# Improved Delay-Dependent Stability Analysis for Neural Networks with Interval Time-Varying Delays

Authors: Jun-kang Tian; Yan-min Liu
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The problem of delay-dependent asymptotic stability analysis for neural networks with interval time-varying delays is considered based on the delay-partitioning method. Some less conservative stability criteria are established in terms of linear matrix inequalities (LMIs) by constructing a new Lyapunov-Krasovskii functional (LKF) in each subinterval and combining with reciprocally convex approach. Moreover, our criteria depend on both the upper and lower bounds on time-varying delay and its derivative, which is different from some existing ones. Finally, a numerical example is given to show the improved stability region of the proposed results.


---

## Body

## 1. Introduction

In the past decades, neural networks have been paid much attention due to their strong capability of information processing such as pattern recognition, image processing, fault diagnosis, and associative memories. Meanwhile, due to the finite switching speed of amplifiers, time-delay is inevitably encountered in real-world neural networks. Time-delays often cause instability and oscillation in neural networks. Therefore, many effects have been paid to delay-dependent stability analysis of neural networks with time-delays.For the delay-dependent stability criteria of neural networks with time delays, the main purpose is to obtain a maximum value of the admissible delay such that the concerned systems are asymptotically stable. To enhance the feasible region of stability criteria, by using the free weighting matrix method, a new delay-dependent stability criterion for delayed neural networks was obtained in [1]. But some useful terms were ignored in [1] when estimating the upper bound on the derivative of the Lyapunov functionals. Then, the result was further improved in [2] by considering the relationships among some useful terms adequately. In order to reduce the conservatism, researchers have introduced the augmented Lyapunov functional method [3] and the reciprocally convex approach [4]. Recently, the delay-partitioning approach which divides delay interval into some subintervals was employed to investigate the delay-dependent stability problem of delayed neural networks [5–12]. In [5, 6], the delay-partitioning number was chosen as 2. In [5], some improved stability criteria were obtained by utilizing different free-weighting matrices in each delay subinterval for neural networks with time-varying delays. In [7–12], the delay-partitioning number was chosen as m, where m⩾1. In [8], a weighting delay based method was introduced to derive the stability criteria for recurrent neural networks with time-varying delay. Different from some existing results, by employing weighting delays, the delay interval [0,d(t)] was divided into some variable subintervals. In [10], complete delay-decomposing approach was introduced to derive asymptotic stability criterion for neural networks with time-varying delays by using reciprocally convex technique. Among those above methods, the delay-partitioning method is proven to be more effective than the free-weighting matrix approach and augmented Lyapunov functional method.In the above mentioned results [1–12], the lower bound of time-varying delay for delayed neural networks is restricted to be 0. However, in the real world, the time-varying delay may be an interval delay; that is, the lower bound of the delay is not restricted to be 0. Therefore, many effects have been paid to delay-dependent stability analysis of neural networks with interval time-varying delay [13–17]. In [13], A piecewise delay method is introduced to derive several novel delay-dependent stability criteria for neural networks with interval time-varying delay. By using the delay decomposition method and a new convex combination technique, some new delay-dependent stability criteria were obtained in [14]. Recently, by constructing a new Lyapunov functional and dividing the lower bound of the time-varying delay, the asymptotic stability for cellular neural networks with interval time-varying delays is investigated in [15]. By dividing the lower and upper bounds of the time-varying delay and constructing an improved Lyapunov-Krasovskii functional, [16] obtained some delay-dependent stability criteria in terms of LMIs to reduce the conservatism. The proposed stability conditions are less conservative due to the novel delay-partitioning method and convex combination technique considered. By considering the sufficient information of neuron activation functions, [17] obtained some improved delay-dependent stability criteria for neural networks with interval time-varying delay. However, there are rooms for further improvement. Firstly, in [15], only the delay interval [0,h1] has been divided into m segments. In this case, there is no connection between time-varying delay and each subinterval, which may lead to much conservatism. In [16], the delay intervals [0,h1] and [h1,h2] have been divided into some segments and the information τ(t)∈[h1,h2] is considered, but the relation between the time-varying delay and each subinterval is not adequately considered, which may lead to considerable conservatism. Secondly, many stability conditions ignore the information of the lower bound of delay derivative. In fact, the lower bound of delay derivative plays an important role in improving stability region of the proposed results. Thirdly, the information of neuron activation functions is not adequately considered, which may lead to much conservatism.In this paper, to show the merits of our new delay-partitioning method, we only need to divide the delay intervals[0,h1] and [h1,h2] into 2 segments; that is, [0,h1]=[0,h1/2]∪[h1/2,h1], [h1,h2]=[h1,(h1+h2)/2]∪[(h1+h2)/2,h2], respectively. Some existing results [14–16] only consider the information τ(t)∈[h1,h2] and do not consider the information τ(t)∈[h1,(h1+h2)/2] or τ(t)∈[(h1+h2)/2,h2], which may lead to considerable conservatism. Thus, when dealing with the time derivative of V(zt), not only the information τ(t)∈[h1,h2] but also the information τ(t)∈[h1,(h1+h2)/2] and τ(t)∈[(h1+h2)/2,h2] is considered in our paper. So the information between time-varying delay τ(t) and each subinterval is considered adequately, which may play an important role in reducing conservatism of derived results. By taking more information of the lower bound of delay derivative, an augmented LKF is introduced to derive two stability criteria for neural networks with interval time-varying delay. Then, by employing reciprocally convex approach, some less conservative delay-dependent stability criteria are presented in terms of LMI. Finally, a numerical example is given to show the effectiveness of the proposed method.


## 2. Problem Formulation

Consider the following neural networks with interval time-varying delay:(1)x˙t=-Cxt+Agxt+Bgxt-τt+μ,where x(t)=[x1(t),x2(t),…,xn(t)]T∈Rn is the neuron state vector, g(x(·))=[g1(x1(·)),g2(x2(·)),…,gn(xn(·))]T∈Rn denotes the neuron activation function, and μ=(μ1,μ2,…,μn)T∈Rn is a constant input vector. A∈Rn×n is the connection weight matrix and B∈Rn×n is the delayed connection weight matrix. C=diag(C1,C2,…,Cn) is a diagonal matrix with Ci>0, i=1,2,…,n. τ(t) is time-varying continuous function that satisfies 0⩽h1⩽τ(t)⩽h2, u1⩽τ˙(t)⩽u2, where h1, h2 and u1, u2 are constants. In addition, it is assumed that each neuron activation function gi(·), i=1,2,…,n, satisfies the following condition:(2)ki-⩽gix-giyx-y⩽ki+,∀x,y∈R,x≠y,i=1,2,…,n,where ki-, ki+, i=1,2,…,n are constants.Assuming thatx∗=[x1∗,x2∗,…,xn∗]T is the equilibrium point of (1). By the transformation z(·)=x(·)-x∗, (1) can be converted to the following form:(3)z˙t=-Czt+Afzt+Bfzt-τt,where z(t)=[z1(t),z2(t),…,zn(t)]T, f(z(·))=[f1(z1(·)),f2(z2(·)),…,fn(zn(·))]T, and fi(zi(·))=gi(zi(·)+xi∗)-gi(xi∗), i=1,2,…,n. According to inequality (2), we obtain(4)ki-⩽fizitzit⩽ki+fi0=0,i=1,2,…,n.Lemma 1 (see [18]).
For any constant matrixZ∈Rn×n, Z=ZT>0, scalars h2>h1>0; then(5)-h2-h1∫t-h2t-h1‍xTsZxsds⩽-∫t-h2t-h1‍xTsdsZ∫t-h2t-h1‍xsds.Lemma 2 (see [19]).
Letf1,f2,…,fN:Rm→R have positive values in an open subset D of Rm. Then, the reciprocally convex combination of fi over D satisfies(6)minαi∣αi>0,∑iαi=1∑i‍1αifit=∑i‍fit+maxgi,jt∑i≠j‍gi,jtsubject to(7)gi,j:Rm⟶R,gj,itΔ__gi,jt,fitgi,jtgi,jtfjt⩾0.


## 3. Main Results

Theorem 3.
For given positive scalarsh1,h2, h=h2-h1, and any scalars u1,u2, diagonal matrices K1=diag(k1-,k2-,…,kn-), K2=diag(k1+,k2+,…,kn+), and system (3) is globally asymptotically stable for 0⩽h1⩽τ(t)⩽h2, u1⩽τ˙(t)⩽u2 if there exist symmetric positive matrices P, X=Xij2×2, Q=Qij2×2, P¯=Pij2×2, Ri(i=1,2,3,4), positive diagonal matrices T1, T2, T3, Δ=diag(δ1,δ2,…,δn), Λ=diag(λ1,λ2,…,λn), and any matrices S1, S2 with appropriate dimensions, satisfying the following LMIs:(8)Σ+Φ+Π1ATR∗-R<0,(9)Σ+Φ+Π2ATR∗-R<0,(10)R1S1∗R1>0,(11)R2S2∗R2>0,where(12)
Σ
=
Σ
11
X
12
0
0
0
0
0
Q
12
-
P
12
-
Y
12
∗
X
22
-
X
11
-
X
12
0
0
0
0
0
0
∗
∗
-
X
22
-
Y
0
0
0
0
0
0
∗
∗
∗
Σ
44
0
0
-
1
-
u
2
Q
12
0
0
∗
∗
∗
∗
Σ
55
Σ
56
Σ
57
0
Y
12
∗
∗
∗
∗
∗
Σ
66
Λ
-
Δ
B
0
0
∗
∗
∗
∗
∗
∗
-
1
-
u
2
Q
22
0
0
∗
∗
∗
∗
∗
∗
∗
Q
22
-
P
22
0
∗
∗
∗
∗
∗
∗
∗
∗
Y
22
-
Y
11
,
Φ
=
-
2
K
2
T
3
K
1
0
0
0
0
0
0
T
3
K
1
+
K
2
0
∗
0
0
0
0
0
0
0
0
∗
∗
0
0
0
0
0
0
0
∗
∗
∗
-
2
K
2
T
2
K
1
0
0
T
2
K
1
+
K
2
0
0
∗
∗
∗
∗
-
2
K
2
T
1
K
1
T
1
K
1
+
K
2
0
0
0
∗
∗
∗
∗
∗
-
2
T
1
0
0
0
∗
∗
∗
∗
∗
∗
-
2
T
2
0
0
∗
∗
∗
∗
∗
∗
∗
-
2
T
3
0
∗
∗
∗
∗
∗
∗
∗
∗
0
,
Π
1
=
-
R
1
-
R
4
S
1
0
R
1
-
S
1
0
0
0
0
R
4
∗
-
R
1
-
R
2
R
2
R
1
-
S
1
T
0
0
0
0
0
∗
∗
-
R
2
0
0
0
0
0
0
∗
∗
∗
-
2
R
1
+
S
1
+
S
1
T
0
0
0
0
0
∗
∗
∗
∗
-
R
3
0
0
0
R
3
∗
∗
∗
∗
∗
0
0
0
0
∗
∗
∗
∗
∗
∗
0
0
0
∗
∗
∗
∗
∗
∗
∗
0
0
∗
∗
∗
∗
∗
∗
∗
∗
-
R
3
-
R
4
,
Π
2
=
-
R
1
-
R
4
R
1
0
0
0
0
0
0
R
4
∗
-
R
1
-
R
2
S
2
R
2
-
S
2
0
0
0
0
0
∗
∗
-
R
2
R
2
-
S
2
T
0
0
0
0
0
∗
∗
∗
-
2
R
2
+
S
2
+
S
2
T
0
0
0
0
0
∗
∗
∗
∗
-
R
3
0
0
0
R
3
∗
∗
∗
∗
∗
0
0
0
0
∗
∗
∗
∗
∗
∗
0
0
0
∗
∗
∗
∗
∗
∗
∗
0
0
∗
∗
∗
∗
∗
∗
∗
∗
-
R
3
-
R
4
,
Σ
11
=
X
11
+
Q
11
-
P
11
-
Y
22
,
Σ
44
=
-
1
-
u
2
Q
11
+
1
-
u
1
Y
,
Σ
55
=
-
P
C
-
C
T
P
-
K
2
Δ
-
K
1
Λ
C
-
C
T
K
2
Δ
-
K
1
Λ
+
P
11
+
Y
11
,
Σ
56
=
P
A
-
C
Λ
-
Δ
+
K
2
Δ
-
K
1
Λ
A
+
P
12
,
Σ
57
=
P
B
+
K
2
Δ
-
K
1
Λ
B
,
Σ
66
=
Λ
-
Δ
A
+
A
T
Λ
-
Δ
+
P
22
,
A
=
0
0
0
0
-
C
A
B
0
0
,
R
=
h
2
4
R
1
+
R
2
+
h
1
2
4
R
3
+
R
4
.Proof.
We construct a new LKF as(13)Vzt=∑i=15‍Vizt,where (14)
V
1
z
t
=
∫
t
-
h
1
-
h
/
2
t
-
h
1
‍
z
s
z
s
-
h
2
T
X
11
X
12
∗
X
22
h
h
h
h
h
h
h
h
·
z
s
z
s
-
h
2
d
s
h
h
h
h
h
h
h
+
∫
t
-
h
1
/
2
t
‍
z
s
z
s
-
h
1
2
T
Y
11
Y
12
∗
Y
22
h
h
h
h
h
h
h
·
z
s
z
s
-
h
1
2
d
s
,
V
2
z
t
=
z
T
t
P
z
t
+
2
∑
i
=
1
n
‍
∫
0
z
i
t
‍
λ
i
f
i
s
-
k
i
-
s
d
s
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
+
∫
0
z
i
t
‍
δ
i
k
i
+
s
-
f
i
s
d
s
,
V
3
z
t
=
∫
t
-
τ
t
t
-
h
1
‍
z
s
f
z
s
T
Q
11
Q
12
∗
Q
22
z
s
f
z
s
d
s
h
h
h
h
h
h
h
+
∫
t
-
h
1
t
‍
z
s
f
z
s
T
P
11
P
12
∗
P
22
z
s
f
z
s
d
s
,
V
4
z
t
=
∫
t
-
h
2
t
-
τ
t
‍
z
T
s
Y
z
s
d
s
,
V
5
z
t
=
h
2
∫
-
h
1
+
h
2
/
2
-
h
1
‍
∫
t
+
θ
t
‍
z
˙
T
s
R
1
z
˙
s
d
s
d
θ
h
h
h
h
h
h
h
+
h
2
∫
-
h
2
-
h
1
+
h
2
/
2
‍
∫
t
+
θ
t
‍
z
˙
T
s
R
2
z
˙
s
d
s
d
θ
h
h
h
h
h
h
h
+
h
1
2
∫
-
h
1
/
2
0
‍
∫
t
+
θ
t
‍
z
˙
T
s
R
3
z
˙
s
d
s
d
θ
h
h
h
h
h
h
h
+
h
1
2
∫
-
h
1
-
h
1
/
2
‍
∫
t
+
θ
t
‍
z
˙
T
s
R
4
z
˙
s
d
s
d
θ
.
Remark
4. When constructing the V1(zt), we not only divide the delay interval [0,h1] but also divide the delay interval [h1,h2] into 2 segments; that is, [0,h1]=[0,h1/2]∪[h1/2,h1], [h1,h2]=[h1,(h1+h2)/2]∪[(h1+h2)/2,h2], respectively, which may play an important role in reducing conservatism of derived results.
Remark
5. Since V4(zt) considers the information of the lower bound of delay derivative, the LKF in this paper is more general than that in [14–16]. So the obtained stability criteria may be less conservative than the existing ones.
The time derivative ofV(zt) along the trajectories of system (3) yields (15)V˙zt=∑i=15‍V˙izt,where(16)
V
˙
1
z
t
=
z
t
-
h
1
z
t
-
h
1
+
h
2
2
T
X
11
X
12
∗
X
22
·
z
t
-
h
1
z
t
-
h
1
+
h
2
2
-
z
t
-
h
1
+
h
2
2
z
t
-
h
2
T
X
11
X
12
∗
X
22
·
z
t
-
h
1
+
h
2
2
z
t
-
h
2
-
z
t
z
t
-
h
1
2
T
Y
11
Y
12
∗
Y
22
z
t
z
t
-
h
1
2
-
z
t
-
h
1
2
z
t
-
h
1
T
Y
11
Y
12
∗
Y
22
z
t
-
h
1
2
z
t
-
h
1
,
V
˙
2
z
t
=
2
z
T
t
P
z
˙
t
+
2
f
z
t
-
K
1
z
t
T
Λ
z
˙
t
+
2
K
2
z
t
-
f
z
t
T
Δ
z
˙
t
=
2
z
T
t
P
-
C
z
t
+
A
f
z
t
+
B
f
z
t
-
τ
t
+
2
f
z
t
-
K
1
z
t
T
Λ
+
B
f
z
t
-
τ
t
-
C
z
t
+
A
f
z
t
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
+
B
f
z
t
-
τ
t
+
2
K
2
z
t
-
f
z
t
T
Δ
+
B
f
z
t
-
τ
t
-
C
z
t
+
A
f
z
t
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
+
B
f
z
t
-
τ
t
,
V
˙
3
z
t
⩽
z
t
-
h
1
f
z
t
-
h
1
T
Q
11
Q
12
∗
Q
22
z
t
-
h
1
f
z
t
-
h
1
-
1
-
u
2
z
t
-
τ
t
f
z
t
-
τ
t
T
·
Q
11
Q
12
∗
Q
22
z
t
-
τ
t
f
z
t
-
τ
t
+
z
t
f
z
t
T
P
11
P
12
∗
P
22
z
t
f
z
t
-
z
t
-
h
1
f
z
t
-
h
1
T
P
11
P
12
∗
P
22
z
t
-
h
1
f
z
t
-
h
1
,
V
˙
4
z
t
⩽
1
-
u
1
z
T
t
-
τ
t
Y
z
t
-
τ
t
-
z
T
t
-
h
2
Y
z
t
-
h
2
.
By (16), we obtain(17)V˙1zt+V˙2zt+V˙3zt+V˙4zt⩽ζTtΣζt,where Σ is defined in (8). Consider the following:(18)ζTt=zTt-h1zTt-h1+h22·zTt-h2zTt-τt·zTtfTztfTzt-τt·fTzt-h1fTzt-h12,V˙5zt=z˙Tth22R1+R2+h122R3+R4z˙t-h2∫t-h1+h2/2t-h1‍z˙TsR1z˙sds-h2∫t-h2t-h1+h2/2‍z˙TsR2z˙sds-h12∫t-h1/2t‍z˙TsR3z˙sds-h12∫t-h1t-h1/2‍z˙TsR4z˙sds.Furthermore, there exist positive diagonal matrices T1, T2, T3, such that the following inequalities hold based on (4):(19)0⩽-2fTztT1fzt+2zTtT1K1+K2fzt-2zTtK2T1K1zt-2fTzt-τtT2fzt-τt+2zTt-τtT2K1+K2fzt-τt-2zTt-τtK2T2K1zt-τt-2fTzt-h1T3fzt-h1+2zTt-h1T3K1+K2fzt-h1-2zTt-h1K2T3K1zt-h1=ζTtΦζt.For h1⩽τ(t)⩽(h1+h2)/2, by use of Lemma 1, we obtain(20)-h2∫t-h1+h2/2t-h1‍z˙TsR1z˙sds=-h2∫t-h1+h2/2t-τt‍z˙TsR1z˙sds-h2∫t-τtt-h1‍z˙TsR1z˙sds.Let(21)e1t=∫t-h1+h2/2t-τt‍z˙sds,e2t=∫t-τtt-h1‍z˙sds;then(22)-h2∫t-h1+h2/2t-h1‍z˙TsR1z˙sds⩽-h/2h1+h2/2-τte1TtR1e1t-h/2τt-h1e2TtR1e2t=-e1TtR1e1t-τt-h1h1+h2/2-τte1TtR1e1t-e2TR1e2t-h1+h2/2-τtτt-h1e2TtR1e2t.If R1S1∗R1>0, then, by using Lemma 2, we obtain(23)τt-h1h1+h2/2-τte1t-h1+h2/2-τtτt-h1e2tTR1S1∗R1·τt-h1h1+h2/2-τte1t-h1+h2/2-τtτt-h1e2t⩾0,which implies(24)-τt-h1h1+h2/2-τte1TtR1e1t-h1+h2/2-τtτt-h1e2TtR1e2t⩽-e1TtS1e2t-e2TtS1Te1t.Then, from (19)–(24), we obtain(25)
-
h
2
∫
t
-
h
1
+
h
2
/
2
t
-
h
1
‍
z
˙
T
s
R
1
z
˙
s
d
s
⩽
-
e
1
T
t
R
1
e
1
t
-
e
2
T
t
R
1
e
2
t
-
e
1
T
t
S
1
e
2
t
-
e
2
T
t
S
1
T
e
1
t
=
-
z
t
-
τ
t
-
z
t
-
h
1
+
h
2
2
T
·
R
1
z
t
-
τ
t
-
z
t
-
h
1
+
h
2
2
-
z
t
-
h
1
-
z
t
-
τ
t
T
·
R
1
z
t
-
h
1
-
z
t
-
τ
t
-
z
t
-
τ
t
-
z
t
-
h
1
+
h
2
2
T
·
S
1
z
t
-
h
1
-
z
t
-
τ
t
-
z
t
-
h
1
-
z
t
-
τ
t
T
·
S
1
T
z
t
-
τ
t
-
z
t
-
h
1
+
h
2
2
,
(26)
-
h
2
∫
t
-
h
2
t
-
h
1
+
h
2
/
2
‍
z
˙
T
s
R
2
z
˙
s
d
s
⩽
-
z
t
-
h
1
+
h
2
2
-
z
t
-
h
2
T
·
R
2
z
t
-
h
1
+
h
2
2
-
z
t
-
h
2
,
(27)
-
h
1
2
∫
t
-
h
1
/
2
t
‍
z
˙
T
s
R
3
z
˙
s
d
s
-
h
1
2
∫
t
-
h
1
t
-
h
1
/
2
‍
z
˙
T
s
R
4
z
˙
s
d
s
⩽
z
t
-
z
t
-
h
1
2
T
R
3
z
t
-
z
t
-
h
1
2
-
z
t
-
h
1
2
-
z
t
-
h
1
T
·
R
4
z
t
-
h
1
2
-
z
t
-
h
1
.Therefore(28)V˙5zt⩽ζTtΠ1+ATRAζt,where Π1, A, R are defined in (8).
From (16)–(28), we obtain(29)V˙zt⩽ζTtΣ+Φ+Π1+ATRAζt.
IfΣ+Φ+Π1+ATRA<0, then there exists a scalar ɛ>0, such that(30)V˙zt⩽-ɛζTtζt⩽-ɛzTtzt<0,∀zt≠0.Thus, according to [20], system (3) is globally asymptotically stable for h1⩽τ(t)⩽(h1+h2)/2. By Schur complement, Σ+Φ+Π1+ATRA<0 is equivalent to (8).
For(h1+h2)/2⩽τ(t)⩽h2, similar to h1⩽τ(t)⩽(h1+h2)/2, we obtain(31)-h2∫t-h1+h2/2t-h1‍z˙TsR1z˙sds⩽-zt-h1-zt-h1+h22T·R1zt-h1-zt-h1+h22.
IfR2S2∗R2>0, then we obtain(32)-h2∫t-h2t-h1+h2/2‍z˙TsR2z˙sds⩽-e3TtR2e3t-e4TtR2e4t-e3TtS2e4t-e4TtS2Te3t=-zt-h1+h22-zt-τtT·R2zt-h1+h22-zt-τt-zt-τt-zt-h2T·R2zt-τt-zt-h2-zt-h1+h22-zt-τtT·S2zt-τt-zt-h2-zt-τt-zt-h2T·S2Tzt-h1+h22-zt-τt,where(33)e3t=∫t-τtt-h1+h2/2‍z˙sds,e4t=∫t-h2t-τt‍z˙sds.Therefore, by (27), (31), and (32), we obtain(34)V˙5zt⩽ζTtΠ2+ATRAζt,where Π2,A,R are defined in (9).
From (16)–(19) and (34) we obtain(35)V˙zt⩽ζTtΣ+Φ+Π2+ATRAζt.If Σ+Φ+Π2+ATRA<0, then there exists a scalar ε1>0, such that(36)V˙zt⩽-ε1ζTtζt⩽-ε1zTtzt<0,∀zt≠0.Thus, according to [20], system (3) is globally asymptotically stable for (h1+h2)/2⩽τ(t)⩽h2. By Schur complement, Σ+Φ+Π2+ATRA<0 is equivalent to (9). This completes the proof.Remark 6.
From (25) and (32), the information τ(t)∈[h1,(h1+h2)/2] and τ(t)∈[(h1+h2)/2,h2] is considered adequately in our paper, which may lead to less conservative results.Remark 7.
The reciprocally convex optimization technique was utilized in each subinterval[h1,(h1+h2)/2] and [(h1+h2)/2,h2], which has potential to yield less conservative conditions.Remark 8.
Based on the results of Theorem3, an improved stability criterion for system (3) is introduced by utilizing the idea of [17] which divided the bounding of activation function ki-⩽fi(zi(t))/zi(t)⩽ki+ into two subintervals, such as ki-⩽fi(zi(t))/zi(t)⩽(ki-+ki+)/2 and ki-+ki+/2⩽fi(zi(t))/zi(t)⩽ki+. Finally, a numerical example is given to show that Theorem 9 significantly improves the feasible region of stability criterion comparing with some previous literatures.Theorem 9.
For given positive scalarsh1,  h2, h=h2-h1, and any scalars u1, u2, diagonal matrices K1=diag(k1-,k2-,…,kn-), K2=diag(k1+,k2+,…,kn+), and system (3) is globally asymptotically stable for 0⩽h1⩽τ(t)⩽h2, u1⩽τ˙(t)⩽u2, if there exist symmetric positive matrices P, X=Xij2×2, Q=Qij2×2, P¯=Pij2×2, Ri(i=1,2,3,4), positive diagonal matrices T11,T12,T13,T21,T22,T23, Δ=diag(δ1,δ2,…,δn),Λ=diag(λ1,λ2,…,λn), and any matrices S1, S2 with appropriate dimensions, satisfying the following LMIs:(37)Σ+Φ1+Π1ATR∗-R<0,(38)Σ+Φ1+Π2ATR∗-R<0,(39)Σ+Φ2+Π1ATR∗-R<0,(40)Σ+Φ2+Π2ATR∗-R<0,(41)R1S1∗R1>0,(42)R2S2∗R2>0,where(43)Φ1=Φ1110000003K1+K22T130∗00000000∗∗0000000∗∗∗-K1+K2T12K1003K1+K22T1200∗∗∗∗-K1+K2T11K13K1+K22T11000∗∗∗∗∗-2T11000∗∗∗∗∗∗-2T1200∗∗∗∗∗∗∗-2T130∗∗∗∗∗∗∗∗0,Φ2=Φ211000000K1+3K22T230∗00000000∗∗0000000∗∗∗-K1+K2T22K200K1+3K22T2200∗∗∗∗-K1+K2T21K2K1+3K22T21000∗∗∗∗∗-2T21000∗∗∗∗∗∗-2T2200∗∗∗∗∗∗∗-2T230∗∗∗∗∗∗∗∗0,where(44)Φ111=-K1+K2T13K1,Φ211=-K1+K2T23K2.The Σ,Π1,Π2 are defined in Theorem 3.Proof.
Let us choose the same Lyapunov functional proposed in Theorem3.
Case
1. Consider(45)ki-⩽fizitzit⩽ki-+ki+2.Similar to (19), there exist positive diagonal matrices T11, T12, T13, such that the following inequalities hold based on (45):(46)0⩽-2fTztT11fzt+2zTtT113K1+K22fzt-zTtK1+K2T11K1zt-2fTzt-τtT12fzt-τt+2zTt-τtT123K1+K22fzt-τt-zTt-τtK1+K2T12K1zt-τt-2fTzt-h1T13fzt-h1+2zTt-h1T133K1+K22fzt-h1-zTt-h1K1+K2T13K1zt-h1=ζTtΦ1ζt.Therefore, by the use of proof of Theorem 3, when ki-⩽fi(zi(t))/zi(t)⩽(ki-+ki+)/2, h1⩽τ(t)⩽(h1+h2)/2, from (16)–(18), (34), and (46), we obtain(47)V˙zt⩽ζTtΣ+Φ1+Π1+ATRAζt.If Σ+Φ1+Π1+ATRA<0, then there exists a scalar ε2>0, such that(48)V˙zt⩽-ε2ζTtζt⩽-ε2zTtzt<0,∀zt≠0.Thus, according to [20], system (3) is globally asymptotically stable for ki-⩽fi(zi(t))/zi(t)⩽(ki-+ki+)/2, h1⩽τ(t)⩽(h1+h2)/2. By Schur complement, Σ+Φ1+Π1+ATRA<0 is equivalent to (37).
Case 2. Consider(49)ki-+ki+2⩽fizitzit⩽ki+.Similar to (19), there exist positive diagonal matrices T21, T22, T23, such that the following inequalities hold based on (49):(50)0⩽-2fTztT21fzt+2zTtT213K2+K12fzt-zTtK1+K2T21K2zt-2fTzt-τtT22fzt-τt+2zTt-τtT223K2+K12fzt-τt-zTt-τtK1+K2T22K2zt-τt-2fTzt-h1T23fzt-h1+2zTt-h1T233K2+K12fzt-h1-zTt-h1K1+K2T23K2zt-h1=ζTtΦ2ζt.Therefore, by the use of proof of Theorem 3, when (ki-+ki+)/2⩽fi(zi(t))/zi(t)⩽ki+, h1⩽τ(t)⩽(h1+h2)/2, from (16)–(18), (34), and (50), we obtain(51)V˙zt⩽ζTtΣ+Φ2+Π1+ATRAζt.If Σ+Φ2+Π1+ATRA<0, then there exists a scalar ε3>0, such that (52)V˙zt⩽-ε3ζTtζt⩽-ε3zTtzt<0,∀zt≠0.Thus, according to [20], system (3) is globally asymptotically stable for (ki-+ki+)/2⩽fi(zi(t))/zi(t)⩽ki+, h1⩽τ(t)⩽(h1+h2)/2. By Schur complement, Σ+Φ2+Π1+ATRA<0 is equivalent to (38).


## 4. Numerical Examples

In this section, an example is given to demonstrate the effectiveness of the derived method.Example 1.
Consider the stability of neural networks (3) with the following parameters:(53)C=1.276900000.623100000.923000000.4480,A=-0.03730.4852-0.33510.2336-1.60330.5988-0.32241.23520.3394-0.0860-0.3824-0.5785-0.13110.3253-0.9534-0.5015,B=0.8674-1.2405-0.53250.02200.0474-0.91640.03600.98161.84952.6117-0.37880.8428-2.04130.51791.1734-0.2775,K1=diag0,0,0,0,K2=diag0.1137,0.1279,0.7994,0.2368f1s=0.05685s+1-s-1,f2s=0.06395s+1-s-1,f3s=0.3997s+1-s-1f4s=0.1184s+1-s-1.For given h1=3, u1=0, the upper bounds of h2 for different u2 are derived by Theorems 3 and 9 and those results in [15–17] are listed in Table 1. According to Table 1, our obtained results are much less conservative than those in the literature.Table 1
Allowable upper bound ofh2 for different u2.
Method
u
2
=
0.1
u
2
=
0.5
u
2
=
0.9
[15]
3.65
3.32
3.26
[16]
3.78
3.45
3.39
[17]
4.19
3.62
3.59
Theorem3
4.97
4.01
3.90
Theorem9
5.45
4.65
4.57


## 5. Conclusions

The problem of delay-dependent asymptotic stability analysis for neural networks with interval time-varying delays is considered based on the delay-partitioning method. Some less conservative stability criteria are established in terms of linear matrix inequalities (LMIs) by constructing a new L-K functional in each subinterval and combining with reciprocally convex approach. Finally, a numerical example is given to show the improved stability region of the proposed results.



---

*Source: 705367-2015-09-16.xml*