International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1571
ISSN 2229-5518
Stochastically bounded Solutions of stochastic integrodifferential equations modeling neural
networks
Zhenkun Huang, Honghua Bin
Abstract— In these paper, by using stochastic integral properties about solutions of homogeneous linear equations and fixed-point theorem, we investigate bounded dynamics of neural networks with stochastic effects and distributed delays. Some new criteria for the existence of a unique stochastic bounded solution of stochastic newworks are given.
Keywords — Stochastically bounded; stochastic neural networks; stochastic integral; distributed delays.
—————————— ——————————
Tochastic differential equations and relative applications have recently been studied intensively [1, 2, 3]. It is of great
supv∈� | f
j (v) |≤ B f j
< +∞,
supv∈� | g
j (v) |≤ Bg j
< +∞,
† ≤ B
< +∞
Moreover,
h (⋅),g
(⋅) and
interest to discuss qualitative behavior such as stochastic
boundedness, exponential stability, periodicity or almost peri-
supv∈ | hij (v) | † ; j j
ij
odicity [6, 7, 8, 11] and so on. It is well known that stochastic boundedness of stochastic differential equations depend on
h† (⋅)
are Lipschitz-continuous with Lipschitz constant
IJSER
it’s linearized homogeneous equations [9]. For this research
L > 0, L > 0
j j
and
L † > 0,
hij
respectively.
direction, II’chenko [5] established existence of a unique sto-
chastically bounded solution of a linear nonhomogeneous dif-
h† ( x) := h ( x) - α
x, where αij ≠ 0 if i =
j and αij =0 if
ferential eqution. Later, Luo [10] extended relative results to a
class nonlinear stochastic differential eqution and reported
i ≠ j .
some criteria foe existence of a stochastically bounded solution.
kij (⋅)
is in
It is shown that such stochastically bounded solutions can in-
L1 (0, +∞) and satisfy with
+ ∞
k (v)dv = k *
∈ (0, +∞) .
herit properties of the coefficients of the equation if they are
either stationary or periodic.
Meanwhile, there will exist interest qualitative behavior
about stochastic boundedness for neural networks with sto-
chastic perturbation. However, so far little is known about the
existence of a unique stochastically bounded solutions of ne-
rual networks and the aim of this paper to is close this gap.
In the present paper, we consider the following stochastic
∫0 ij ij
Based on some stochastic integral properties and fixed-point theorem, we establish new criteria for the existence of a unique stochastically bounded solution for (1.1) The nonautonomous cases are alse considered.
neural networks with distributed delays [4]
Let (� M ,|| ⋅ ||)
be a Banach space. The collection of all
M measurable, square-integrable random variables, denoted by
dxi (t ) = [-ai xi (t ) + ∑ bij f j ( x j (t ))
j =1
L2 (P, � M ) , equipped with norm
|| X || = (E || X ||2 )1/ 2 ,
L2 ( P ,� M )
M t where and the expectation E is defined by
+ ∑ ∫-∞
j =1
kij (t - u) g j ( x j (u))du + Ii ]dt
(1.1)
E[ g ] = ∫Ω
g (ω)dP(ω) . Define
B (� , L2 (P, � M ))
to be
M the collection of all stochastic process x : �
→ L2 (P, � M ) ,
+ ∑ hij ( x j (t ))dw j (t ),
which are continuous and bounded in quadratic mean. It is
j =1
then easy to check that B (� , L2 (P, �
M )) is a Banach space
Where
i ∈M := {1, 2,, M }, w(t ) = (w1 (t ), wM (t ))
is when it is equipped with the norm
M-dimensional independent Wiener processes with respect to a
probability space (Ω, F , Ft , -∞ < t < ∞, P) .Throughout this
|| X ||∞
= sup(E || X ||2 )1/ 2 .
t∈�
paper, for each i, j ∈ M ,we suppose some basic assumptions:
For any given i ∈ M ,It oˆ -type homogeneous linear eqution [1]
dxi (t ) = -αi xi (t )dt + αij xi (t )dwi (t )
ai > 0, bij
and Ii
are real constants;
has a solution
i (t ) = exp{ai (t - s) + αii [ωi (t ) - ωi (s)]}
IJSER © 2013 http://www.ijser.org
International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1572
ISSN 2229-5518
which satisfys with the initial condition xi (s) = 1, where
s -∞
lim ∫ (t )ψ (u)du =∫
u (t )ψ (u)du
a* := -a
- 2-1 a2 < 0 . For any arbitrary p ∈ � ,
s →-∞ i i
i i ii
E( s (t )) p := exp{[a* + p2-1 a2 ](t - s) p}.
(2.1)
lim ∫
s -∞
i (t )ψ (u)dωi (u) =∫
i (t )ψ (u)dωi (u)
i i ii
s →-∞ t t
By It oˆ ’s formula, we can check that xs (t ) of (1.1) can be represented in the following form
exist almost surely for all t ∈ � , respectively.
Lemma 2.3. ([12]) Let {Xt ,Ft ;0 ≤ t ≤ +∞} be a
i i i
t
∫s i
-1 M
∑
ij j j
submartingale whose every path is right-
xs (t ) = s (t )[ x (s) +
M u
( s (u)) [
j =1
b f ( xs (u))
(2.2)
continuous, let [σ ,τ ]
be a suninterval of
+ ∑ ∫-∞
j =1
M t
k (u - v) g ( xs (v))dv + I - α h† ( xs (u))]du
[0, +∞). Then Doob’s maximal inequality holds:
E( sup X ) p ≤ ( p )E( X p ), p > 1
+ ∑ ∫s
i =1
( s (u))-1 h† ( xs (u))dw (u)],
i ∈M
σ ≤t ≤τ
t p -1 r
The following basic definition and three lemmas
Provided
X t ≥ 0 a.s. P for every t ≥ 0 , and
are essential in the proof of our main results.
Definition 2.1. A solution x(t ), t ∈ � , of (1.1) is
said to be stochastically bounded if
lim sup P{| xi (t ) |> N} = 0
E( X p ) < +∞.
For simplicity, for real constant c , denote c :=| c | . Then our main result follows as:
N→+∞ t∈�
holds for each i ∈ M.
solution x-∞
(t ), t ∈ � , of (1.1) if 2a > a2 , i ∈ M , and
2 2 * L 2
Lemma 2.1. ([5]) Assume that ψ (t ) is a continu-
M (L f bij ) (Lg )
kij
( h† )
ous bounded function on t ∈ � . Then the follow-
Θ := max{3M ∑[
* *†††
]} < 1,
j j ij
IJSER
i∈M
j = a a a a a
ing reverse integral formula holds for s ≤ t :
1
†* 1 2 -
i i1
i i1 i 2
†* 2
s t s -1
Where ai1 := αi + 2
αii and ai 2 := 2(αi
+ αii ) . In this case
i (t )∫s ( i (u))
ψ (u)dwi (u) =
of the existence, we have
M
s u s u
-∞ -∞ u -∞
- ∫ i (t )ψ (u)dwi (u) - αii ∫
i (t )ψ (u)du,
x (t ) := -∫t
i (t )[∑ bij f j ( x j
(u))
where
t t
i ∈ M.
j =1
M u
Lemma 2.2. ([5]) Assume that ψ (t ) is a continu-
+ k (u - v) g
( x-∞ (u))dv + I ]du
(2.3)
ous function with supt∈� |ψ (t ) |≤ K < +∞. For
∑ ∫-∞
j =1
M
ij j j i
each i ∈ M,one gets
(i) for ∀r ∈ � and
p ∈ (0,1 + 2a α -2 ),
there
-∞
+ ∫t
j =1
u (t )h† ( x-∞ (u))dω
(u),
i ∈ M.
are constants
0 < q < 1 such that
T > 0, L = L(K ) > 0
and
Moreover,
sup E || x-∞ (t ) ||2 < +∞.
t∈�
t -Tn u r r - n
- P n
P{∫t -T ( n +1)
( i (t ) |ψ (u) |)
dr > N
2 } ≤ LN q
M s
s s u - s
i i i
∑ ∫t
i ij j j
for all t ∈ � , N > 0 and n ∈ � ;
x (t ) =
(t ) x (s) -
j =1
( (t )) 1 h† ( x (u))dω
(u)
(ii) the following two limits
s
- (t ))-1[b f
( xs (u))
∫t i ij j j
M u
∑ ∫-∞
ij j j i
+
j =1
k (u - v) g
( xs (v))dv + I ]du
————————————————
Putting
x(s) = 0 in the above and approaching the limit as
• Honghua Bin is a professor in Jimei University, Xiamen 361021, P. R. China.
(This information is optional; change it according to your need.)
s → -∞ , due to Lemma 2.2, we get the limit x-∞ (t ) which is a solution of (1.1). The process x-∞ (t ) , t ∈ � , is measurable with respect to the flow
Γt := σ {ωk (s2 ) - ωk (s1 ): s1 ≤ s2 ≤ t, k ∈ M} .
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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1573
ISSN 2229-5518
For any given
z-∞ (t) ∈ B (� , L2 (P, � M ))
, define
-∞ u M -∞
E
-∞ -∞ -∞ T
P{| ∫t
[ i (t )(∑ bij f j ( x j
(u))
ϕ(z
(t)) := (ϕ1 (z
(t)),,ϕM (z
-∞
(t)))
M
by j =1
M u
-∞ r
ϕ ( z -∞ (t )) := -
(t )[
b f ( z -∞ (u))
+∑ ∫
k (u - v) g ( x
(v))dv + I ] du |
(2.5)
i ∫t i
∑
j =1
ij j j
-∞
j =1
ij j j i
M u
+ ∑ ∫-∞
kij (u - v) g j ( z j
(v))dv + Ii ]du
> N r } ≤ L N - p ,
where L < +∞ and the integral is defined for all trajectories.
j =1
M -∞
1
In fact, by Lemma 2.2, we have
M
∑ ∫t
i ij j j
P{| ∫
[ i (t )(∑ bij f j ( x j
(u))
-
j =1
(t )h† ( z -∞ (u))dω
(u),
-∞ u
t
-∞
j =1
where i ∈ M. Now we need three steps to complete our proof.
Step1: We will prove that ϕ ( z -∞ (t )) is continuous. Since
M u
+∑ ∫-∞ kij (u - v) g j ( x j
(v))dv + Ii ] du |> N }
-∞ M
j =1
+∞
t -Tn M
∫t + t i
∑ ij j j
≤ P{| ∫
[ i (t )(∑ bij f j ( x j
(u))
E | (t +�t )[
�
b f
j =1
( z -∞ (u))
n =0
u
t -T ( n +1)
-∞
j =1
M u
+ ∑ ∫-∞
kij (u - v) g j ( z j
(v))dv + Ii ]du
M u -∞
+ ∫-∞ kij (u - v) g j ( x j
(v))dv + Ii ] du |> N
2- n }
j =1
j =1
-∞ M
t -Tn M
-∞
- (t )[
b f ( z -∞ (u))
≤ ∑ P{∫
[ i (t )(∑ bij | f j ( x j
(u)) |
∫t i
∑
j =1
ij j j
n =0
t -T ( n +1)
j =1
M IJSEM u R-∞
r r - n
+ k (u - v) g
( z -∞ (v))dv + I ]du |2
+∑ ∫-∞ kij (u - v)| g j ( x j
(v))| dv + Ii )] du |> N 2 }
∑ ∫-∞
j =1
ij j j i
j =1
+∞
t -Tn M M
-∞
∫t + t i
M
∑ ij j j
≤ ∑ P{∫
n =0
[u (t )(∑ b B
j =1
+∑ k * B
j =1
+ I ]r du > N r 2- n }
= (t +�t )[
�
b f
j =1
( z -∞ (u))
t -T ( n +1)
i ij f j
ij g j i
M u
+ ∑ ∫-∞
kij (u - v) g j ( z j
(v))dv + Ii ]du
+∞
≤ ∑ LN - p qn
n =0
≤ L
1 - q
N - p .
j =1
Second, we claim that
Step 2: We will show the process x-∞ (t ) is stochastically
-∞ u
† -∞
r - p
bounded. For any N > 0 ,
P{| ∫t i (t )hij ( x j
(u))dω j (u) |> N
t
} ≤ L2 N ,
(2.6)
-∞
i ∫t i
M
∑ ij j j
for r ∈ � and L2 < +∞ . Let
ω j (u) := ω j (t - u) - ω j (u).
P{| x-∞ (t ) |> N} ≤ P{|
(t )[
b f
j =1
( x-∞ (u))
Since
t -v u (t )h† ( x-∞ (u))dω
(u) =
M u -∞ N
∫t i ij j j
+∑ ∫-∞ kij (u - v) g j ( x j
(v))dv + Ii ]du > }
M +
(2.4)
v t -u
† -∞ t
j =1
1 ∫0 i
(t )hij ( x j
(t - u))dω j (u)
M
∑ ∫t
-∞ -∞ N
i ij j j
is a martingale. Apply Doob’s maximal inequality, there exist a constant c < +∞ such that
+
j =1
P{|
(t )h ( x
(u))dω
(u) |>
},
M + 1
-∞ u
† -∞ r
i ∈ M.
It is sufficient to prove that every term in the right side of (2.4) is
P{| ∫t
i (t )hij ( x j
t -v
(u))dω j (u) |> N }
u † -∞ r
stochastically bounded.
≤ lim P{ sup | ∫
i (t )hij ( x j
(u))dω j (u) |> N }
Let r ∈ � and
p ∈ (0, min {1 + 2a α -2 }).
First, we
V →∞
0<v<V t
i∈M
i ii
-2 r2
t -v u
† -∞
claim that
≤ lim N E[ sup | ∫
i (t )hij ( x j
(u))dω j (u) |]
V →∞
- r
0<v<V t
t -v u -∞
≤ lim N
V →∞
2 2cE[∫ [
(t )h† ( x
(u))] du]
≤ lim N -2 r cE[-
V →∞
-∞
∫t i (t )hij ( x j
(u))] du].
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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1574
ISSN 2229-5518
So it remains to show that
-∞ -∞
lows from Step 1 that ϕ maps B∞ (� , L2 (P, � M )) into
E[-∫t
[u (t )h†2( x
(u))] du] < +∞
itself. To complete the proof, we will prove that ϕ has a
for t ∈ � . Taking r = 2 and
p = 2 + δ (0 < δ < min {2a α 2 -1})
int the prood
unique fix-point. For any
x-∞ (t ), y -∞ (t ) ∈ B∞ (� , L2 (P, � M )), we get
of (2.5), we have
-∞ -∞
i∈M
i ii
ϕi ( x
-∞ (t )) - ϕ ( y -∞
M
(t )) =
E[-∫t
[u (t )h†2( x
(u))] du]
-∞ u
(t )
b [ f
( x-∞ (u)) -( y-∞
(u))]du
+∞ -∞
≤ 1+∑ 2 P{4 ≤ -∫t
n =0
[ i (t )hij ( x j (u))] du ≤ 4 }
∫t i
M u
∑ ∫
∑
j =1
ij j j j
-∞ -∞
+ - -
+∞ -∞
≤ 1+∑ 2 P{4 ≤ -∫t
n =0
[ i (t )hij ( x j (u))] du}
j =1
M
∑
kij (u v)[ g j ( x j
-∞
(v))
g j ( y j
(v))]dv]du
+∞ ∫ u
†† -∞ -∞
≤ 1+ 22( n +1) L 2- n ( 2+δ ) = 1 +
4L1 .
i (t )[hij ( x j
(u))
hij ( y j
(u))]dω j (u),i
- - ∈M
∑ 1
n=0
1 - 2-δ
j =1
Thus, (2.6) holds and we have proved that the solution is
It is obviously that
-∞ -∞
stochastically bounded. Together with (2.5) and (2.6), one gets that
ϕi ( x
M
(t )) - ϕi ( y
-∞
(t ))
P{| x
(t ) |> N} ≤ L3 N ,
(2.7)
≤ ∑ bij ∫
i (t ) | f j ( x j
(u)) - f j ( y j
(u)) | du
-∞ - p
where L3 < +∞ . Using (2.7) for
u
t j =1
M u
-∞ -∞
-∞ -∞
p = 2 + δ (0 < δ < min {2a α 2 -1}) , we get
+∑ ∫
kij (u - v)[ g j ( x j
(v)) - g j ( y j
(v))]dv]du
IJSER
i∈M
+∞
i ii
-∞
j =1
E | x-∞ (t ) |2 ≤ 1 + ∑ 22( n+1) P{2n ≤| x-∞ (t ) |≤ 2n+1}
n=0
+∞
M -∞
+∑ ∫t
j =1
i (t )[hij ( x j (u)) - hij ( y j (u))]dω j
(u),i ∈M
≤ 1 + ∑ 22( n+1) P{2n ≤| x-∞ (t ) |}
n=0
(2.8)
Since (∑i =1 ri )
n1 2
i =1 1 i
, we can write:
+∞ E|ϕ ( x-∞ (t )) - ϕ ( y -∞ (t ))|2
≤ 1 +
∑
n=0
2( n+1) - n ( 2+δ )
3
M
≤ E[∑ 3M (L f bij ) [
-∞
i (t )| x j
(u) - y j
(u)| du]
=1+
4L3
< +∞
j =1
M
2 u
j t
-∞ u u
-∞ -∞ 2
-∞ -∞
1 - 2-δ
+ 3M (L
)2 [
(t )[
k (u - v) |x
(v) - y
(v)| dv]du]2
∞ 2 M
∑
j =1
g j ∫t i
∫-∞
ij j j
Step 2: Let BE (� , L (P, �
)) be the collection of all
M -∞
stochastic bounded process
x : �
→ L2 (P, � M )
with
+∑ 3M (
h† ) [∫
i ( )( j ( )
j ( ))
j ( ) .
j =1
L 2
ij t
t x u
- y -∞ u dω
u ] 2,]
i ∈M
E | x-∞ (t ) | p ≤
4L3
. Obviously, B∞ (� , L2 (P, � M ))
Let p s (t ) = exp{a* (t - s) + 2α
[ω (t ) - ω (s)]}. Then,
1 - 2-δ E
i i ii i i
⊂ B (� , L2 (P, � M )) is a Banach space. Define
E(p s (t )) p := exp{[a* + 2 pα
](t - s) p}
for any
E
-∞ -∞ -∞ T
p ∈ � . Now, using Cauchy-Swhwarz inequality we can
ϕ ( x
(t )) := (ϕ1 ( x
-∞
(t )),ϕM ( x
M
(t ))) by
write:
i ∫t i
∑ ij j j
ϕ ( x-∞ (t )) := -
M u
(t )[
b f
j =1
( x-∞ (u))
+ ∑ ∫-∞
kij (u - v) g j ( x j
(v))dv + Ii ]du
j =1
M -∞
- ∑ ∫t
j =1
i (t )hij ( x j (u))dω j
(u),
where i ∈ M and
x-∞ (t ) ∈ B∞ (� , L2 (P, � M )).
It fol-
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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1575
ISSN 2229-5518
E|ϕ ( x-∞ (t)) - ϕ ( y-∞ (t))|2
M -∞ -∞
Young Professors of Jimei University, the Excellent Youth
Foundation of Fujian Province 2012J06001, NCETFJ
≤ E[∑3M (Lf j bij ) [∫t exp{ai (t - u)}du][
2 *
j =1
∫t i (t) | x j (u) - y j (u) | du]
JA11144 and the Foundation of Fujian Higher Education
JA10184 and JA11154.
M -∞ -∞ u
∑ g j ∫t i
∫t i
∫-∞ ij j j
+
j =1
M
3M (L
)2 [ exp{a* (t - u)}du][
-∞
(t)[
k (u - v) |x-∞ (v) - y-∞ (v) | dv]du]2
∑ h† ∫ i
j j j
[1] L. Arnold, “Stochastic Differential Equations: Theory and Applications,” New
+
j =1
3M (L )2 [
ij t
(t)(x (u) - y (u))dω (u)] ]
York, JOHN WILEY & SONS, 1974.
M
≤ 3M
(L b )2
j
-∞
Ep u (t)du sup E | x-∞ (u) - y-∞ (u) |2
[2] N. Ikeda, S. Vatanabe, “Stochastic Differential Equations and Diffusion Pro-
∑
j =1
M
* ∫t
i
(L )2 -∞
i j j
t∈�
u
cesses,” NorthHoland, Amsterdam, 1981.
[3] Mao, X. “Exponential Stability of Stochastic Differentiao Equations,”
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k (u - v)dv]2 du]sup E | x-∞ (u) - y-∞ (u) |2
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∑
j =1
M
∑
* ∫t i
i
-∞
h† ∫ i
∫-∞ ij
j j
t∈�
j j
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j =1
3M (L )2 [
ij t
Ep u (t)2 du]sup E | x-∞ (u) - y-∞ (u) |2 .
t∈�
318.
It follows from (2.1) that
E|ϕ ( x-∞ (t )) - ϕ ( y -∞ (t ))|2
M (L b )2
[5] O. V. IL’chenko, “Stochastically bounded solutions of a linear homo-
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≤ 3M
f j ij
sup E | x-∞ (u) - y -∞ (u) |2
∑
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* †
i i1
j j
t∈�
Lyapunov function, stability and ultimate boundedness,” J. Math.
Anal. Appl., 212(1997)537-553.
M
+ 3M
(L )2
j (
+∞
k (v)dv)2 sup E | x-∞ (u) - y -∞ (u) |2
[7] Ichikawa, A., “Semelinear stochastic evolution equations: bound-
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* † ∫0 ij
i i1
j j
t∈�
edness, stability and invariant measures,” Stochastics, 12(1984)1-39.
[8] Chow, P.L. “Stability of nonlinear stochastic evolution equations,” J.
M (L )2
IJSER
+∑ 3M
j =1
hij
a†
i 2
sup E | x-∞ (u) - y -∞ (u) |2
t∈�
2 2
Math, Anal, Appl, 89(1982)400-419.
[9] R.Z. Khasminskii, “Stability of Systems of Differential Equations
Under Random Perturbations of their Parameters,” Sijitoff and
M (L b )2
(L )
j
(L † )
hij
-∞ -∞ 2
Noordhoff, Alphen ann Rijn, 1980.
∑ a*a†* ††
a a a t∈ j j
[10] J. Luo, “Stochastically bounded solutions of a nonlinear stochastic
≤ 3M
j =1
[ j
i i1
+ +
i i1
]sup E || x
i 2 �
(u) - y
(u) ||
Thus, it follows that
E||ϕ ( x-∞ (t )) - ϕ ( y -∞ (t ))||2
differential equations,” Journal of Computational and Applied Math-
ematics, 196(2006)87-93.
[11] F. Bedouhene, O. Mellah, P.R. Defitte, “Bochner-almost periodicity
M 2 2
(L )2
for stochastic processes,” Stochastic Analysis and Application
(L f bij ) (Lg ) h†
30(2012)322-342.
i∈M
∑ a*a*†††
a a a
[12] I. Karatzas, S.E. Shreve, “Brownian Motion and Stochastic Calculus,”
≤ max{3M
[
j =1
j
i i1
+ j
i i1
+ ij ]}
i 2
New York, SpringerVerlag, 1988.
× sup E || x-∞ (u) - y -∞ (u) ||2
t∈�
≤ Θ sup E || x-∞ (u) - y -∞ (u) ||2 .
t∈�
Consequently, if Θ < 1 , then (1.1) has a unique fixed- point in B∞ (� , L2 (P, � M )) , which can be express ex- plicitly by (2.3). The proof is complete.
By using stochastic integral properties of homogeneous linear equations and fixed-point theorem, we investigate bounded dynamics of delayed neural networks with sto- chastic effects. Some new criteria for the existence of a unique stochastically bounded solution of stochastic net- works are given. Our results can be generalized to nonau- tonomous cases.
This research was supported by the National Natural Sci- ence Foundation of China 11101187, the Foundation for
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