t  0
M(t) = 2

t
0
e
c
1
s
x(s)
T
σ(s)dW (s)
t = 0.
k = 1, 2, ε > 0
c
2
e
λτ

c
1
− λ
+
2εc
3
c
1
− λ
< 1.
P (ω : sup
0tk
[M(t) −
1
2
εM(t)] > 2ε
−1
log k) 
1
k
2
.
ω ∈ Ω
k
0
= k
0
(ω)
sup
0tk

[M(t) −
1
2
εM(t)]  2ε
−1
log k
k  k
0
M(t) 
1
2
εM(t) + 2ε
−1
log k
0  t  k, k  k
0
M(t) = 4

t
0
e
2c
1
s
|x
T
(s)|
2
|σ(s)|
2

dt.
M(t)  4

t
0
e
2c
1
s
|x(s)|
2
trace(σ(s)σ(s)
T
)ds  4c
3

t
0
e
c
1
s
|x(s)|
2
ds.
M(t)  2εc
3

t
0

e
c
1
s
|x(s)|
2
ds + 2ε
−1
log k,
0  t  k, k  k
0
e
c
1
t
|x(t)|
2
 |ξ(0)|
2
+ c
3
t + 2ε
−1
log k
+ c
2

t
0
e

c
1
s
|x(s − τ)|
2
ds + 2εc
3

t
0
e
c
1
s
|x(s)|
2
ds
0  t  k, k  k
0
|x(t)|
2
 [|ξ(0)|
2
+ c
3
k + 2ε
−1
log k]e
−c
1

t
+ c
2
e
−c
1
t

t
0
e
c
1
s
|x(s − τ)|
2
ds + 2εc
3
e
−c
1
t

t
0
e
c
1
s
|x(s)|

2
ds
0  t  k, k  k
0
ω ∈ Ω 0  T 
k, k  k
0

T
0
e
λt
|x(t)|
2
dt 

T
0
e
λt
[(|ξ(0)|
2
+ c
3
k + 2ε
−1
log k)e
−c
1
t

+ c
2
e
−c
1
t

t
0
e
c
1
s
|x(s − τ)|
2
ds + 2εc
3
e
−c
1
t

t
0
e
c
1
s
|x(s)|
2

ds]dt
= [|ξ(0)|
2
+ c
3
k + 2ε
−1
log k]

T
0
e
−(c
1
−λ)t
dt + J
1
+ J
2
=
1
c
1
− λ
[|ξ(0)|
2
+ c
3
k + 2ε
−1

log k](1 − e
−(c
1
−λ)
) + J
1
+ J
2

1
c
1
− λ
[|ξ(0)|
2
+ c
3
k + 2ε
−1
log k] + J
1
+ J
2
,
J
1
= c
2

T

0
e
−(c
1
−λ)t

t
0
e
c
1
s
|x(s − τ)|
2
dsdt,
J
2
= 2εc
3

T
0
e
−(c
1
−λ)t

t
0
e

c
1
s
|x(s)|
2
dsdt.
J
1
= c
2

T
0
e
c
1
s
|x(s − τ)|
2

T
s
e
−(c
1
−λ)t
dtds
=
c
2

c
1
− λ

T
0
e
c
1
s
|x(s − τ)|
2
|(e
−(c
1
−λ)s
− e
−(c
1
−λ)T
)ds
=
c
2
c
1
− λ

T
0

|x(s − τ)|
2
(e
λs
− e
c
1
s−(c
1
−λ)T
)ds

c
2
c
1
− λ

T
0
|x(s − τ)|
2
e
λs
ds

c
2
c
1

− λ

τ
0
|ξ(s − τ)|
2
e
λs
ds +
c
2
e
λτ
c
1
− λ

T ∨τ
τ
e
λ(s−τ)
|x(s − τ)|
2
ds

c
2
c
1
− λ


τ
0
|ξ(s − τ)|
2
e
λs
ds +
c
2
e
λτ
c
1
− λ

T
0
e
λs
|x(s)|
2
ds.
J
2
= 2εc
3

T
0

e
c
1
s
|x(s)|
2

T
s
e
−(c
1
−λ)t
dtds
=
2εc
3
c
1
− λ

T
0
e
c
1
s
|x(s)|
2
(e

−(c
1
−λ)s
− e
−(c
1
−λ)T
)ds
=
2εc
3
c
1
− λ

T
0
|x(s)|
2
(e
λs
− e
c
1
s−(c
1
−λ)T
)ds

2εc

3
c
1
− λ

T
0
e
λs
|x(s)|
2
ds.

T
0
e
λt
|x(t)|
2
dt 
1
c
1
− λ
[|ξ(0)|
2
+ c
3
k + 2ε
−1

log k] +
2εc
3
c
1
− λ

T
0
e
λs
|x(s)|
2
ds
+
c
2
c
1
− λ

τ
0
e
λs
|ξ(s − τ)|
2
ds +
c
2

e
λτ
c
1
− λ

T
0
e
λs
|x(s)|
2
ds
=
1
c
1
− λ
[|ξ(0)|
2
+ c
3
k + 2ε
−1
log k + c
2

τ
0
e

λs
|ξ(s − τ)|
2
ds]
+
c
2
e
λτ
+ 2εc
3
c
1
− λ

T
0
e
λs
|x(s)|
2
ds.
e
λt
|x(t)|
2
= |x(0)|
2
+ N(t)
+


t
0
e
c
1
s
(λ|x(s)|
2
+ 2x(s)
T
f(x(s), x(s − τ), s) + trace(σ(s)σ(s)
T
)ds
 |x(0)|
2
+ N(t) +

t
0
e
λs
(λ|x(s)|
2
− c
1
|x(s)|
2
+ c
2

|x(s − τ)|
2
+ c
3
e
−c
1
s
)ds
= |x(0)|
2
+ N(t) − (c
1
− λ)

t
0
e
λs
|x(s)|
2
ds
+ c
2

t
0
e
λs
|x(s − τ)|

2
ds + c
3

t
0
e
−(c
1
−λ)s
ds
= |x(0)|
2
+ N(t) − (c
1
− λ)

t
0
e
λs
|x(s)|
2
ds
+ c
2

t
0
e

λs
|x(s − τ)|
2
ds +
c
3
c
1
− λ
(1 − e
−(c
1
−λ)t
)
 |ξ(0)|
2
+ N(t) +
c
3
c
1
− λ
+ c
2

t
0
e
λs
|x(s − τ)|

2
ds, ∀t  0,
N(t) = 2

t
0
e
λs
x(s)
T
σ(s)dW (s).
t = 0
P (ω : sup
0tk
[N(t) −
1
2
N(t)] > 2 log k) 
1
k
2
.
ω ∈ Ω
k
1
= k
1
(ω)
sup
0tk

[N(t) −
1
2
N(t)]  2 log k
0  t  k k  k
1
N(t) 
1
2
N(t) + 2 log k 0  t  k; k  k
1
N(t)  4

t
0
e
2λs
|x(s)|
2
trace(σ(s)σ(s)
T
)ds
 4c
3

t
0
e
2λs−c
1

s
|x(s)|
2
ds
 4c
3

t
0
e
λs
|x(s)|
2
ds.
N(t) 
1
2
N(t) + 2 log k  2c
3

t
0
e
λs
|x(s)|
2
ds + 2 log k
0  t  k, k  k
1
.

e
λt
|x(t)|
2
 |ξ(0)|
2
+ 2 log k +
c
3
c
1
− λ
+ c
2

t
0
e
λs
|x(s − τ)|
2
ds
+ 2c
3

t
0
e
λs
|x(s)|

2
ds
= |ξ(0)|
2
+ 2 log k +
c
3
c
1
− λ
+ c
2

τ
0
e
λs
|x(s − τ)|
2
ds
+ c
2

t∨τ
τ
e
λs
|x(s − τ)|
2
ds + 2c

3

t
0
e
λs
|x(s)|
2
ds
= |ξ(0)|
2
+ 2 log k +
c
3
c
1
− λ
+ c
2

τ
0
e
λs
|x(s − τ)|
2
ds
+ c
2
e

λτ

t∨τ
τ
e
λ(s−τ)
|x(s − τ)|
2
ds + 2c
3

t
0
e
λs
|x(s)|
2
ds
= |ξ(0)|
2
+ 2 log k +
c
3
c
1
− λ
+ c
2

τ

0
e
λs
|ξ(s − τ)|
2
ds
+ (2c
3
+ c
2
e
λτ
)

t
0
e
λs
|x(s)|
2
ds
0  t  k, k  k
1
e
λt
|x(t)|
2
 |ξ(0)|
2
+2 log k+

c
3
c
1
− λ
+c
2

τ
0
e
λs
|ξ(s−τ)|
2
ds+(2c
3
+c
2
e
λτ
)Ck
0  t  k, k  k
0
∨ k
1
ω ∈ Ω k − 1  t  k k  k
0
∨ k
1
1

t
log(e
λt
|x(t)|
2
) 
1
k − 1
log(|ξ(0)|
2
+ 2 log k +
c
3
c
1
− λ
+ c
2

τ
0
e
λs
|ξ(s − τ)|
2
ds + (2 + c
2
)Ck).
lim sup
t→∞

1
t
log(e
λt
|x(t)|
2
)  0
lim sup
t→∞
1
t
log(e
λt
|x(t)|
2
) = λ + 2 lim sup
t→∞
1
t
log |x(t)|.
lim sup
t→∞
1
t
log |x(t)| 
−λ
2
c
1
÷ c

3
Q d × d
∀x, y ∈ R
d
t  0
x
T
(Q + Q
T
)f(x, y, t)  −c
1
x
T
Qx + c
2
y
T
Qy
(σ(t), σ(t)
T
)  c
3
e
−c
1
t
c
1
> c
2

ξ x(t, ξ) = x(t). c
1
> c
2
λ ∈ (0, c
1
)
c
2
e
λτ
c
1
− λ
< 1.
ω ∈ Ω k
0
= k
0
(ω)
C = C(ω) k  k
0

T
0
e
λt
x
T
(t)Qx(t)dt  Ck 0  T  k.

V (x, t) = e
c
1
t
x
T
Qx
V (x(t), t) = V (x(0), 0) + M(t) +

t
0
e
c
1
s
(c
1
x(s)
T
Qx(s)
+ x(s)
T
(Q + Q
T
)f(x(s), x(s − τ), s) +
1
2
trace(σ(s)(Q + Q
T
)σ(s)

T
)ds
 λ
max
(Q)|x(0)|
2
+M(t)+
1
2
c
3
λ
max
(Q+Q
T
)t+c
2

t
0
e
c
1
s
x
T
(s−τ)Qx(s−τ)ds
t  0
M(t) =


t
0
e
c
1
s
x(s)
T
(Q + Q
T
)σ(s)dW (s)
t = 0. k = 1, 2,
ε > 0
c
2
e
λτ
+ εc
3
c
1
− λ
< 1.
P (ω : sup
0tk
[M(t) −
1
2
εM(t)] > 2ε
−1

log k) 
1
k
2
.
ω ∈ Ω
k
0
= k
0
(ω)
sup
0tk
[M(t) −
1
2
εM(t)]  2ε
−1
log k
k  k
0
M(t) 
1
2
εM(t) + 2ε
−1
log k
0  t  k, k  k
0
M(t) 


t
0
e
2c
1
s
x
T
(s)(Q + Q
T
)x(s)trace(σ(s)σ
T
(s))ds
 2c
3

t
0
e
c
1
s
x
T
(s)Qx(s)ds.
M(t)  εc
3

t

0
e
c
1
s
x
T
(s)Qx(s)ds + 2ε
−1
log k,
0  t  k, k  k
0
V (x(t), t)  λ
max
(Q)|ξ(0)|
2
+
1
2
c
3
λ
max
(Q + Q
T
)t + 2ε
−1
log k
+ εc
3


t
0
e
c
1
s
x
T
(s)Qx(s)ds + c
2

t
0
e
c
1
s
x
T
(s − τ)Qx(s − τ)ds
0  t  k, k  k
0
ω ∈ Ω 0  T 
k, k  0

T
0
x(t)
T

Qx(t)e
λt
dt  [λ
max
(Q)|ξ(0)|
2
+
1
2
c
3
λ
max
(Q + Q
T
)k
+ 2ε
−1
log k]

T
0
e
−(c
1
−λ)t
dt + J
1
+ J
2

J
1
= c
2

T
0
e
−(c
1
−λ)t

t
0
e
c
1
s
x
T
(s − τ)Qx(s − τ)dsdt,
J
2
= εc
3

T
0
e
−(c

1
−λ)t

t
0
e
c
1
s
x(s)
T
Qx(s)dsdt.
J
1
= c
2

T
0
e
c
1
s
x
T
(s − τ)Qx(s − τ)

T
s
e

−(c
1
−λ)t
dtds
=
c
2
c
1
− λ

T
0
e
c
1
s
x
T
(s − τ)Qx(s − τ)(e
−(c
1
−λ)s
− e
−(c
1
−λ)T
)ds
=
c

2
c
1
− λ

T
0
x
T
(s − τ)Qx(s − τ)(e
λs
− e
c
1
s−(c
1
−λ)T
)ds

c
2
c
1
− λ

T
0
x
T
(s − τ)Qx(s − τ)e

λs
ds

c
2
c
1
− λ

τ
0
x
T
(s − τ)Q(s − τ)e
λs
ds
+
c
2
e
λτ
c
1
− λ

T ∨τ
τ
e
λ(s−τ)
x

T
(s − τ)Qx(s − τ)ds

c
2
c
1
− λ

τ
0
x
T
(s − τ)Qx(s − τ)e
λs
ds +
c
2
e
λτ
c
1
− λ

T
0
e
λs
x
T

(s)Qx(s)ds.
J
2
= εc
3

T
0
e
c
1
s
x
T
(s)Qx(s)

T
s
e
−(c
1
−λ)t
dtds
=
εc
3
(c
1
− λ)


T
0
e
c
1
s
x
T
(s)Qx(s)(e
−(c
1
−λ)s
− e
−(c
1
−λ)T
)ds
=
εc
3
(c
1
− λ)

T
0
x(s)Qx(s)(e
λs
− e
c

1
s−(c
1
−λ)T
)ds

εc
3
(c
1
− λ)

T
0
e
λs
x
T
(s)Qx(s)ds.
J
1
J
2

T
0
e
λt
x
T

(t)Qx(t)dt
 [λ
max
(Q)|ξ(0)|
2
+
1
2
c
3
λ
max
(Q + Q
T
)k + 2ε
−1
log k]

T
0
e
−(c
1
−λ)t
dt
+
c
2
c
1

− λ
λ
max
(Q)

τ
0
x
T
(s − τ)Qx(s − τ)e
λs
ds +
c
2
e
λτ
c
1
− λ

T
0
e
λs
x(s)
T
Qx(s)ds
+
εc
3

(c
1
− λ)

T
0
e
λs
x
T
(s)Qx(s)ds