TELK
OMNIKA
Indonesian
Journal
of
Electrical
Engineering
V
ol.
12,
No
.
7,
J
uly
2014,
pp
.
5430
5437
DOI:
10.11591/telk
omnika.v12.i7.5505
5430
Stoc
hastic
Sync
hr
onization
of
Neutral-type
Chaotic
Mark
o
vian
Neural
Netw
orks
with
Impulsive
Eff
ects
Cheng-De
Zheng*
and
Xixi
Lv
School
of
Science
,
Dalian
Jiaotong
Univ
ersity
No
.
794,
Huanghe
Road,
Dalian,
116028,
P
.
R.
China
*Corresponding
author
,
e-mail:
15566913851@163.com
Abstract
This
paper
studies
the
globally
stochastic
synchronization
prob
lem
f
or
a
class
of
neutr
al-type
chaotic
neur
al
netw
or
ks
with
Mar
k
o
vian
jumping
par
ameters
under
impulsiv
e
per
turbatio
ns
.
By
vir
tue
of
dr
iv
e-
response
concept
and
time-dela
y
f
eedbac
k
control
techniques
,
b
y
using
the
L
y
a
puno
v
functional
method,
Jensen
integ
r
al
inequality
,
a
no
v
el
reciprocal
co
n
v
e
x
lemma
and
the
free-w
eight
matr
ix
method,
a
no
v
el
sufficient
condition
is
der
iv
ed
to
ensure
t
he
asymptotic
synchronization
of
tw
o
identi
cal
Mar
k
o
vian
jumping
chaotic
dela
y
ed
neur
al
netw
or
ks
with
impulsiv
e
per
turbation.
The
proposed
results
,
which
do
not
require
the
diff
erentiability
and
monoton
icity
of
the
activ
ation
functions
,
can
be
easily
chec
k
ed
via
Ma
tlab
softw
are
.
Finally
,
a
n
umer
ical
e
xample
with
their
sim
ulations
is
pro
vided
to
illustr
ate
the
eff
ectiv
eness
of
the
presented
synchronization
scheme
.
K
e
yw
or
ds:
Stochastically
asymptotic
synchronization,
chaotic
neur
al
netw
or
ks
,
Mar
k
o
vian
jump
,
impulse
,
reciprocal
con
v
e
x
Cop
yright
c
2014
Institute
of
Ad
v
anced
Engineering
and
Science
.
All
rights
reser
v
ed.
1.
Intr
oduction
The
prob
lem
of
synchronization
ar
ises
in
n
umerous
pr
actical
prob
lems
in
ph
ysics
,
ecol-
ogy
,
and
ph
ysiology
.
In
1990,
the
pioneer
ing
w
or
k
of
P
ecor
a
and
Carroll
[6]
brought
attention
to
the
impor
tance
of
control
and
synchronization
of
chaotic
systems
.
In
their
seminal
paper
,
P
ec-
or
a
and
Carrol
proposed
the
dr
iv
e-response
concept
f
or
constr
ucting
synchronization
of
coupled
chaotic
systems
.
The
idea
is
to
use
the
output
of
the
dr
iving
system
to
control
the
response
sys-
tem
so
that
the
y
oscillate
in
a
synchronization
manner
.
Since
then,
chaos
synchronization
has
been
widely
in
v
estigated
with
a
vie
w
to
its
applications
in
secure
comm
unication
systems
[8].
Mar
k
o
vian
jump
system,
introduced
b
y
Kr
aso
vskii
and
Lidskii
in
1961,
is
a
special
class
of
h
ybr
id
systems
.
In
a
Mar
k
o
vian
jump
system,
the
r
andom
jump
of
par
ameters
is
go
v
er
ned
b
y
a
Mar
k
o
v
process
which
tak
es
v
alues
in
a
finite
set.
Thus
,
Mar
k
o
v
jump
systems
can
descr
ibe
some
ph
ysical
systems
with
abr
upt
v
ar
iations
v
er
y
w
ell,
e
.g.,
solar
ther
mal
centr
al
receiv
ers
,
economic
systems
[5],
and
so
on.
Recently
,
a
lot
of
research
results
on
the
stability
analysis
f
or
dela
y
ed
neur
al
netw
or
ks
with
Mar
k
o
vian
jumping
par
ameters
ha
v
e
been
repor
ted,
see
,
f
or
instance
,
[8].
Impulsiv
e
eff
ect
is
lik
ely
to
e
xist
in
a
wide
v
ar
iety
of
e
v
olutionar
y
processes
in
which
states
are
changed
abr
uptly
at
cer
tain
moments
of
time
in
the
fields
such
as
medicine
and
biology
,
eco-
nomics
,
electronics
and
telecomm
unications
.
Neur
al
netw
or
ks
are
often
subject
to
impulsiv
e
per-
turbation
that
in
tur
n
af
f
ect
dynamical
beha
viors
of
systems
.
Theref
ore
,
it
is
necessar
y
to
consider
both
the
impulsiv
e
eff
ect
and
dela
y
eff
ect
when
in
v
estigating
the
stability
of
neur
al
netw
or
ks
.
So
f
ar
,
se
v
er
al
interesting
results
ha
v
e
been
repor
ted
that
ha
v
e
f
ocused
on
the
impulsiv
e
eff
ect
of
dela
y
ed
neur
al
netw
or
ks
[1,
7].
Motiv
ated
b
y
af
orementioned
discussion,
this
paper
in
v
estigates
the
globally
stochastic
synchronization
of
a
class
of
neutr
al-type
chaotic
neur
al
netw
or
ks
with
Mar
k
o
vian
jumping
pa-
r
ameters
under
impulsiv
e
per
turbations
.
The
mix
ed
dela
ys
consists
of
discrete
and
distr
ib
uted
time-v
ar
ying
dela
ys
.
By
vir
tue
of
dr
iv
e-response
concept
and
time-d
ela
y
f
eedbac
k
control
tech-
niques
,
b
y
using
the
L
y
apuno
v
functional
method,
Jensen
integ
r
al
inequality
,
a
no
v
el
reciprocal
Receiv
ed
December
25,
2013;
Re
vised
March
5,
2014;
Accepted
March
28,
2014
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
2302-4046
5431
con
v
e
x
lemma
and
the
free-w
eight
matr
ix
method,
a
no
v
el
sufficient
condition
is
der
iv
ed
to
as-
sure
the
stochastic
synchronization
of
tw
o
identical
Mar
k
o
vian
jumping
chaotic
dela
y
ed
neur
al
netw
or
ks
with
impulsiv
e
per
turbation.
The
proposed
results
,
which
do
not
require
the
diff
eren-
tiability
and
monotonicity
of
the
activ
ation
functions
,
can
be
easily
chec
k
ed
via
Matlab
softw
are
.
Finally
,
a
n
umer
ical
e
xample
with
their
sim
ulations
is
pro
vided
to
illustr
ate
the
eff
ectiv
eness
of
the
presented
synchronization
scheme
.
Notations
:
Throughout
this
paper
,
W
T
;
W
1
denote
the
tr
anspose
and
the
in
v
erse
of
a
square
matr
ix
W
;
respectiv
ely
.
W
>
0(
<
0)
denotes
a
positiv
e
(negativ
e)
definite
symmetr
ic
matr
ix,
I
denotes
the
identity
matr
ix
with
compatib
le
dimension,
the
symbol
“*”
denotes
a
b
loc
k
that
is
readily
inf
erred
b
y
symmetr
y
.
The
shor
thand
col
f
M
1
;
M
2
;
:::;
M
k
g
denotes
a
column
matr
ix
with
the
matr
ices
M
1
;
M
2
;
:::;
M
k
:
sym
(
A
)
is
defined
as
A
+
A
T
;
diag
fg
stands
f
or
a
diagonal
or
b
loc
k-diagonal
matr
ix.
F
or
>
0
;
C
[
;
0];
R
n
denotes
the
f
amily
o
f
contin
uous
functions
from
[
;
0]
to
R
n
with
the
nor
m
jj
jj
=
sup
s
0
j
(
s
)
j
:
Moreo
v
er
,
let
(
;
F
;
P
)
be
a
complete
probability
space
with
a
filtr
ation
f
F
t
g
t
0
satisfying
the
usual
conditions
and
E
fg
representing
the
mathematical
e
xpectation.
Denote
b
y
C
p
F
0
[
;
0];
R
n
the
f
amily
of
all
bounded,
F
0
-measur
ab
le
,
C
[
;
0];
R
n
-v
alued
r
andom
v
ar
iab
les
=
f
(
s
)
:
s
0
g
such
that
sup
s
0
E
j
(
s
)
j
p
<
1
:
jj
jj
stands
f
or
the
Euclidean
nor
m;
Matr
ices
,
if
not
e
xplicitly
stated,
are
assumed
to
ha
v
e
compatib
le
dimensions
.
2.
Pr
ob
lem
description
and
preliminaries
In
this
paper
,
w
e
consider
the
f
ollo
wing
neutr
al-type
chaotic
neur
al
netw
or
ks
with
Mar
k
o-
vian
jumping
par
ameters
under
impulsiv
e
per
turbations
8
<
:
_
x
(
t
)
=
C
(
(
t
))
x
(
t
)
+
A
(
(
t
))
g
(
x
(
t
))
+
B
(
(
t
))
g
(
x
(
t
(
t;
(
t
)))
+
D
(
(
t
))
R
t
t
(
t
)
g
(
x
(
s
))d
s
+
E
(
(
t
))
_
x
(
t
(
t
))
+
J
;
x
(
t
)
=
'
1
(
t
)
;
s
2
[
^
;
0]
;
(1)
where
x
(
t
)
=
(
x
1
(
t
)
;
x
2
(
t
)
;
:::;
x
n
(
t
))
T
2
R
n
is
the
state
v
ector
associated
with
n
neurons
,
real
con-
stant
matr
ices
C
(
(
t
))
;
A
(
(
t
))
;
B
(
(
t
))
;
D
(
(
t
))
;
E
(
(
t
))
are
the
interconnection
matr
ices
repre-
senting
the
w
eight
coefficient
s
of
the
neurons
.
g
(
x
(
t
))
=
g
1
(
x
1
(
t
))
;
g
2
(
x
2
(
t
))
;
:::;
g
n
(
x
n
(
t
))
T
2
R
n
denotes
the
neur
al
activ
ation
function.
The
bounded
functions
(
t
)
;
(
t
)
represent
un
kno
wn
time-
v
ar
ying
dela
ys
with
0
(
t;
(
t
))
(
(
t
))
;
_
(
t;
(
t
))
0
(
(
t
))
0
<
1
;
0
(
t
)
;
_
(
t
)
0
<
1
;
0
(
t
)
;
_
(
t
)
0
<
1
;
where
;
;
are
positiv
e
scalars
,
^
=
max
f
;
;
g
:
J
is
an
e
xter
nal
input,
'
1
(
t
)
is
a
real-v
alued
initial
v
ector
function
that
is
contin
uous
on
the
inter
v
al
[
^
;
0]
.
f
(
t
)
;
t
0
g
is
a
homogeneous
,
fin
ite-state
Mar
k
o
vian
process
with
r
ight
contin
uous
tr
ajector
ies
and
taking
v
alues
in
finite
set
N
=
f
1
;
2
;
:::;
N
g
based
on
giv
en
probability
space
(
;
F
;
P
)
with
and
the
initial
model
0
:
Let
=
[
ij
]
N
N
denote
the
tr
ansition
r
ate
matr
ix
with
tr
ansition
probability:
P
(
(
t
+
)
=
j
j
(
t
)
=
i
)
=
ij
+
o
(
)
;
i
6
=
j
;
1
+
ii
+
o
(
)
;
i
=
j
;
where
>
0
;
lim
!
0
+
o
(
)
=
0
and
ij
is
the
tr
ansition
r
ate
from
mode
i
to
mode
j
satisfying
ij
0
f
or
i
6
=
j
with
ii
=
N
X
j
=1
;j
6
=
i
ij
;
i;
j
2
N
:
F
or
c
o
n
v
enience
,
each
possib
le
v
alue
of
(
t
)
is
denoted
b
y
(
2
N
)
in
the
sequel.
Then
w
e
ha
v
e
A
=
A
(
(
t
))
;
B
=
B
(
(
t
))
;
C
=
C
(
(
t
))
;
D
=
D
(
(
t
))
;
E
=
E
(
(
t
))
:
Throughout
this
paper
,
w
e
mak
e
the
f
ollo
wing
assumptions:
Synchronization
of
Neutr
al-type
Chaotic
Mar
k
o
vian
Impulsiv
e
...
(Cheng-De
Zheng)
Evaluation Warning : The document was created with Spire.PDF for Python.
5432
ISSN:
2302-4046
Assumption
1
.
Each
neur
al
activ
ation
function
g
j
(
)(
j
=
1
;
2
;
:::;
n
)
is
b
ounded,
diff
eren-
tiab
le
and
satisfies
the
f
ollo
wing
condition
j
g
j
(
)
g
j
(
)
+
j
;
8
;
2
R
;
6
=
;
where
j
;
+
j
are
kno
wn
real
constants
.
F
or
simplicity
,
w
e
denote
1
=
diag
1
;
2
;
;
n
;
2
=
diag
+
1
;
+
2
;
;
+
n
;
3
=
diag
1
+
1
;
2
+
2
;
;
n
+
n
g
;
4
=
1
2
diag
1
+
+
1
;
2
+
+
2
;
;
n
+
+
n
.
The
system
(1)
is
considered
as
a
dr
iv
e
system,
the
correspon
ding
response
system
of
(1)
is
giv
en
in
the
f
ollo
wing
f
or
m:
8
>
>
<
>
>
:
_
y
(
t
)
=
C
y
(
t
)
+
A
g
(
y
(
t
))
+
B
g
(
y
(
t
(
t
))
+
D
R
t
t
(
t
)
g
(
y
(
s
))d
s
+
E
_
y
(
t
(
t
))
+
J
+
u
(
t
)
;
t
>
0
;
t
6
=
t
k
;
y
(
t
k
)
=
y
(
t
k
)
y
(
t
k
)
=
k
y
(
t
k
)
x
(
t
k
)
;
k
2
Z
+
;
y
(
t
)
=
'
2
(
t
)
;
s
2
[
^
;
0]
;
(2)
where
y
(
t
)
=
(
y
1
(
t
)
;
y
2
(
t
)
;
:::;
y
n
(
t
))
T
2
R
n
is
the
state
v
ector
associated
with
n
neurons
,
u
(
t
)
=
(
u
1
(
t
)
;
:::;
u
n
(
t
))
T
2
R
n
is
the
state
f
eedbac
k
controller
giv
en
to
achie
v
e
the
e
xponential
synchro-
nization
betw
een
the
dr
iv
e
and
response
systems
,
k
is
a
kno
wn
matr
ix,
'
2
(
t
)
is
a
real-v
alued
contin
uous
v
ector
function
on
the
inter
v
al
[
^
;
0]
:
In
order
to
in
v
estigate
the
synchronization
f
or
t
he
chaotic
dela
y
ed
neur
al
netw
or
ks
with
impulsiv
e
per
turbation,
e
j
(
t
)
=
y
j
(
t
)
x
j
(
t
)
is
defined
as
the
synchronization
error
,
where
x
j
(
t
)
and
y
j
(
t
)
are
the
i
-th
state
v
ar
iab
les
of
dr
iv
e
system
(1)
and
response
system
(2),
respectiv
ely
.
Theref
ore
,
the
error
dynamical
system
betw
een
(1)
and
(2)
is
giv
en
as
f
ollo
ws:
8
>
>
<
>
>
:
_
e
(
t
)
=
C
e
(
t
)
+
A
f
(
e
(
t
))
+
B
f
(
e
(
t
(
t
))
+
D
R
t
t
(
t
)
f
(
e
(
s
))d
s
+
E
_
e
(
t
(
t
))
+
u
(
t
)
;
t
>
0
;
t
6
=
t
k
;
e
(
t
k
)
=
e
(
t
k
)
e
(
t
k
)
=
k
e
(
t
k
)
;
k
2
Z
+
;
e
(
t
)
=
'
(
t
)
:
=
'
2
(
t
)
'
1
(
t
)
;
t
2
[
^
;
0]
;
(3)
where
e
(
t
)
=
(
e
1
(
t
)
;
e
2
(
t
)
;
:::;
e
n
(
t
))
T
;
f
j
(
e
j
(
t
))
=
g
j
(
y
j
(
t
))
g
j
(
x
j
(
t
))
:
In
this
paper
,
the
control
input
v
ector
with
state
f
eedbac
k
is
designed
as
f
ollo
ws:
u
(
t
)
=
Y
1
e
(
t
)
+
Y
2
e
(
t
(
t
))
:
(4)
Theref
ore
,
it
f
ollo
ws
from
[2]
that
system
(3)
admits
a
tr
ivial
solution
e
(
t
)
=
0
:
The
de
v
elopment
of
the
w
or
k
in
this
paper
requires
the
f
ollo
wing
lemmas
.
Lemma
1
(see
[4]).
Let
z
(
t
)
2
R
n
has
contin
uous
der
iv
ed
function
_
z
(
t
)
on
inter
v
al
[
a;
a
+
!
]
;
then
f
or
an
y
n
n
matr
ix
>
0
;
the
f
ollo
wing
inequality
holds:
Z
a
+
!
a
_
z
T
(
s
)
_
z
(
s
)d
s
2
!
1
!
Z
a
+
!
a
z
(
s
)d
s
z
(
a
)
T
1
!
Z
a
+
!
a
z
(
s
)d
s
z
(
a
)
:
Lemma
2
(see
[9]).
Assume
th
at
;
;
#
;
#
are
real
scalars
such
that
1
;
+
4
;
and
#
<
#:
Let
#
:
R
!
(
#
;
#
)
be
a
real
function.
Then
f
or
an
y
non-negativ
e
scalars
a;
b;
the
f
ollo
wing
inequality
holds
a
#
(
t
)
#
b
#
#
(
t
)
1
#
#
max
f
a
b;
a
b
g
:
TELK
OMNIKA
V
ol.
12,
No
.
7,
J
uly
2014
:
5430
5437
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
2302-4046
5433
3.
Main
result
No
w
,
w
e
begin
to
state
our
result
f
or
error
system
(3)
with
input
(4).
Theorem
1
.
Assume
that
Assumpt
ion
1
hold,
the
dr
iv
e
system
(1)
and
the
respon
se
system
(2)
with
(4)
can
be
stochastically
asymptotically
synchroniz
ed
in
mean
square
if
there
e
xist
positiv
e
definite
matr
ices
P
;
Q
;
R
;
S
;
Z
;
U
i
(
i
=
1
;
:::;
7)
;
positiv
e
diagonal
matr
ices
;
;
T
;
W
;
real
matr
ices
X
1
;
X
2
of
appropr
iate
dimensions
such
that
N
X
j
=1
j
Q
j
<
U
1
;
N
X
j
=1
j
R
j
<
U
2
;
(5)
N
X
j
=1
j
S
j
<
U
3
;
N
X
j
=1
j
j
S
<
U
4
;
(6)
P
(
I
k
)
P
P
0
;
k
2
Z
+
;
(7)
(
I
k
)
T
P
(
I
k
)
P
l
;
l
6
=
;
l
;
2
N
;
(8)
4(
$
8
$
2
)
T
S
(
$
8
$
2
)
<
0
;
(9)
4(
$
9
$
3
)
T
S
(
$
9
$
3
)
<
0
;
(10)
where
=
[
ij
]
9
9
;
$
i
=
0
(
i
1)
n
n
I
0
(10
i
)
n
n
;
i
=
1
;
2
;
:::;
10
;
with
1
;
1
=
Q
+
U
1
+
U
6
W
3
+
X
N
j
=1
j
P
j
;
1
;
4
=
W
4
;
1
;
7
=
P
1
+
2
C
Z
+
X
1
;
2
;
2
=
(1
0
)
Q
T
3
2
S
+
X
N
j
=1
j
j
Q
;
2
;
5
=
T
4
;
2
;
7
=
X
2
;
2
;
8
=
2
S
;
3
;
3
=
U
6
2
S
;
3
;
9
=
2
S
;
4
;
4
=
R
+
U
2
+
2
U
5
W
;
4
;
7
=
+
A
T
Z
;
5
;
5
=
(1
0
)
R
T
+
X
N
j
=1
j
j
R
;
5
;
7
=
B
T
Z
;
6
;
6
=
U
5
;
6
;
7
=
D
T
Z
;
7
;
7
=
2
S
+
2
2
U
3
+
U
4
+
U
7
2
Z
;
7
;
10
=
Z
E
;
8
;
8
=
2
S
;
9
;
9
=
2
S
;
10
;
10
=
(1
0
)
U
7
;
j
=
max
f
j
;
0
g
;
and
the
control
gain
matr
ices
Y
1
and
Y
2
in
(4)
are
giv
en
as
Y
T
1
=
X
1
Z
1
;
Y
T
2
=
X
2
Z
1
:
Pr
oof
.
Constr
uct
a
L
y
apuno
v-Kr
aso
vskii
functional
in
the
f
ollo
wing
f
or
m
V
(
t;
e
(
t
))
=
e
(
t
)
T
P
e
(
t
)
+
3
X
i
=1
V
i
(
t;
e
(
t
))
;
where
V
1
(
t;
e
(
t
))
=2
n
X
j
=1
Z
e
i
(
t
)
0
n
i
f
i
(
s
)
i
s
+
i
+
i
s
f
i
(
s
)
o
d
s
+
Z
t
t
(
t
)
e
(
s
)
T
Q
e
(
s
)
+
f
(
e
(
s
))
T
R
f
(
e
(
s
))
d
s
+
Z
t
t
Z
t
_
e
(
s
)
T
S
_
e
(
s
)d
s
d
;
V
2
(
t;
e
(
t
))
=
Z
t
t
Z
t
e
(
s
)
T
U
1
e
(
s
)
+
f
(
e
(
s
))
T
U
2
f
(
e
(
s
))
d
s
d
+
Z
t
t
Z
t
Z
t
_
e
(
s
)
T
U
3
_
e
(
s
)d
s
d
d
+
Z
t
t
Z
t
_
e
(
s
)
T
U
4
_
e
(
s
)d
s
d
;
V
3
(
t;
e
(
t
))
=
Z
t
t
(
t
)
Z
t
f
(
e
(
s
))
T
U
5
f
(
e
(
s
))d
s
d
+
Z
t
t
e
(
s
)
T
U
6
e
(
s
)d
s
+
Z
t
t
(
t
)
_
e
(
s
)
T
U
6
_
e
(
s
)d
s:
Synchronization
of
Neutr
al-type
Chaotic
Mar
k
o
vian
Impulsiv
e
...
(Cheng-De
Zheng)
Evaluation Warning : The document was created with Spire.PDF for Python.
5434
ISSN:
2302-4046
Denoting
e
=
e
(
t
(
t
))
,
calculating
the
w
eak
infinitesimal
oper
ator
along
the
system
(3)
giv
es
L
V
(
t;
e
(
t
))
=2
e
(
t
)
T
P
_
e
(
t
)
+
N
X
j
=1
j
e
(
t
)
T
P
j
e
(
t
)
+
3
X
i
=1
L
V
i
(
t;
e
(
t
))
;
(11)
where
L
V
1
(
t;
e
(
t
))
=2
_
e
(
t
)
T
[
f
(
e
(
t
))
1
e
(
t
)]
+
[
2
e
(
t
)
f
(
e
(
t
))]
+
e
(
t
)
T
Q
e
(
t
)
+
f
(
e
(
t
))
T
R
f
(
e
(
t
))
(1
_
(
t
))
e
T
Q
e
+
f
(
e
)
T
R
f
(
e
)
+
N
X
j
=1
j
Z
t
t
(
t
)
e
(
s
)
T
Q
j
e
(
s
)
+
f
(
e
(
s
))
T
R
j
f
(
e
(
s
))
d
s
+
N
X
j
=1
j
j
(
t
)
e
T
Q
e
+
f
(
e
)
T
R
f
(
e
)
+
2
_
e
(
t
)
T
S
_
e
(
t
)
Z
t
t
_
e
(
s
)
T
S
_
e
(
s
)d
s
+
N
X
j
=1
j
Z
t
t
Z
t
_
e
(
s
)
T
S
_
e
(
s
)d
s
d
+
N
X
j
=1
j
j
Z
t
t
_
e
(
s
)
T
S
_
e
(
s
)d
s;
(12)
L
V
2
(
t;
e
(
t
))
=
e
(
t
)
T
U
1
e
(
t
)
+
f
(
e
(
t
))
T
U
2
f
(
e
(
t
))
Z
t
t
e
(
s
)
T
U
1
e
(
s
)
+
f
(
e
(
s
))
T
U
2
f
(
e
(
s
))
d
s
+
2
2
_
e
(
t
)
T
U
3
_
e
(
t
)
Z
t
t
Z
t
_
e
(
s
)
T
U
3
_
e
(
s
)d
s
d
+
_
e
(
t
)
T
U
4
_
e
(
t
)
Z
t
t
_
e
(
s
)
T
U
4
_
e
(
s
)d
s;
(13)
L
V
3
(
t;
e
(
t
))
=
(
t
)
f
(
e
(
t
))
T
U
5
f
(
e
(
t
))
Z
t
t
(
t
)
f
(
e
(
s
))
T
U
5
f
(
e
(
s
))d
s
+
e
(
t
)
T
U
6
e
(
t
)
e
(
t
)
T
U
6
e
(
t
)
+
_
e
(
t
)
T
U
7
_
e
(
t
)
(1
_
(
t
))
_
e
(
t
(
t
))
T
U
4
_
e
(
t
(
t
))
:
(14)
F
or
0
<
(
t
)
;
define
1
(
t
)
=
1
(
t
)
R
t
t
(
t
)
e
(
s
)d
s:
It
is
easy
to
see
tha
t
1
(
t
)
!
e
(
t
)
while
(
t
)
!
0
:
Theref
ore
w
e
can
define
1
(
t
)
=
e
(
t
)
when
(
t
)
=
0
:
Similar
ly
,
f
or
0
(
t
)
<
;
define
2
(
t
)
=
1
(
t
)
R
t
(
t
)
t
e
(
s
)d
s
;
when
(
t
)
=
;
define
2
(
t
)
=
e
(
t
)
:
F
or
0
<
(
t
)
<
;
utilizing
Lemma
1
giv
es
Z
t
t
_
e
(
s
)
T
S
_
e
(
s
)d
s
=
Z
t
t
(
t
)
_
e
(
s
)
T
S
_
e
(
s
)d
s
Z
t
(
t
)
t
_
e
(
s
)
T
S
_
e
(
s
)d
s
2
(
t
)
1
2
(
t
)
2
;
(15)
where
1
=
[
1
(
t
)
e
]
T
S
[
1
(
t
)
e
]
;
2
=
[
2
(
t
)
e
(
t
)]
T
S
[
2
(
t
)
e
(
t
)]
:
It
is
easy
to
see
that
inequality
(15)
holds
f
or
an
y
t
>
0
with
(
t
)
=
0
or
(
t
)
=
:
F
rom
inequality
(15)
and
Lemma
2,
w
e
get
the
f
ollo
wing
inequality
Z
t
t
_
e
(
s
)
T
S
_
e
(
s
)d
s
2
max
1
3
2
;
3
1
2
:
(16)
Fur
ther
more
,
from
the
Jensen
inequality
w
e
ha
v
e
Z
t
t
(
t
)
f
(
e
(
s
))
T
U
5
f
(
e
(
s
))d
s
Z
t
t
(
t
)
f
(
e
(
s
))d
s
!
T
U
5
Z
t
t
(
t
)
f
(
e
(
s
))d
s
!
:
(17)
TELK
OMNIKA
V
ol.
12,
No
.
7,
J
uly
2014
:
5430
5437
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
2302-4046
5435
Moreo
v
er
,
based
on
(H2),
the
f
ollo
wing
matr
ix
inequalities
hold
f
or
an
y
positiv
e
diagonal
matr
ices
L
;
T
0
e
(
t
)
T
W
3
e
(
t
)
+
2
e
(
t
)
T
W
4
f
(
e
(
t
))
f
(
e
(
t
))
T
W
f
(
e
(
t
))
;
(18)
0
e
T
T
3
e
+
2
e
T
T
4
f
(
e
)
f
(
e
)
T
T
f
(
e
)
:
(19)
Fur
ther
more
,
the
f
ollo
wing
equality
is
tr
ue
f
or
an
y
real
matr
ix
Z
0
=
2
_
e
(
t
)
T
Z
T
_
e
(
t
)
C
e
(
t
)
+
A
f
(
e
(
t
))
+
B
f
(
e
(
t
(
t
))
+
D
Z
t
t
(
t
)
f
(
e
(
s
))d
s
+
E
_
e
(
t
(
t
))
+
Y
1
e
(
t
)
+
Y
2
e
(
t
(
t
))
:
(20)
Denoting
Z
Y
1
=
X
T
1
;
Z
Y
2
=
X
T
2
;
substituting
(12)-(20)
into
(11)
and
taking
mathemati-
cal
e
xpectation
giv
es
d
E
V
(
t;
e
(
t
))
d
t
=
E
(
t
)
T
(
t
)
;
t
2
[
t
k
1
;
t
k
)
;
k
2
Z
+
:
(21)
where
(
t
)
=col
(
e
(
t
)
;
e
;
e
(
t
)
;
f
(
e
(
t
))
;
f
(
e
)
;
Z
t
t
(
t
)
f
(
e
(
s
))d
s;
_
e
(
t
)
;
1
(
t
)
;
2
(
t
)
;
_
e
(
t
(
t
))
)
;
=
+
4
max
(
$
8
$
2
)
T
S
(
$
8
$
2
)
;
(
$
9
$
3
)
T
S
(
$
9
$
3
)
:
W
e
deduce
that
in
equality
<
0
is
equiv
alent
to
in
equalities
(9)
and
(10)
respectiv
ely
.
Theref
ore
,
if
inequalities
(9)
and
(10)
hold,
then
from
(21)
w
e
der
iv
e
that
d
E
V
(
t;
e
(
t
))
d
t
<
0
;
8
t
2
[
t
k
1
;
t
k
)
;
k
2
Z
+
:
(22)
When
t
=
t
k
;
k
2
Z
+
;
from
the
condition
(H5),
w
e
ha
v
e
V
(
t
k
;
e
(
t
k
))
=
V
(
t
k
;
e
(
t
k
))
+
e
(
t
k
)
T
(
I
k
)
T
P
(
I
k
)
P
e
(
t
k
)
:
(23)
On
the
other
hand,
it
f
ollo
ws
from
(7)
that
I
0
0
P
1
P
(
I
k
)
P
P
I
0
0
P
1
0
;
that
is
P
I
k
P
1
0
:
F
rom
the
Schur
complement,
w
e
ha
v
e
P
(
I
k
)
T
P
(
I
k
)
0
:
(24)
Combining
(23)
with
(24),
w
e
can
deduce
that
V
(
t
k
;
e
(
t
k
))
V
(
t
k
;
e
(
t
k
))
;
k
2
Z
+
:
By
simple
calculation,
it
can
be
v
er
ified
from
(8)
that
V
(
t
k
;
e
(
t
k
))
V
l
(
t
k
;
e
(
t
k
))
:
(25)
F
or
t
2
[
t
k
1
;
t
k
]
;
k
2
Z
+
,
in
vie
w
of
(22)
and
(25),
w
e
ha
v
e
V
(
t
k
;
e
(
t
k
))
V
l
(
t
k
;
e
(
t
k
))
V
l
(
t
k
1
;
e
(
t
k
1
))
:
(26)
By
the
similar
proof
and
Mathematical
induction,
w
e
can
der
iv
e
that
(26)
is
tr
ue
f
or
an
y
m;
l
;
(0)
=
0
2
N
;
k
2
Z
+
V
(
t
k
;
e
(
t
k
))
V
l
(
t
k
;
e
(
t
k
))
V
l
(
t
k
1
;
e
(
t
k
1
))
V
0
(
t
0
;
e
(
t
0
))
:
Theref
ore
,
the
system
(4)
is
asymptotically
stab
le
in
mean
square
.
This
completes
the
proof
of
Theorem
1.
Synchronization
of
Neutr
al-type
Chaotic
Mar
k
o
vian
Impulsiv
e
...
(Cheng-De
Zheng)
Evaluation Warning : The document was created with Spire.PDF for Python.
5436
ISSN:
2302-4046
4.
Illustrative
e
xample
In
this
section,
w
e
giv
e
a
e
xample
to
demonstr
ate
the
eff
ectiv
eness
of
our
theoretical
results
.
Example
1
.
Consider
system
(1)
with
n
=
N
=
2
and
the
f
ollo
wing
par
ameters:
A
1
=
1
:
9
0
:
18
4
:
2
3
:
0
;
A
2
=
2
:
0
0
:
23
4
:
1
3
:
1
;
B
1
=
1
:
8
0
:
3
0
:
4
2
:
7
;
B
2
=
1
:
9
0
:
4
0
:
3
2
:
8
;
C
1
=
2
:
7
0
0
2
:
2
;
C
2
=
2
:
8
0
0
2
:
1
;
D
1
=
2
I
;
D
2
=
1
:
8
0
0
2
:
1
;
E
1
=
0
:
25
I
;
E
2
=
0
:
5
0
:
4
1
0
:
6
;
J
=
0
:
The
activ
ation
functions
are
g
1
(
x
)
=
g
2
(
x
)
=
tanh(
x
)
;
and
the
time-v
ar
ying
dela
ys
are
1
(
t
)
=
0
:
8
+
0
:
3
sin
t;
2
(
t
)
=
0
:
65
+
0
:
25
sin
t;
(
t
)
=
0
:
5
+
0
:
3
cos
t;
(
t
)
=
0
:
6
+
0
:
2
cos
t:
Then
Assumption
1
is
satisfied
with
1
=
0
;
2
=
I
;
3
=
0
;
4
=
0
:
5
I
and
=
1
=
1
:
1
;
2
=
0
:
9
;
0
1
=
0
:
3
;
0
2
=
0
:
25
;
=
0
:
8
;
0
=
0
:
3
;
=
0
:
8
;
0
=
0
:
2
:
In
this
paper
,
the
tr
ansition
r
ate
matr
ix
is
giv
en
as
f
ollo
ws
=
0
:
7
0
:
7
0
:
3
0
:
3
:
Solving
the
LMIs
(5)-(10)
in
Theorem
1
b
y
resor
ting
to
the
Matlab
LMI
Control
T
oolbo
x,
w
e
can
obtain
one
f
easib
le
solution.
The
control
input
v
ector
with
state
f
eedbac
k
is
designed
as
(4)
with
Y
11
=
15
:
4038
I
;
Y
12
=
12
:
7158
I
;
Y
21
=
2
:
0655
I
;
Y
22
=
2
:
4185
I
:
Theref
ore
,
w
e
conclude
that
system
(1)
and
(2)
with
(4)
can
be
stochastically
asymptotically
syn-
chroniz
ed.
−1
.
5
−1
−0
.
5
0
0
.
5
1
1
.
5
2
2
.
5
3
−4
−3
−2
−1
0
1
2
3
4
x1
(t
)
x2
(t
)
Fig.
1.
Chaotic
attr
actor
of
Example
1.
Fig.
1
sho
ws
the
neur
al
netw
or
k
model
has
a
chaotic
attr
actor
with
initial
v
alues
x
1
(
t
)
=
0
:
3
;
x
2
(
t
)
=
0
:
4
;
t
2
[
1
;
0]
:
The
initial
v
alues
of
the
response
system
are
tak
en
as
y
1
(
t
)
=
1
:
1
;
y
2
(
t
)
=
1
:
6
;
t
2
[
1
;
0]
:
Fig.
2
sho
ws
the
error
states
.
By
n
umer
ical
sim
ulation,
w
e
can
see
that
the
dynamical
beha
viors
of
response
system
(2)
synchroniz
e
with
master
system
(1).
5.
Conc
lusion
This
paper
deals
with
the
synchronization
prob
lem
f
or
a
class
of
neutr
al-type
chaotic
neur
al
netw
or
ks
with
both
leakage
dela
y
and
Mar
k
o
vian
jumping
par
ameters
under
impulsiv
e
TELK
OMNIKA
V
ol.
12,
No
.
7,
J
uly
2014
:
5430
5437
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
2302-4046
5437
Fig.
2.
The
error
state
of
t
e
1
(
t
)
e
2
(
t
)
:
per
turbations
.
By
vir
tue
of
dr
iv
e-response
concept
and
time-dela
y
f
eedbac
k
control
techniques
,
b
y
using
the
L
y
apuno
v
functional
method,
Jensen
integ
r
al
inequality
,
a
no
v
el
reciprocal
con
v
e
x
lemma
and
the
free-w
eight
mat
r
ix
method,
a
no
v
el
sufficient
condition
is
der
iv
ed
to
assure
the
stochastic
synchronization
of
tw
o
identical
Mar
k
o
vian
jumping
chaotic
dela
y
ed
neur
al
netw
or
ks
with
impulsiv
e
per
turbation.
The
proposed
results
,
which
do
not
require
the
diff
erentiability
and
monotonicity
of
the
activ
ation
functions
,
can
be
easily
chec
k
ed
via
Matlab
softw
are
.
Finally
,
a
n
umer
ical
e
xample
with
their
sim
ulations
is
pro
vided
to
illustr
ate
the
eff
ectiv
eness
of
the
presented
synchronization
scheme
.
Ac
kno
wledg
ement
This
w
or
k
w
as
suppor
ted
b
y
the
National
Natur
al
Science
F
oundation
of
China
No
.
61273022.
Ref
erences
[1]
M.
Dong,
H.
Zhang,
and
Y
.
W
ang,
“Dynamics
analysis
of
impulsiv
e
stochastic
Cohen-
Grossberg
neur
al
netw
or
ks
with
Mar
k
o
vian
jumping
and
mix
ed
time
dela
ys
,
”
Neurocomputing,
2009,72:1999-2004.
[2]
A.
F
r
iedman,
Stochastic
diff
erential
equations
and
applications
,
Ne
w
Y
or
k:
Academic
Press
,
1976.
[3]
K.
Gu,
“An
integ
r
al
inequality
in
the
stability
prob
lem
of
time-dela
y
systems
,
”
in:
Proc.
39th
IEEE
Conf
.
Decision
and
Control,
Sydne
y
,
A
ustr
alia,
2000,
pp
.
2805-2810.
[4]
Z.
Liu,
J
.
Y
u,
and
D
.
Xu,
“V
ector
Wir
tinger-type
inequality
and
the
stability
analysis
of
dela
y
ed
neur
al
netw
or
k,
”
Comm
un.
Nonlinear
Sci.
Numer
.
Sim
ulat.,
2013,18:
1246-1257.
[5]
M.
Mar
iton,
J
ump
Linear
Systems
in
A
utomatic
Control,
Ne
w
Y
or
k:
Dekk
er
,
1990.
[6]
L.
P
ecor
a,
and
T
.
Carroll,
“Synchronization
in
chaotic
systems
,
”
Ph
ys
.
Re
v
.
Lett.,
1990,64:821-
824.
[7]
X.
Song,
X.
Xin,
and
W
.
Huang,
“Exponential
stability
of
dela
y
ed
and
impulsiv
e
cellular
neu-
r
al
netw
or
ks
with
par
tially
Lipschitz
contin
uous
activ
ation
functions
,
”
Neur
al
Netw
.,
2012,29-
30:80-90.
[8]
Y
.
T
ang,
J
.
F
ang,
and
Q.
Miao
,
“On
the
e
xponential
synchronization
o
f
stochastic
jumping
chaotic
neur
al
netw
or
ks
with
mix
ed
dela
ys
and
sector-bounded
non-linear
ities
,
”
Neurocom-
puting,
2009,72:1694-1701.
[9]
C
.-D
.
Zheng,
Q.-H.
Shan,
H.
Zhang,
and
Z.
W
ang,
“On
stabilization
of
stochastic
Cohen-
Grossberg
neur
al
netw
or
ks
with
mode-dependent
mix
ed
time-dela
ys
and
mar
k
o
vian
s
witch-
ing,
”
IEEE
T
r
ans
.
Neur
al
Netw
.
Lear
n.
Syst.,
2013,(5):800-811.
Synchronization
of
Neutr
al-type
Chaotic
Mar
k
o
vian
Impulsiv
e
...
(Cheng-De
Zheng)
Evaluation Warning : The document was created with Spire.PDF for Python.