JOURNAL OF SCIENCE OF HNUE

Natural Sci., 2013, Vol. 00, No. 0, pp. 1-11

Exponential stability of nonlinear neutral systems with time-varying delay
Le Van Hien† and Hoang Van Thi ††
†
Hanoi National University of Education
††
Hong Duc University, Thanh Hoa
E-mail:
Abstract.
In this paper, the problem of exponential stability for a class of nonlinear neutral systems with interval time-varying delay is studied. Based on improved Lyapunov-Krasovskii functionals combine with Leibniz-Newton’s formula, new delay-dependent sufficient conditions for the exponential stability
of the systems are established in terms of linear matrix inequalities (LMIs),
which allows to compute the maximal bound of the exponential stability rate
of the solution. Numerical examples are also given to show the effectiveness
of the obtained results.
Keywords: Neutral systems; interval time-varying delay; nonlinear uncertainty; exponential stability; linear matrix inequality

1.

Introduction

Time-delay occurs in most of practical models, such as, aircraft stabilization, chemical engineering systems, inferred grinding model, manual control, neural network, nuclear
reactor, population dynamic model, ship stabilization, and systems with lossless transmission lines. The existence of this time-delay may be the source for instability and bad performance of the system. Hence, the problem of stability analysis for time-delay systems
has received much attention of many researchers in recent years, see [4, 5, 7, 11, 12, 14,
17] and references therein.
In many practical systems, the system models can be described by functional differential equations of neutral type, which depend on both state and state derivatives. Neutral
system examples include distributed networks, heat exchanges, and processes involving
steam. Recently, the stability analysis of neutral systems has been widely investigated
by many researchers, see [3, 7] for time-varying delay, and [8, 10-12] for interval timevarying delay. The main approach is Lyapunov-Krasovskii functional method and linear

matrix inequality technique. However, in most of this results, the time-varying delay is
assumed to be differentiable, which makes stability conditions more conservatism.
1


L.V. Hien & H.V. Thi

In this paper, we consider exponential stability problem for a class of nonlinear
neutral systems with interval time-varying delay. By using improved Lyapunov-Krasovskii
functionals combined with LMIs technique, we propose new criteria for the exponential
stability of the system. The delay-dependent conditions are formulated in terms of LMIs,
being thus solvable by utilizing Matlab’s LMI Control Toolbox available in the literature
to date. Compared to the existing results, our result has its own advantages. First, it deals
with the neutral system considered in this paper is subjected to nonlinear uncertainties.
Second, the time delay is assumed to be a time-varying continuous function belonging to
a given interval, which means that the lower and upper bounds for the time-varying delay
are available, but the delay function is bounded but not necessary to be differentiable.
This allows the time-delay to be a fast time-varying function and the lower bound is not
restricted to being zero. Third, our approach allows us to obtain novel exponential stability
conditions established in terms of LMIs, which allows to compute the maximal bound of
the exponential stability rate of the solution. Therefore, our results are more general than
the related previous results.
The paper is organized as follows: Section 2 presents definitions and some wellknown technical propositions needed for the proof of the main results. Delay-dependent
exponential stability conditions of the system is presented in Section 3. Numerical examples are given in Section 4. The paper ends with conclusions and cited references.
Notations. The following notations will be used throughout this paper. R+ denotes the set
of all nonnegative real numbers; Rn denotes the n−dimensional Euclidean space with the
norm . and scalar product xT y of two vectors x, y; λmax (A) (λmin (A), resp.) denotes
the maximal (the minimal, resp.) number of the real part of eigenvalues of A; AT denotes
the transpose of the matrix A and I denote the identity matrix. A matrix Q ≥ 0 (Q > 0,
resp.) means that Q is semi-positive definite (positive definite, resp.) i.e. Qx, x ≥ 0 for

all x ∈ Rn (resp. Qx, x > 0 for all x = 0); A ≥ B means A − B ≥ 0; C 1 ([a, b], Rn)
denotes the set of all continuously differentiable functions on [a, b]. The segment of the
trajectory x(t) is denoted by xt = {x(t + s) : s ∈ [−¯h, 0]}.

2.

Preliminaries
Consider a nonlinear neutral system with interval time-varying delay of the form

x(t)
˙ − Dx(t
˙ − τ ) = A0x(t) + A1 x(t − h(t)) + f (t, x(t), x(t − h(t)), x(t
˙ − τ )) , t ≥ 0,
x(t) = φ(t), t ∈ [−¯h, 0],
(2.1)
n
where x(t) ∈ R is the system state; A0, A1, D are given real matrices; time varying delay
¯ = max{τ, hM }; nonlinear
h(t) satisfies 0 ≤ hm ≤ h(t) ≤ hM , constant τ ≥ 0 and h
+
n
n
n
n
uncertainty function f : R × R × R × R → R satisfies
f(t, x, y, z)
2

2


≤ a20 x

2

+ a21 y

2

+ a22 z 2 ,

∀(x, y, z),

t ≥ 0,

(2.2)


Exponential stability of nonlinear neutral systems with time-varying delay

where, a0, a1, a2 are given nonnegative constants. The initial function φ ∈ C 1([−¯h, 0], Rn )
2.
˙
φ(t) 2 + φ(t)
with its norm φ s = sup ¯
−h≤t≤0

Definition 2.1. System (2.1) is said to be globally exponentially stable if there exist constants α > 0, γ ≥ 1 such that all solution x(t, φ) of the system satisfies the following
condition
x(t, φ) ≤ γ φ s e−αt, ∀t ≥ 0.
We introduce the following technical well-known propositions, which will be used

in the proof of our results.
Proposition 2.1. (Schur Complement, see Boyd et. al. [1]) For given matrices X, Y, Z
with appropriate dimensions satisfying X = X T , Y T = Y > 0. Then X + Z T Y −1 Z < 0
if and only if
X ZT
−Y Z
< 0 or
< 0.
Z −Y
ZT X
Proposition 2.2. (Completing square) Let S be a symmetric positive definite matrix. Then
for any x, y ∈ Rn and matrix F , we have
2 F y, x − Sy, y ≤ F S −1F T x, x .
The proof of the above proposition is easily derived from completing square:
S(y − S −1 F T x), y − S −1 F Tx ≥ 0.
Proposition 2.3. (See, Gu [2]) For any symmetric positive definite matrix W , scalar
ν > 0 and vector function w : [0, ν] −→ Rn such that the concerned integrals are well
defined, then
T

ν

w(s)ds

ν

3.

wT (s)W w(s)ds.


w(s)ds ≤ ν

W

0

ν

0

0

Main results

Consider system (2.1), where the delay function h(t) satisfies 0 ≤ hm ≤ h(t) ≤
hM , constant τ ≥ 0 and ¯h = max{τ, hM } and the nonlinear perturbation function f(.) satisfies the condition (2.2). For given symmetric positive definite matrices P, Q, R, S, T, Z, W
we set
ρ(α) = 2αλmax (P ) + 1 − e−2ατ (λmax (R) + λmax (S))
+ 1 − e−2αhm λmax (Q) + h2M e2αhM − 1 λmax (T )
+ hM − hm

2

e2αhM − 1 λmax (Z) + τ 2 e2ατ − 1 λmax (W ).
3


L.V. Hien & H.V. Thi

Note that, the scalar function ρ(α) is continuous and strictly increasing function in

α ∈ [0, ∞), ρ(0) = 0, ρ(α) → ∞ as α → ∞. Hence, for any λ0 > 0, there is a unique
positive solution α∗ of the equation ρ(α) = λ0 , and ρ(α) < λ0 for all α ∈ (0, α∗). Let us
set λ1 = λmin (P ), and
1
λ2 = λmax (P ) + hm λmax (Q) + τ λmax (R) + λmax (S) + h3M e2α∗hM λmax (T )
2
1
1
+ (hM − hm )2 (hM + hm )e2α∗hM λmax (Z) + τ 3e2α∗τ λmax (W ).
2
2
The exponential stability of system (2.1) is summarized in the following theorem.
Theorem 3.1. Assume that, for system (2.1), there exist matrices Uk , (k = 1, . . . , 7),
symmetric positive definite matrices P, Q, R, S, T, Z, W, and positive number , such that
the following linear matrix inequality hold:


Ξ11 AT0 U2 + W Ξ13
AT0 U4
−U1T + AT0 U5
Ξ16
Ξ17
 ∗

−R − W U2T A1
0
−U2T
U2T D
U2T



T
T
T
T
T
T
T
 ∗

∗
Ξ
Z
+
A
U
−U
+
A
U
U
D
+
A
U
U
+
A
U
33

4
5
6
7
1
3
1
3
1
3
1


T
T
T

 < 0,
Ξ= ∗
∗
∗
−Q − Z
−U4
U4 D
U4

T
T
 ∗


∗
∗
∗
Ξ
U
D
−
U
U
−
U
55
6
7
5
5


T
T
 ∗
∗
∗
∗
∗
Ξ66
U6 + D U7 
∗
∗
∗

∗
∗
∗
Ξ77
(3.1)
where
Ξ11 = AT0 (P + U1 ) + (P + U1T )A0 + a20I + Q + R − T − W ;
Ξ13 = P A1 + U1T A1 + AT0 U3 + T ;
Ξ16 = P D + U1T D + AT0 U6 ;

Ξ17 = AT0 U7 + P + U1T ;

Ξ33 = −T − Z + AT1 U3 + U3T A1 + a21I;
Ξ55 = −U5 − U5T + S + h2M T + (hM − hm )2Z + τ 2 W ;
Ξ66 = −S + DT U6 + U6T D + a22I;
Ξ77 = − I + U7T + U7 .
Then the system (2.1) is globally exponentially stable. Moreover, every solution
x(t, φ) of the system satisfies
x(t, φ) ≤

λ2
φ s e−αt ,
λ1

∀α ∈ (0, α∗ ], ∀t ≥ 0.

Chứng minh. Let λ0 = λmin −Ξ > 0 (due to (3.1)). Taking any α > 0 from the interval
(0, α∗ ], we consider the following Lyapunov-Krasovskii functional for the system (2.1)
7


V (t, xt) =

Vk ,
i=1

4

(3.2)


Exponential stability of nonlinear neutral systems with time-varying delay

where,
V1 = xT (t)P x(t),
t

e2α(s−t)xT (s)Qx(s)ds

V2 =
t−hm
t

e2α(s−t)xT (s)Rx(s)ds,

V3 =
t−τ
t

e2α(s−t)x˙ T (s)S x(s)ds,
˙


V4 =
t−τ

t

t

e2α(θ−t+hM ) x˙ T (θ)T x(θ)dθds,
˙

V5 = hM
t−hM

s
t−hm

t

e2α(θ−t+hM ) x˙ T (θ)Z x(θ)dθds,
˙

V6 = (hM − hm )
t−hM
t

s

t


e2α(θ+τ −t)x˙ T (θ)W x(θ)dθds.
˙

V7 = τ
t−τ

s

Taking the derivative of V1 along the solution of system (2.1) we have
V˙1 = 2xT (t)P x(t)
˙
= xT (t) P A0 + AT0 P x(t)
+ 2xT (t)P A1x(t − h(t)) + Dx(t
˙ − τ ) + f(t) ,
where, for convenient, we denote f(t) =: f(t, x(t), x(t − h(t)), x(t
˙ − τ )).
From (2.2) we obtain
a20xT (t)x(t) + a21xT (t − h(t))x(t − h(t)) + a22x˙ T (t − τ )x(t
˙ − τ ) − f T (t)f(t) ≥ 0,
for any > 0. Therefore, the derivative of V1 satisfies
V˙1 ≤ xT (t) P A0 + AT0 P + a20I x(t)
+ 2xT (t)P A1x(t − h(t)) + Dx(t
˙ − τ ) + f(t)
+

(3.3)

a21xT (t − h(t))x(t − h(t)) + a22x˙ T (t − τ )x(t
˙ − τ ) − f T (t)f(t) .
5



L.V. Hien & H.V. Thi

Next, the derivatives of Vk , k = 2, . . . , 7 give
V˙2 = xT (t)Qx(t) − e−2αhm xT (t − hm )Qx(t − hm ) − 2αV2 ;
V˙3 = xT (t)Rx(t) − e−2ατ xT (t − τ )Rx(t − τ ) − 2αV3 ;
V˙4 = x˙ T (t)S x(t)
˙ − e−2ατ x˙ T (t − τ )S x(t
˙ − τ ) − 2αV4 ;
t

V˙5 = h2M e2αhM x˙ T (t)T x(t)
˙ − hM

e2α(s−t+hM ) x˙ T (s)T x(s)ds
˙
− 2αV5

(3.4)

t−hM
t

≤ h2M e2αhM x˙ T (t)T x(t)
˙ − hM

x˙ T (s)T x(s)ds
˙
− 2αV5 ;

t−hM

and
V˙6 = (hM − hm )2e2αhm x˙ T (t)Z x(t)
˙
t−hm

e2α(s−t+hM ) x˙ T (s)Z x(s)ds
˙
− 2αV6

− (hM − hm )
t−hM

˙
≤ (hM − hm )2 e2αhm x˙ T (t)Z x(t)
t−hm

(3.5)

x˙ T (s)Z x(s)ds
˙
− 2αV6 ;

− (hM − hm )
t−hM

t

V˙7 = τ 2e2ατ x˙ T (t)W x(t)

˙ −τ

e2α(s+τ −t)x˙ T (s)W x(s)ds
˙
− 2αV7
t−τ
t

≤ τ 2e2ατ x˙ T(t)W x(t)
˙ −τ

x˙ T (s)W x(s)ds
˙
− 2αV7 ;
t−τ

Applying Proposition 3 and the Leibniz-Newton formula, we have
t

t

x˙ T (s)T x(s)ds
˙
≤ −h(t)

−hM
t−hM

x˙ T (s)T x(s)ds
˙

t−h(t)

T

t

≤−

x(s)ds
˙

t

t−h(t)

t−h(t)

≤ − x(t) − x(t − h(t))

T

T x(t) − x(t − h(t)) ;

t−hm

t−hm
T

−(hM − hm )


x˙ T (s)Z x(s)ds
˙

x˙ (s)Z x(s)ds
˙
≤ −(h(t) − hm )
t−hM

t−h(t)
T

t−hm

≤−

x(s)ds
˙
t−h(t)

t−hm

x(s)ds
˙

Z
t−h(t)
T

≤ − x(t − hm ) − x(t − h(t))
6


(3.6)

x(s)ds
˙

T

Z x(t − hm ) − x(t − h(t)) ;

(3.7)


Exponential stability of nonlinear neutral systems with time-varying delay

and
t

T

t
T

−τ

x˙ (s)W x(s)ds
˙
≤−
t−τ


x(s)ds
˙

t

x(s)ds
˙

W

t−τ

t−τ

(3.8)

T

≤ − x(t) − x(t − τ ) W x(t) − x(t − τ ) .
By using the following identity relation
−x(t)
˙ + Dx(t
˙ − τ ) + A0x(t) + A1x(t − h(t)) + f(t) = 0,
we obtain
2 xT (t)U1T + xT (t − τ )U2T + xT (t − h(t))U3T
+ xT (t − hm )U4T + x˙ T(t)U5T + x˙ T (t − τ )U6T + f T (t)U7T

(3.9)

× −x(t)

˙ + Dx(t
˙ − τ ) + A0x(t) + A1x(t − h(t)) + f(t) = 0.
Therefore, from (3.3)-(3.9) we have
V˙ (t, xt) + 2αV (t, xt) ≤ η T (t)Φη(t),

(3.10)

where,
η T (t) = xT (t) xT (t − τ ) xT (t − h(t)) xT (t − hm ) x˙ T (t) x˙ T (t − τ ) f T (t) ,


Φ11 AT0 U2 + W Φ13
AT0 U4
−U1T + AT0 U5
Φ16
Φ17

 ∗
Φ22
U2T A1
0
−U2T
U2T D
U2T


T
T
T
T

T
T
T

 ∗
∗
Φ
Z
+
A
U
−U
+
A
U
U
D
+
A
U
U
+
A
U
33
4
5
6
7
1

3
1
3
1
3
1


T
T
T
,
∗
∗
Φ44
−U4
U4 D
U4
Φ=

 ∗
T
T

 ∗
∗
∗
∗
Φ
U

D
−
U
U
−
U
55
6
7
5
5


T
T
 ∗
∗
∗
∗
∗
Φ66
U6 + D U7 
∗
∗
∗
∗
∗
∗
Φ77
and


Φ11 = (A0 + αI)T P + P (A0 + αI) + AT0 U1 + U1T A0 + a20I + Q + R − W − T ;
Φ13 = P A1 + U1T A1 + AT0 U3 + T ;
Φ17 = P + U1T + AT0 U7 ;

Φ16 = P D + U1T D + AT0 U6 ;

Φ22 = −e−2ατ R − W ;

Φ33 = a21I − T − Z + AT1 U3 + U3T A1;

Φ44 = −e−2αhm Q − Z;

Φ55 = S + h2M e2αhM T + (hM − hm )2e2αhM Z + τ 2 e2ατ W − U5 − U5T ;
Φ66 = −e−2ατ S + U6T D + DT U6 + a22I;
Φ77 = − I + U7 + U7T .
7


L.V. Hien & H.V. Thi

Observe that Φ = Ξ + Ψ, where,
Ψ = diag 2αP, (1 − e−2ατ )R, 0, (1 − e−2αhm )Q, h2M (e2αhM − 1)T
+ (hM − hm )2 (e2αhM − 1)Z + τ 2(e2ατ − 1)W, (1 − e−2ατ )S, 0 .
hence
V˙ (t, xt) + 2αV (t, xt) ≤ η T (t)(Ξ + Ψ)η(t).

(3.11)

Taking (3.11) into account, we finally obtain

V˙ (t, xt) + 2αV (t, xt) ≤ ρ(α) − λ0

η(t)

2

≤ 0,

(3.12)

which implies V (t, xt) ≤ V (0, x0 )e−2αt, t ≥ 0. To estimate the exponential stability rate
of the solution, we use (3.2) that
λ1 x(t)

2

≤ V (t, xt) ≤ λ2 xt 2s ,

t ∈ R+ .

and from the differential inequality (3.12), we obtain
x(t, φ) ≤

λ2
φ s e−αt ,
λ1

t≥0

which completes the proof of the theorem.

Remark 3.1. The exponential convergence rate α in Theorem 1 can be obtained by solving a nonlinear scalar equation ρ(α) = λ0 . For this equation, many algorithms and computational methods can be used, e.g., iterative or Newton’s method [9]. However, for a
more explicit condition, we estimate the exponential rate α as follow: From the fact that,
¯
¯ we have ρ(α) ≤ γ e2α¯h − 1 , where, γ = λmax (P ) + λmax (Q) +
e2αh − 1 ≥ 2αh,
¯
h
λmax (R) + λmax (S) + h2M λmax (T ) + (hM − hm )2 λmax (Z). Therefore, system (2.1) is
1
λ0
exponentially stable with the exponential rate 0 < α ≤ ¯ ln 1 +
γ
2h

.

Remark 3.2. Theorem 1 gives conditions for the exponential stability of neutral systems
with nonlinear uncertainties and interval-time varying state delay. These conditions are
derived in terms of linear matrix inequalities which can be solved effectively by various
computation tools [1]. Different from [5, 6, 12, 13], where the α-exponential stability
problem is considered, the exponential rate α is given and enters as nonlinear terms in the
stability conditions. In this paper, the exponential convergence rate is determined in terms
of linear matrix inequalities.
8


Exponential stability of nonlinear neutral systems with time-varying delay

4.


Numerical examples

In this section, we give some numerical examples to illustrate the effectiveness of
our obtained results in comparison with the existing results.
Example 4..1. Consider neutral system (2.1), where
A0 =

−2 0
,
1 −4

A1 =

0.1 −1
,
0 −0.1

0.1 0
,
0 0.1

D=

a0 = 0.2, a1 = 0.2, a2 = 0.1, τ = 1,
and h(t) = 1 + ψ(t), where, ψ(t) = 0.5 sin(t) if t ∈ I = ∪k≥0 [2kπ, (2k + 1)π] and
ψ(t) = 0 if t ∈ R+ \I.
Note that, the delay function h(t) is continuous, but non-differentiable on R+ .
Therefore, the stability results obtained in [3, 10-12, 16, 18-21] are not applicable. By
using LMI toolbox of Matlab, we can verify that, the LMI (3.1) is satisfied with hm =
1, hM = 1.5, = 10 and

P =

11.1343 −8.6199
,
−8.6199 33.9554

Q=

6.4391 −2.7561
,
−2.7561 23.0890

R=

5.9522 −1.4762
,
−1.4762 10.8871

S=

1.1647 −0.1417
,
−0.1417 2.0581

T =

0.4364 −0.0872
,
−0.0872 0.8021


W =

0.5411 −0.0846
,
−0.0846 1.1756

Z=

2.6249 −1.2851
,
−1.2851 12.4711

U1 =

U3 =

0.3302 −0.7932
,
0.7994 −8.1438
U6 =

U4 =

−14.3260 41.8769
,
−7.9793 −28.4459
−0.2285 1.8629
,
−0.0296 0.2517


−0.0126 −0.8276
,
0.7984
0.0933

U7 =

U2 =

U5 =

0.1456 0.0452
,
−0.0317 0.2562

3.3512 7.8831
,
−10.1070 7.1532

1.1847 −8.1379
.
8.1228 1.3701

We have λ0 = 0.3635 and
ρ(α) = 73.6906α + 36.9084 1 − e−2α + 1.1867 e2α − 1 + 5.0081 e3α − 1 .
The unique positive solution of equation ρ(α) = λ0 is α∗ = 0.0022057. Then all solution
x(t, φ) of the system satisfies the following inequality
x(t, φ) ≤ 3.1803 φ s e−0.0022t,

∀t ≥ 0.


Example 4..2. Consider the system studied in ([15, 20]):
d
[x(t) − Dx(t − τ )] = A0 x(t) + A1x(t − τ ) + f(t, x(t), x(t − τ )),
dt

(4.1)

where,
A0 =

−2 0.5
,
0 −1

A1 =

1 0.4
,
0.4 −1

D=

0.2 1
, a0 = 0.2, a1 = 0.1.
0 0.2
9


L.V. Hien & H.V. Thi


Applying Corollary 1 for hm = 0, hM = τ and a2 = 0 we obtain the allowable
value of the delay for the asymptotic stability of system (4.1) is τ = 1.8106, while the
upper bound of value τ given in [15] and [20] is 0.583 and 1.7043, respectively.

5.

Conclusion

In this paper, we have proposed new delay-dependent exponential stability conditions for a class of nonlinear neutral systems with non-differentiable interval time-varying
delay. Based on the improved Lyapunov-Krasovskii functionals and linear matrix inequality technique, new delay-dependent sufficient conditions for the exponential stability of
the systems have been established in terms of LMIs. Numerical examples are given to
show the effectiveness of our results.
Acknowledgments.
This work was partially supported by Hanoi National University of Education and the Ministry of Education and Training, Vietnam.

REFERENCES

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