SCIENCE CHINAInformation Sciences

January 2018, Vol. 61 012203:1–012203:14

doi: 10.1007/s11432-017-9097-1

c© Science China Press and Springer-Verlag Berlin Heidelberg 2017 info.scichina.com link.springer.com

. RESEARCH PAPER .

Impulsive control of unstable neural networks with

unbounded time-varying delays

Xiaodi LI1,2, Shiji SONG3* & Jianhong WU4

1School of Mathematics and Statistics, Shandong Normal University, Ji’nan 250014, China;2Institute of Data Science and Technology, Shandong Normal University, Ji’nan 250014, China;

3Department of Automation, Tsinghua University, Beijing 100084, China;4Department of Mathematics and Statistics, York University, Toronto M3J 1P3, Canada

Received 6 February 2017/Accepted 20 April 2017/Published online 30 August 2017

Abstract This paper considers the impulsive control of unstable neural networks with unbounded time-

varying delays, where the time delays to be addressed include the unbounded discrete time-varying delay

and unbounded distributed time-varying delay. By employing impulsive control theory and some analysis

techniques, several sufficient conditions ensuring µ-stability, including uniform stability, (global) asymptotical

stability, and (global) exponential stability, are derived. It is shown that an unstable delay neural network,

especially for the case of unbounded time-varying delays, can be stabilized and has µ-stability via proper

impulsive control strategies. Three numerical examples and their simulations are presented to demonstrate

the effectiveness of the control strategy.

Keywords unstable neural networks, impulsive control, unbounded discrete time-varying delay, unbounded

distributed time-varying delay, µ-stability

Citation Li X D, Song S J, Wu J H. Impulsive control of unstable neural networks with unbounded time-varying

delays. Sci China Inf Sci, 2018, 61(1): 012203, doi: 10.1007/s11432-017-9097-1

1 Introduction

Neural networks have wide applications in many fields such as pattern recognition, parallel computing,

image processing, association, and optimal computation, see [1–7]. In these applications, time delays

are frequently encountered owing to the finite switching speed of hardware and communication time.

Moreover, sometimes it is necessary to introduce time delays to acquire a desired performance. For

example, time delays are introduced in signals transmitted between cells to process moving images [8,9].

Time delays are often introduced in neural networks to exhibit chaotic phenomena that can be applied

to secure communication [10]. Hence, neural networks with time delays have attracted wide interest, and

a large number of results have been reported in recent years, for example [11–14].

In fact, in many practical delayed neural networks, time delays are variant and sometimes may depend

deeply on the information of the history so that they are unbounded, or are known only to be time

varying but nothing else [15–18]. Moreover, neural networks with time delays admit infinite dimensional

structures. In such cases, the essential problem of dynamics analysis for neural networks with unbounded

time-varying delays becomes technically quite difficult. Recently, some interesting results on the dynam-

ics of neural networks with unbounded time-varying delays were reported, see [19–24]. In particular,

*Corresponding author (email: shijis@mail.tsinghua.edu.cn)

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:2

Ref. [19] introduced a new concept of µ-stability and investigated the global dynamics of neural networks

with unbounded time-varying delays. Ref. [21] investigated the global asymptotic stability and global

exponential stability of neural networks with unbounded time-varying delays via some novel analysis

techniques. Ref. [22] studied the global µ-stability of complex-valued neural networks with unbounded

time-varying delays via Lyapunov-Krasovskii functional coupled with linear matrix inequality (LMI)

techniques. Recently, Ref. [24] derived some conditions that ensured the multiple µ-stability of neural

networks with unbounded time-varying delays.

It is noted that in many practical applications, impulsive control as a type of hybrid control method is

attractive because it allows the control action on a plant only at some discrete instances, see [25–29]. For

example, in chaos-based secure communication systems [29], impulsive control allows for the stabilization

of a chaotic system using only small control impulses, which may reduce the amount of information to be

transmitted and increase the robustness against the disturbance. In population models [25,27], to control

the population of a type of insect by using its natural enemies, it is natural to release enemies at proper

instances but not in a continuous release. In recent years, many researchers focused their attention

on impulsive control (impulsive stabilization) of delayed neural networks. This is quite different from

the cases of impulsive perturbation, which is a type of robustness problem, and some attractive results

have been presented in the literature, for example [30–41]. The main idea of impulsive control in neural

networks is to introduce an impulse input into the topological structure of the networks and then control

the states of systems. In [33], impulsive control for the global exponential stability of high-order Hopfield-

type neural networks with bounded time-varying delays were studied via the Lyapunov-Razumikhin

technique. Ref. [34] studied the global exponential control of impulsive neural networks with unbounded

continuously distributed delays using an inequality technique from the control point of view. Ref. [35]

introduced a new concept of control topology and studied the synchronization of complex dynamical

networks with bounded time-varying delays via distributed impulsive control. Recently, Ref. [37] studied

the impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks via

time-varying Lyapunov functionals coupled with a convex combination technique. However, most of the

existing results, such as those in [30–41], focus mainly on the impulsive control of dynamics analysis of

neural networks with bounded discrete delays (constant delays or time-varying delays) and unbounded

continuously distributed delays. There are few results dealing with cases of unbounded time-varying

delays owing to the fact that such a model not only depends on the state information of the entire

history, which is time-varying, but is also discontinuous and admits mixed features of continuous and

discrete systems. It is difficult to introduce impulse input into this structure.

In this paper, we investigate the impulsive control of neural networks with unbounded time-varying

delays. Some control results for µ-stability, which includes uniform stability, (global) asymptotical sta-

bility, and (global) exponential stability, are presented by employing impulsive control theory and some

analysis techniques. An outline of this paper is given as follows. In Section 2, we introduce some prelim-

inary knowledge. In Section 3, some impulsive control results for the µ-stability of neural networks with

unbounded discrete time-varying delays and unbounded distributed time-varying delays are presented.

In Section 4, three numerical examples are provided, and a conclusion is given in Section 5.

2 Preliminaries

Let R denote a set of real numbers, R+ a set of nonnegative real numbers, Rn and Rn×m the n-dimensional

and n×m-dimensional real spaces equipped with the Euclidean norm || · ||, respectively, and Z+ a set of

positive integer numbers. For a symmetric A , let λmax(A ) and λmin(A ) be the maximum and minimum

eigenvalues of matrix A , respectively. Let α∨ β and α∧ β denote the maximum and minimum values of

α and β, respectively. A > 0 or A < 0 denotes that matrix A is a symmetric and positive or negative

definite matrix. ⋆ denotes the symmetric block in one symmetric matrix, and I denotes the identity matrix

with appropriate dimensions. Λ = {1, 2, . . . , n}. For any w = (wij)n×n ∈ Rn×n, [w]

+= (|wij |)n×n. For

any A ⊆ R, B ⊆ Rk (1 6 k 6 n), C(A,B) = {ϕ : A → B is continuous}, C1(A,B) = {ϕ : A → B

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:3

0 10

−4

0

4

(a) (b)

t

x

=0

0 100−5

0

5

t

=3

0 800−5

0

5

t

x

=8

0 800−5

0

5

t

=0.5t

τ

τ τ

τ

(c) (d)

Figure 1 (Color online) State trajectories x = (x1, x2)T of model (3) with different delays τ , where red lines denote the

trajectories of x1, blue lines denote the trajectories of x2. (a) τ=0; (b) τ= 3; (c) τ=8; (d) τ=0.5t.

is continuously differentiable}, PC(A,B) = {ϕ : A → B is continuous everywhere except at a finite

number of points t, at which ϕ(t+), ϕ(t−) exist and ϕ(t+) = ϕ(t)} and PCB(A,B) = {ϕ ∈ PC(A,B) : ϕis bounded }. For any α > 0 (or ∞), let PCBα = PCB([−α, 0],Rn), and for each ϕ ∈ PCBα, the norm

is defined by ‖ϕ‖α = sup−α6s60 ‖ϕ(s)‖. Define set Φ = {µ ∈ C1(R+, [1,∞]) : µ is nondecreasing on

[0,∞)}.Consider the following delayed neural networks:

x(t) = −Cx(t) +Af(x(t)) +Bg(x(t− τ(t))

)+ J, t > 0,

x(s) = φ(s), s ∈ [−α, 0],(1)

subject to impulsive control:

∆x(tk) = Γk

(x(t−k )− x⋆

), k ∈ Z+, (2)

where x(t) = (x1(t), . . . , xn(t))T is the state vector of the neural network; φ ∈ PCBα and the impulse

times tk satisfy 0 6 t0 < t1 < · · · < tk → +∞ as k → +∞; C =diag(c1, . . . , cn) denote the net self-

inhibition satisfying ci > 0, i ∈ Λ; A ∈ Rn×n, and B ∈ Rn×n are the connection weight matrix and the

delayed weight matrix, respectively. J is the external input, Γk ∈ Rn×n is the impulsive control matrix,

f = (f1, . . . , fn)T and g = (g1, . . . , gn)

T represent neuron activation functions satisfying

|fj(u)− fj(v)| 6 lfj |u− v|, |gj(u)− gj(v)| 6 lgj |u− v|,

for j ∈ Λ, u, v ∈ R with lfj , lgj being some positive constants. Define Lf .

=diag(lf1 , . . . , lfn) and Lg .

=diag(lg1,

. . . , lgn). τ(t) is the transmission delay, and satisfies

0 6 τ(t) 6 α, α 6 +∞.

We note that when α = ∞, the interval [0, α] is understood to be replaced by [0,∞). It has been shown

that the existence of time delays in a neural network may lead to oscillation, instability, or some complex

phenomenon [42, 43]. This assertion can be illustrated by the following example:

x(t) = −x(t) +

(

−0.8 0.2

0.4 −0.8

)

f(x(t− τ(t))) + J, t > 0, (3)

where x = (x1, x2)T, J = 0 f1 = f2 = tanh(s), and τ is the time delay. Figure 1 shows the trajectories

of network (3) with delays τ = 0, 3, 8 and 0.5t, respectively. One may find that the network (3) will be

gradually destabilized along with the increasing of the time delay. Especially when τ = 8 and 0.5t, the

network becomes unstable.

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:4

Assume that x⋆ = (x⋆1, . . . , x

⋆n)

T is an equilibrium point of model (1), and that the impulses in (2) are

viewed as a control of model (1) with respect to the equilibrium point x⋆ at discrete instances. Then, let

u = x− x⋆. One may transform model (1) and (2) into the following form:

u(t) = −Cu(t) +AF (u(t)) +BG(

u(t− τ(t)))

, t > 0, t 6= tk,

∆u(tk) = Γku(t−k ), k ∈ Z+,

u(s) = ϕ(s), s ∈ [−α, 0],

(4)

where ϕ(s) = φ(s)− x⋆, F (u) = f(u+ x⋆)− f(x⋆) and G(u) = g(u+ x⋆)− g(x⋆).

In fact, the existence of the equilibrium points of model (1) can be easily derived using a contraction

mapping theorem, topological degree theory, Brouwer’s fixed point theorem, see [11, 15, 21] for detailed

information. Here, we present only some of the existing results directly without proofs.

Lemma 1. Model (1) has a unique equilibrium point if

√√√√

n∑

i,j∈Λ

p2ij · lf +

√√√√

n∑

i,j∈Λ

q2ij · lg < 1,

where lf = maxj∈Λ lfj , lg = maxj∈Λ lgj , (pij)n×n = C−1A and (qij)n×n = C−1B.

Lemma 2. Model (1) has at least one equilibrium point if C − [A]+Lf − [B]+Lg is an M matrix.

Lemma 3. Model (1) has at least one equilibrium point if f is bounded on Rn.

Definition 1. Let u = u(t, ϕ) be the solution of model (4) through (0, ϕ). Then, the model (4) is said

to be µ-stable. That is, model (1) is µ-stabilized via impulsive control (2) if there exists a scalar M > 0

such that

uTu 6Mµ(t)

‖ϕ‖2α, t > 0,

where ϕ ∈ PCBα and function µ ∈ Φ.

Remark 1. Note that here, the definition of µ-stability includes some classical stabilities such as uniform

stability, asymptotical stability, exponential stability, Lipschitz stability, and practical stability. In this

paper, we focus on the design of an impulsive control strategy {tk,Γk}k∈Z+such that system (1), which

may be unstable without impulses, can be µ-stabilized via impulsive control (2).

3 Main results

In this section, the problems of impulsive control of the stabilization of delayed neural networks are

addressed. The following lemma is needed to facilitate the analysis.

Lemma 4. Assume that there exists a function µ ∈ Φ and scalars µ1 > 1, µ2 > 0 such that

µ(tk)

µ(tk−1)6 µ1,

∫ tk

tk−1

µ(t)

µ⋆(t− τ(t))dt 6 µ2, ∀k ∈ Z+, (5)

where t0 = 0 and

µ⋆(t) =

{

µ(t), t > 0,

1, −α 6 t < 0.

In addition, there exists an n × n matrix P > 0, two n × n diagonal matrices Q1 > 0 and Q2 > 0, and

three scalars σ1 > 0, σ2 > 0, and > 1 such that

Π PA PB

⋆ −Q1 0

⋆ ⋆ −Q2

6 0, LgQ2L

g6 σ2P, (6)

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:5

and

lnµ1 + σ1η + µ2σ2 < ln , (7)

where Π = −PC − CP + LfQ1Lf − σ1P, η = supk∈Z+

{tk − tk−1}.Consider an auxiliary function

LP (t) = µ⋆(t)uTPu, t > −α,

where u = u(t, ϕ) is a solution of model (4) through (0, ϕ). If there exists n ∈ Z+ and some ǫ1 < ǫ2 ∈[tn−1, tn) such that for every t ∈ [ǫ1, ǫ2], we have

LP (s) 6 LP (t), ∀s ∈ [−α, t], (8)

then

LP (ǫ2) < LP (ǫ1).

Proof. For t ∈ [ǫ1, ǫ2], it can be deduced that

D+LP (t) = µ⋆′(t)uTPu+ 2µ⋆(t)uTPu′

6 µ′(t)uTPu+ µ(t)[

uT(−PC − CP )u+ uTPAQ−11 ATPu+ uTPBQ−1

2 BTPu

+ uTLfQ1Lfu+ uT(t− τ(t))LgQ2L

gu(t− τ(t))]

= µ′(t)/µ(t)LP (t) + µ(t)uT[

− PC − CP + PAQ−11 ATP + LfQ1L

f + PBQ−12 BTP

]

u

+ µ(t)uT(t− τ(t))LgQ2Lgu(t− τ(t))

(6)

6 µ′(t)/µ(t)LP (t) + µ(t)σ1uTPu+ µ(t)σ2u

T(t− τ(t))Pu(t − τ(t))

(8)

6

{

µ′(t)

µ(t)+ σ1 + σ2

µ(t)

µ⋆(t− τ(t))

}

LP (t),

which leads to

ln

(

LP (ǫ2)

LP (ǫ1)

)

6

∫ ǫ2

ǫ1

{

µ′(t)

µ(t)+ σ1 + σ2

µ(t)

µ⋆(t− τ(t))

}

dt

6 lnµ(tn)

µ(tn−1)+ σ1η + σ2

∫ tn

tn−1

µ(t)

µ⋆(t− τ(t))dt

(5)

6 lnµ1 + σ1η + µ2σ2

(7)< ln .

This completes the proof of Lemma 4.

Remark 2. It should be noted from Lemma 4 that scalar in condition (7) has a sufficient and necessary

condition. That is, there is > 1 such that Eq. (7) holds if and only if the following condition holds:

µ1µ2σ2 < exp(

− 1− σ1η)

.

Theorem 1. Under conditions (5)–(7), assume that

[

−1/P (I + Γk)TP

⋆ −P

]

6 0, k ∈ Z+. (9)

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:6

Then, the model (4) is µ-stable. Moreover, any solution u = u(t, ϕ) of model (4) through (0, ϕ) satisfies

uTu 6 µ(0)λmax(P )

µ(t)λmin(P )‖ϕ‖2α, t > 0. (10)

Proof. Without loss of generality, assume that ϕ 6≡ 0. We only need to prove that

LP (t) 6 M, t ∈ [tn−1, tn), n ∈ Z+, (11)

where M = µ(0)λmax(P )‖ϕ‖2α. First, we show that Eq. (11) holds for n = 1. Note that

LP (t) = u(t)TPu(t) 6 M < M, −α 6 t 6 0.

If the above assertion is false, then there exists ǫ2 ∈ (0, t1) such that

LP (ǫ2) = M, LP (t) 6 M, −α 6 t 6 ǫ2. (12)

One may further choose ǫ1 ∈ [0, ǫ2) such that

LP (ǫ1) = M, M 6 LP (t) 6 M, ǫ1 6 t 6 ǫ2. (13)

It follows from (12) and (13) that for any t ∈ [ǫ1, ǫ2],

LP (s) 6 M 6 LP (t), ∀s ∈ [−α, t].

By Lemma 4, one obtains

M = LP (ǫ2) < LP (ǫ1) = M,

which is a contradiction. Thus, Eq. (11) holds for n = 1.

Now, we assume that Eq. (11) holds for any n 6 N, for some N ∈ Z+, that is,

LP (t) 6 M, t ∈ [tn−1, tn), n 6 N.

Next, we show that

LP (t) 6 M, t ∈ [tN , tN+1). (14)

Suppose on the contrary that there exists ǫ⋆2 ∈ [tN , tN+1) such that

LP (ǫ⋆2) = M, LP (t) 6 M, −α 6 t 6 ǫ⋆2. (15)

Notice from (9) that

[

−1/P (I + Γk)TP

⋆ −P

]

6 0 ⇔[

(I + Γk)TP (I + Γk)− 1/P 0

⋆ −P

]

6 0

⇔ (I + Γk)TP (I + Γk)− 1/P 6 0,

which implies that

LP (tN ) =µ⋆(tN )uT(tN )Pu(tN)

=µ⋆(tN )uT(t−N )(I + ΓN)TP (I + ΓN )u(t−N )

61/LP (t−N ) = M.

Thus, there exists ǫ⋆1 ∈ [tN , ǫ⋆2) such that

LP (ǫ⋆1) = M, M 6 LP (t) 6 M, ǫ⋆1 6 t 6 ǫ⋆2. (16)

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By (15) and (16), for any t ∈ [ǫ⋆1, ǫ⋆2],

LP (s) 6 M 6 LP (t), ∀s ∈ [−α, t].

Lemma 4 leads to a contradiction that

M = LP (ǫ⋆2) < LP (ǫ

⋆1) = M.

Thus, Eq. (14) holds, that is, Eq. (11) holds for n = N + 1.

By mathematical induction, we finally obtain

LP (t) 6 µ(0)λmax(P )‖ϕ‖2α, t ∈ [tn−1, tn), n ∈ Z+,

which implies that Eq. (10) holds, and model (4) is µ-stable. The proof is complete.

If Γk ≡ ΓI, where Γ is a given constant, then the following corollary can be derived.

Corollary 1. Under conditions (5)–(7), assume that

− 1− 1/√ 6 Γ 6 −1 + 1/

√.

Then, the model (4) is µ-stable. Moreover, any solution u = u(t, ϕ) of model (4) through (0, ϕ) satisfies

(10).

Consider the case where µ(t) = 1 + δt and τ(t) = τt, where δ > 0 and τ ∈ (0, 1). We get Corollary 2.

Corollary 2. Under conditions (6) and (9), assume that

ln(1 + δη) + σ1η +η

1− τσ2 < ln .

Then, the model (4) is globally asymptotically stable. Moreover, any solution u = u(t, ϕ) of model (4)

through (0, ϕ) satisfies

uTu 6 λmax(P )

(1 + δt)λmin(P )‖ϕ‖2α, t > 0.

Consider the case in which µ(t) ≡ µ, where µ > 1 is a given constant. We get Corollary 3.

Corollary 3. Under conditions (6) and (9), assume that

η <ln

σ1 + σ2µ.

Then, the model (4) is uniformly stable. Moreover, any solution u = u(t, ϕ) of model (4) through (0, ϕ)

satisfies

uTu 6 λmax(P )

λmin(P )‖ϕ‖2α, t > 0.

Consider the case in which µ(t) = exp(ςt) and τ(t) ∈ [0, τ ], where ς and τ are two positive constants.

Note that

µ(t)

µ⋆(t− τ(t))=

µ(t)

µ(t− τ(t))6 exp(ςτ), t− τ(t) > 0,

µ(t)

µ⋆(t− τ(t))6 exp

(

ςτ(t))

6 exp(ςτ), t− τ(t) < 0.

Then, the following result can be derived.

Corollary 4. Under conditions (6) and (9), assume that

η <ln

ς + σ1 + σ2 exp(ςτ).

Then, the model (4) is globally exponentially stable. Moreover, any solution u = u(t, ϕ) of model (4)

through (0, ϕ) satisfies

uTu 6 λmax(P )

λmin(P )‖ϕ‖2α exp(−ςt), t > 0. (17)

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:8

Remark 3. Because −σ1P < 0 in (6), it is possible that model (4) is unstable without impulses.

Theorem 1 presents an impulsive control strategy to stabilize model (4) such that it becomes µ-stable.

In particular, the control conditions for globally asymptotic stability, uniform stability, and globally

exponential stability are presented in Corollaries 2–4, respectively. Here, we should point out that

although the globally exponential stability of model (4) is derived in Corollary 4, it is based on the

assumption that time-varying delay τ(t) is bounded, which is somewhat inconsistent with our research

topic on unbounded time-varying delays. In fact, the developed result in Theorem 1 can be used for

exponential stability of some unbounded cases. For example, when µ(t) = exp(t), to guarantee the

exponential estimate (17), it is necessary that the following restriction on the delay τ holds:

∫ tk

tk−1

exp(

τ(t))

dt 6 µ2, ∀k ∈ Z+, (18)

where µ2 is a constant. Obviously, it is a severe restriction that has limited applications such as

τ(t) = ln(

1 + h(t))

,

where tk = ηk, η > 0 is a given constant, and

h(t) =

(2k + 1)2ηt− (2k + 1)3η2

2+

(2k + 1)η

2, t ∈

[(2k + 1)η

2−( 1

2k + 1∧ η

2

)

,(2k + 1)η

2

]

,

−(2k + 1)2ηt+(2k + 1)3η2

2+

(2k + 1)η

2, t ∈

[(2k + 1)η

2,(2k + 1)η

2+( 1

2k + 1∧ η

2

)]

,

0, others.

In this case, there exists µ2 = 3η/2 such that Eq. (18) holds. Thus, more methods and tools for the

exponential stability of unbounded time-varying delays should be explored and developed.

Remark 4. In the literature, Ref. [19–24] dealt with the stability problems of neural networks with

unbounded time-varying delays, and derived some interesting stability criteria. Compared with those

results, the advantage of this paper is that the effect of impulsive control is greatly emphasized, and some

control results are established to stabilize neural networks with unbounded time-varying delays that may

be originally unstable. This indicates that proper impulsive control may contribute to the stabilization

of unstable systems with unbounded time-varying delays.

Next, we develop the previous ideas for delayed neural networks with unbounded distributed time-

varying delays, and derive some impulsive control conditions for µ-stability.

Consider the following delayed neural networks:

x(t) = −Cx(t) +Af(x(t)) +B

∫ τ(t)

0

ω(s)g(x(t− s))ds+ J(t), t > 0,

x(s) = φ(s), s ∈ [−α, 0],

(19)

subject to impulsive control:

∆x(tk) = Γkx(t−k ), k ∈ Z+, (20)

where ω ∈ C([0, α],R+), J is the external input at time t, and some other conditions and assumptions

needed in the following are the same as those in the previous section.

Remark 5. Note that the time delay considered here is a type of unbounded distributed time-varying

delay, which is different from an unbounded continuous distributed delay as it is not only unbounded

but also depends on the time variable t. In this case, it is clear that model (19) may not admit any

equilibrium point owing to the effect of time-varying delay τ(t), and thus the impulsive control here is

considered as the perturbations of the state at discrete points, i.e., (20), instead of the perturbations with

respect to the equilibrium point x⋆. In addition, we need to redefine the µ-stability for models (19) and

(20).

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:9

Definition 2. Assume that u = u(t, ϕ) and v = v(t, φ) are two any solutions of model (19) through

(0, ϕ) and (0, φ), respectively. Then, the models (19) and (20) are said to be µ-stable, namely, model

(19) is µ-stabilized via impulsive control (20) if there exists a scalar M > 0 such that

(u− v)T(u− v) 6Mµ(t)

‖ϕ− φ‖2α, t > 0,

where φ, ϕ ∈ PCBα and function µ ∈ Φ.

Lemma 5. Assume that there exists a function µ ∈ Φ and scalars µ1 > 1, µ2 > 0 such that

µ(tk)

µ(tk−1)6 µ1,

∫ tk

tk−1

µ(t)

µ⋆(t− τ(t))

(∫ τ(t)

0

ω(s)ds

)2

dt 6 µ2,

∀k ∈ Z+, (21)

where t0 = 0 and

µ⋆(t) =

{

µ(t), t > 0,

1, −α 6 t < 0.

In addition, there exist an n× n matrix P > 0, two n× n diagonal matrices Q1 > 0, Q2 > 0, and three

scalars σ1 > 0, σ2 > 0, and > 1 such that Eqs. (6) and (7) hold.

Consider an auxiliary function

LP (t) = µ⋆(t)zTPz, t > −α,

where z = u−v, u = u(t, ϕ) and v = v(t, φ) are any two solutions of model (19) through (0, ϕ) and (0, φ),

respectively. If there exist ǫ1 and ǫ2 ∈ [tn−1, then tn) for some n ∈ Z+ such that Eq. (8) holds. Then

LP (ǫ2) < LP (ǫ1).

Proof. Similar to Lemma 4, for t ∈ [ǫ1, ǫ2], one may derive that

D+LP (t) 6 µ′(t)zTPz + µ(t)

[

zT(−PC − CP )z + zTPAQ−11 ATPz + zTPBQ−1

2 BTPz + zTLfQ1Lfz

+

∫ τ(t)

0

ω(s)ds

∫ τ(t)

0

ω(s)gT(z(t− s))Q2g(z(t− s))ds]

6 µ′(t)/µ(t)LP (t) + µ(t)zT[

− PC − CP + PAQ−11 ATP + LfQ1L

f + PBQ−12 BTP

]

z

+

∫ τ(t)

0

ω(s)zT(t− s)LgQ2Lgz(t− s)ds · µ(t)

∫ τ(t)

0

ω(s)ds

(6)(8)

6 µ′(t)/µ(t)LP (t) + µ(t)σ1zTPz + µ(t)σ2

∫ τ(t)

0

ω(s)ds

∫ τ(t)

0

ω(s)

µ⋆(t− s)dsLP (t)

6

{

µ′(t)

µ(t)+ σ1 + σ2

µ(t)

µ⋆(t− τ(t))·(∫ τ(t)

0

ω(s)ds

)2}

LP (t),

which implies that

ln

(

LP (ǫ2)

LP (ǫ1)

)(21)

6 lnµ1 + σ1η + µ2σ2(7)< ln .

This completes the proof of Lemma 5.

Theorem 2. Assume that conditions (6), (7), (9), and (21) hold. Then, the models (19) and (20) are

µ-stable. Moreover, any two solutions u = u(t, ϕ) and v = v(t, φ) satisfy

(u− v)T(u− v) 6 µ(0)λmax(P )

µ(t)λmin(P )‖ϕ− φ‖2α, t > 0. (22)

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:10

Based on Theorem 2, it is easy derive the following corollaries.

Corollary 5. Under conditions (6), (7), and (21), assume that Γk ≡ ΓI, where Γ is a given constant

satisfying

− 1− 1/√ 6 Γ 6 −1 + 1/

√.

Then, the models (19) and (20) are µ-stable. Moreover, any two solutions u = u(t, ϕ) and v = v(t, φ)

satisfy (22).

For the case of µ(t) = 1 + δt and τ(t) = τt, where δ > 0 and τ ∈ (0, 1), we have the following result.

Corollary 6. Under conditions (6) and (21), assume that

ln(1 + δη) + σ1η +σ2

1− τω2η < ln ,

where ∫ ∞

0

ω(s)ds.= ω < ∞. (23)

Then, the models (19) and (20) are globally asymptotically stable. Moreover, any two solutions u = u(t, ϕ)

and v = v(t, φ) satisfy

(u − v)T(u− v) 6 λmax(P )

(1 + δt)λmin(P )‖ϕ− φ‖2α, t > 0.

For the case of µ(t) ≡ µ, where µ > 1 is a given constant, we get Corollary 7.

Corollary 7. Under conditions (6), (21), and (23), assume that

η <ln

σ1 + σ2ω2.

Then, the models (19) and (20) are uniformly stable. Moreover, any two solutions u = u(t, ϕ) and

v = v(t, φ) satisfy

(u− v)T(u− v) 6 λmax(P )

λmin(P )‖ϕ− φ‖2α, t > 0.

4 Numerical examples

In this section, three examples are provided to illustrate the validity of our proposed approaches.

Example 1. Consider the following 2D network model:

x(t) = −Cx(t) +Af(x(t)) +Bf(

x(t − τ(t)))

+ J, t > 0,

x(s) = φ(s), s ∈ [−α, 0],

(24)

subject to impulsive control:

∆x(tk) = Γk

(

x(t−k )− x⋆)

, k ∈ Z+,

where τ = 0.2t, J = [0, 0]T, and f(u) = u. The parameter matrices C,A and B are given as follows:

C = I, A =

(

1 1

−1 1

)

, B =

(

1 −0.5

0 1

)

.

Obviously, x⋆ = (0, 0)T is an equilibrium point of (24). By our simulations, we know that model (24)

is unstable, as indicated in Figure 2(a) and (b).

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:11

0 100200300400500600700 800−4

−3

−2

−1

0

1

2

3 ×104

×104

×104t

x

(a)

0

800

−4

0

4−4

0

4

tx2

x1

(b)

0 20

−20

−10

0

40

t

x

(c)

020

4060

−200

2040

−10

0

10

20

t

x1

(d)

0 800−6

0

1

t

x

(e)

0 20

0

t

x1x2

x1x2

x2

×105

x1x2

Figure 2 (Color online) State trajectories and phase plots of system (24) in Example 1. (a) State trajectories of 2D (24)

without impulses; (b) phase plots of 2D (24) without impulses; (c) state trajectories of 2D (24) with impulsive input (25);

(d) phase plots of 2D (24) with input (25); (e) state trajectories of 2D (24) with input (26).

To utilize the impulsive control strategy in Corollary 2, we consider the case where µ(t) = 1+0.2t, and

choose σ1 = 3, σ2 = 1, = 1.25, and η = 0.046 such that ln − ln(1 + δη) − σ1η − η1−τ σ2 = 0.0041 > 0

and the LMIs (6) and (9) hold. This leads to the impulsive controller

Γk =

(

−0.1 0.1

−0.2 −1.9

)

,

tk − tk−1 6 0.046, k ∈ Z+,

(25)

such that Eq. (24) can be stabilized and becomes µ-stable, i.e., globally asymptotically stable. This

can be shown in Figure 2(c) and (d) for the case of tk = 0.04k. Moreover, by Corollary 2, any solution

x = x(t, φ) satisfies

xTx 61.7856

(1 + 0.2t)‖φ‖20, t > 0.

However, if we make a slight change and consider the impulsive controller as follows, then

Γ⋆k = Γk+

perturbation︷ ︸︸ ︷

0.08I ,

tk = 0.04+

perturbation︷︸︸︷

0.01 , k ∈ Z+,

(26)

which is against Corollary 2. Thus, in this case, it is interesting to see from Figure 2(e) that Eq. (24)

cannot be stabilized.

Example 2. Consider 3D network model (24) with τ(t) = t + e − (t + e)56 , J = [1.6 − 3.4 cos 1, 0.1 −

2.2 cos 1,−3.5 cos1]T, f(u) = cosu. The parameter matrices C,A and B are given as follows:

C = 0.2I, A =

1.5 0.4 0

1 0 1

0 −1 2

, B =

2 −1 −0.1

0.2 −0.1 0

0.3 0 1.2

.

In this case, as f is bounded, model (24) has at least one equilibrium point. It is easy to check that

x⋆ = (1, 0,−1)T is an equilibrium point of (24) that is unstable by simulation, as indicated in Figure 3(a).

Let µ(t) = ln(e + t). It is clear that µ ∈ Φ. Choose σ1 = 5.5, σ2 = 2.4, = 1.5, and η = 0.036 such that

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:12

0 2000−20

0

15

t

x

(a) x1 x2 x3 x1x2x3

0 10−15

0

15

t

x

(b)

0 50−15

0

15

t

x

(c)

49 50

0t

x

x1x2x3

Figure 3 (Color online) State trajectories of system (24) in Example 2. (a) State trajectories of 3D (24) without impulses;

(b) state trajectories of 3D (24) with impulsive input (27) and tk = 0.035k, k ∈ Z+; (c) state trajectories of 3D (24) with

tk = 0.065k and Γk = −1.9129I.

0 100−20

0

35

t

x

(a)

0 50

0

2

t

x

(b)

43 50−0.3

0

0.3

0 25−2

0

2

t

x

(c) R1

R2

B1

B2

x1x2

x1 x2x1 x2

Figure 4 (Color online) State trajectories of system (28) in Example 3. (a) State trajectories of (28) without impulses;

(b) state trajectories of (28) with impulsive input (29) in which tk = 0.33k and Γ = −1.4714, k ∈ Z+; (c) state trajectories

of (28) with impulsive input (29) in which tk = 0.5k and Γ = −1.4714, k ∈ Z+.

all conditions in Corollary 1 hold. Then, one may obtain the impulsive controller{

Γk ≡ ΓI, −1.8165 6 Γ 6 −0.1835,

tk − tk−1 6 0.036, k ∈ Z+,(27)

such that Eq. (24) can be stabilized and becomes µ-stable, i.e., globally asymptotically stable. Moreover,

any solution x = x(t, φ) satisfies

xTx 62.1408

ln(e+ t)‖φ‖2α, t > 0,

where α = e − e5/6. In particular, when tk = 0.035k and Γk = −1.8165I, the corresponding simulation

is shown in Figure 3(b). This shows that all solutions will tend to the unique equilibrium point x⋆ =

(1, 0,−1)T under the designed impulsive control. If we let tk = 0.065k and Γk = −1.9129I, which are

against the impulsive controller (27), Figure 3(c) tells us that model (24) cannot be effectively stabilized.

The numerical results greatly reflect the advantages of our development control strategies.

Example 3. Consider the following 2D network model with unbounded distributed time-varying delays:

x(t) = −Cx(t) +Af(x(t)) +B

∫ 0.5t

0

ω(s)g(x(t− s))ds+ J(t), t > 0,

x(s) = φ(s), s ∈ [−α, 0],

(28)

subject to impulsive control: ∆x(tk) = Γkx(t−k ), k ∈ Z+, where J = [sin 0.5t, cos 0.3t]T, ω = 0.6 exp(−s),

and f = g = tanh(s). The parameter matrices C,A and B are given as follows:

C = 0.1I, A =

(

2 0.2

−0.3 1

)

, B =

(

0.2 0.1

0.1 −0.2

)

.

By our simulations, we know that model (28) is unstable, as shown in Figure 4(a). Then, consider

µ(t) = 1 + 0.5t and choose σ1 = 4, σ2 = 0.5, = 4.5 and η = 0.33 such that all conditions in Corollary 1

hold. This leads to the following impulsive controller:

Γk ≡ ΓI, −1.4714 6 Γ 6 −0.5286,

tk − tk−1 6 0.33, k ∈ Z+,(29)

Li X D, et al. Sci China Inf Sci January 2018 Vol. 61 012203:13

such that Eq. (28) can be stabilized and becomes µ-stable, as shown in Figure 4(b), for Γ = −1.4714 and

tk = 0.33k. Moreover, any two solutions u = u(t, ϕ) and v = v(t, φ) satisfy

(u− v)T(u − v) 65.4329

1 + 0.5t‖ϕ− φ‖20, t > 0.

If we keep the same impulsive weight Γ = −1.4714 but make a slight change in control intervals such as

tk = 0.5k, k ∈ Z+, then it is against our control scheme. The corresponding simulations are shown in

Figure 4(c). Note that R1 and R2 are two different curves, as well as B1 and B2. Thus, in such cases,

model (28) cannot be stabilized. This indicates that the designed control strategy is efficient as well as

effective.

5 Conclusion

Based on impulsive control theory and some analysis techniques, this paper presented some analytical

results on the µ-stability of unstable neural networks with unbounded time-varying delays, where the

addressed µ-stability includes uniform stability, (global) asymptotical stability, and (global) exponential

stability. The paper’s key idea is how to introduce impulse inputs to estimate the state information of an

entire history that is time varying, rather than an integral from negative infinity until the present time.

Our method shows that unbounded time-varying delay network models can be controlled and become

µ-stable via a proper impulsive control strategy even if it was originally unstable or divergent. Three

numerical examples and their computer simulations were given to show the effectiveness of the proposed

control strategy. Here, we also point out that although the developing results in this paper can be applied

to control problems of unstable delayed neural networks, we mainly focuses on the rational stability and

admits very limited applications for the exponential control of cases with unbounded time-varying delays

such as the problem mentioned in (18). As we know, in addition, the method of the Lyapunov-Krasovskii

functional is very popular for solving the dynamics of delayed neural networks with or without impulses,

but mainly focuses on impulsive robustness rather than impulsive control. Thus, how to develop a

method to solve the impulsive control problem in delay neural networks, even for constant delays or

bounded time-varying delays, is still an open problem. Hence, more methods and tools for the impulsive

control of delayed neural networks should be developed and explored.

Acknowledgements This work was jointly supported by National Natural Science Foundation of China (Grant

Nos. 11301308, 61673247, 61273233), Outstanding Youth Foundation of Shandong Province (Grant Nos. ZR20170-

2100145, ZR2016J L024), and Natural Sciences and Engineering Research Council of Canada.

Conflict of interest The authors declare that they have no conflict of interest.

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