for all F(t) satisfying F
T
ðtÞFtðÞr I, if and only if there exists some
ε4 0, such that
Q þ
εDD
T
þε
1
E
T
Eo 0: ð11Þ
Lemma 2.3 ([26]). Let f
1
,f
2
, …,f
n
:R
m
-R have positive values in an
open subsets D of R
m
. Then, the reciprocally convex combination of f
i
over D satisfies
min
f
α
i
j
α
i
4 0;
P
i
α
i
¼ 1g
X
i
1
α
i
f
i
ðtÞ¼
X
i
f
i
ðtÞþmax
g
i;j
ðtÞ
X
i a j
g
i;j
ðtÞ;
subject to
g
i;j
: R
m
↦R; g
j;i
ðtÞ9g
i;j
ðtÞ;
f
i
ðtÞ g
i;j
tðÞ
g
j;i
ðtÞ f
j
ðtÞ
"#
Z 0
()
:
Lemma 2.4 ([26]). For k
i
tðÞA ½0 ; 1,
P
N
i ¼ 1
k
i
ðtÞ¼1, and vectors
η
i
which satisfy η
i
¼ 0 with k
i
ðtÞ¼0, matrices R
i
4 0, there exist
matrices S
ij
ði ¼ 1; 2; …N 1; j ¼ iþ1; …; NÞ, satisfies
R
i
n
S
ij
R
j
hi
Z 0 ; such
that the following inequality holds
X
N
i ¼ 1
1
k
i
ðtÞ
η
T
i
R
i
η
i
Z
η
1
η
2
⋮
η
N
2
6
6
6
6
4
3
7
7
7
7
5
T
R
1
… S
1N
n
… S
2N
⋮ … ⋮
n
… R
N
2
6
6
6
4
3
7
7
7
5
η
1
η
2
⋮
η
N
2
6
6
6
6
4
3
7
7
7
7
5
:
Lemma 2.5 ([27]). For any given positive matrix Z 4 0, the following
inequality holds for differentiable function x(t) in ½a; b-R
n
:
Z
b
a
_
x
T
ðsÞZ
_
xðsÞdsZ
1
a b
η
T
Z
4
η;
where
η ¼
xðbÞxðaÞ
xðbÞþxðaÞ
2
b a
R
b
a
xðsÞds
"#
and Z
4
¼ diag½Z; 3Z:
3. Main results
In this section, the main results are going to be obtained by
using LMI approach. Firstly, the following notations are given
Σ
þ
¼ diag½σ
þ
1
; …; σ
þ
n
and Σ
¼ diag½σ
1
; …; σ
n
;
η
1
ðtÞ¼col
xðtÞ
R
t
t h
1
xðsÞds
R
0
h
3
R
t
t þ
θ
f ðxðsÞÞdsdθ
hi
;
η
2
ðtÞ¼col
xðtÞ xðt h
1
Þ … xðtðm 1Þh
1
Þ
hi
;
η
3
ðtÞ¼col
xðt
τ
2
Þ xðt τ
2
h
2
Þ … xðtτ
2
ðl 1Þh
2
Þ
hi
;
f ð
η
2
ðtÞÞ ¼ col
f ðxðtÞÞ f ðxðt h
1
ÞÞ … f ðxðt ðm1Þh
1
ÞÞ
hi
;
f ð
η
3
ðtÞÞ ¼ col
f ðxðt
τ
2
ÞÞ f ðxðt τ
2
h
2
ÞÞ …
f ðxðt
τ
2
ðl 1Þh
2
ÞÞ
;
η
4
ðtÞ¼col
Z
t
t
ρ
1
ðtÞ
f ðxðsÞÞds
Z
t
ρ
1
ðtÞ
t 2
ρ
1
ðtÞ
f ðxðsÞÞds…
"
Z
t q 1ðÞ
ρ
1
ðtÞ
t
τ
3
ðtÞ
f ðxðsÞÞds
#
;
ξ
T
ðtÞ¼
η
T
2
ðtÞ η
T
3
ðtÞ x
T
ðt τ
þ
2
Þ x
T
ðt τ
2
ðtÞÞ η
T
3
ðt ρðtÞÞ
f
T
ðη
2
ðtÞÞf
T
η
3
ðtÞ
f
T
ðxðt τ
þ
2
ÞÞ f
T
ðxðt τ
2
ðtÞÞÞ
_
x
T
ðtÞ η
T
4
ðtÞ
Z
t
τ
3
ðtÞ
t
τ
3
ðtÞ
ρ
1
ðtÞ
f
T
ðxðsÞÞds
Z
t
t h
1
x
T
ðsÞds
1
τ
1
ðtÞτ
1
Z
t
τ
1
t
τ
1
ðtÞ
x
T
ðsÞds
1
τ
þ
1
τ
1
ðtÞ
Z
t
τ
1
ðtÞ
t
τ
þ
1
x
T
ðsÞds
1
τ
2
ðtÞτ
2
Z
t
τ
2
t
τ
2
ðtÞ
x
T
ðsÞds
1
τ
þ
2
τ
2
ðtÞ
Z
t
τ
2
ðtÞ
t
τ
þ
2
x
T
ðsÞds
_
x
T
ðt τ
1
ðtÞÞ x
T
ðt τ
1
Þ
x
T
ðt τ
1
ðtÞÞ x
T
ðt τ
þ
1
Þ
Z
0
h
3
Z
t
t þ
θ
f
T
ðxðsÞÞdsdθ
#
;
e
k
¼
O
nðk 1Þn
I
n
O
nð2m þ 3l þ q þ 16 kÞn
hi
;
r
k
¼
O
nðk 1Þn
I
n
O
nð2m þ 3l þ q þ 10 kÞn
hi
:
Next, the main results are discussed as follows.
Theorem 3.1. Under Assumptions 2.1 – 2.3, the neutral-type uncer-
tain neural networks (6) is robustly stochastically stable in mean
square, if there exist matrices P ¼½P
ij
33
4 0 ,V
j
4 0 j ¼ð1; 2; …; mÞ,
F
k
4 0 ,F
k1
4 0,G
k
,k¼ð1; 2; ‥; lÞ,R
11
,R
12
,R
13
,R
r
,r¼ð2; 3; …; lÞ,
S
1
4 0 ,S
2
4 0,S
1i
4 0,S
2i
4 0 ,U
j
4 0,J
j
4 0 j ¼ð1; 2; 3; 4Þ,U
n
1
,Z
1
4 0,
Z
2
4 0,Z
1i
4 0,Z
2i
4 0 ,T4 0 ,T
i
4 0,
Λ
jp
j ¼ð1; 2; ‥ ;
mÞ; p ¼ð1; 2; …; 2 m þ 3lþqþ 16Þ,N
1i
,N
2i
,N
3i
,N
4i
,M
i1
,M
i2
,M
i3
,M
i4
,
P
1
,P
2
,H
1
,H
2
,Q
1
,Q
2
, positive definite diagonal matrices K
1
; K
2
; L
1
; L
2
and positive scalars ϵ
i
4 0 ,i¼ð1; 2; …NÞ such that the following LMIs
hold
Φ
1
¼
P
1
H
1
n
Q
1
"#
4 0;
Φ
2
¼
P
2
H
2
n
Q
2
"#
4 0; ð12Þ
Φ
3k
¼
F
k
G
T
k
n
F
k
"#
4 0 ; ðk ¼ 1; 2; ‥; lÞ; ð13Þ
Φ
4k
¼
F
k
R
11
R
12
n
F
k
R
13
nn
F
k
2
6
4
3
7
5
4 0; if k ¼ 1;
F
k
R
k
n
F
k
"#
4 0; if k ¼ 2; 3; …; l;
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
ð14Þ
Φ
5
¼
U
1
U
n
1
n
U
1
"#
4 0 ; ð15Þ
Φ
6i
¼
Z
n
2i
M
i
n
Z
n
2i
"#
4 0; M
i
¼
M
i1
M
i2
M
i3
M
i4
"#
; ð16Þ
X
N
j ¼ 1
π
ij
S
1j
S
1
o 0;
X
N
j ¼ 1
π
ij
S
2j
S
2
o 0 ;
X
N
j ¼ 1
π
ij
Z
1j
Z
1
o 0;
X
N
j ¼ 1
π
ij
Z
2j
Z
2
o 0;
X
N
j ¼ 1
π
ij
T
j
T o 0; ð17Þ
Φ
ðkÞ
7i
¼
Ξ
ðkÞ
i
Λ
1
Λ
2
… Λ
m
Δ
T
1i
Δ
T
2i
n
V
1
0 … 000
nn
V
2
… 000
⋮⋮ ⋮… ⋮⋮⋮
nn n
… V
m
00
nn n
…
n
ϵ
1
i
I 0
nn n
…
nn
ϵ
i
I
2
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
5
o 0 ; ð18Þ
where
Ξ
ðkÞ
i
¼ W
1
þW
T
1
þ
X
5
k ¼ 2
W
k
þW
6i
þW
7
þW
8i
þ
X
11
k ¼ 9
W
k
þW
12i
þW
13
þW
T
14i
W
15i
þW
T
15i
W
14i
þ
X
m
j ¼ 1
Λ
j
e
j
þ
X
m
j ¼ 1
e
T
j
Λ
T
j
X
m
j ¼ 1
Λ
j
e
j þ 1
X
m
j ¼ 1
e
T
j þ 1
Λ
T
j
þΥ þ Υ
ðkÞ
þ Υ
ðkÞ
;
Λ
j
¼ col½Λ
j1
; Λ
j2
; …; Λ
j;2m þ 3l þ q þ 16
; ðj ¼ 1; 2; …; mÞ;
Z
n
1i
¼ diagðZ
1i
; 3Z
1i
Þ; Z
n
2i
¼ diagðZ
2i
; 3Z
2i
Þ; Σ
1
¼ Σ
þ
Σ
;
C. Yin et al. / Neurocomputing 207 (2016) 437–449 439