Tính ổn định và ổn định hóa của một số lớp hệ phương trình vi phân phi tuyến có trễ và ứng dụng tt tiếng anh

MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
——————–o0o———————

LE DAO HAI AN

STABILITY OF NONLINEAR TIME-DELAY SYSTEMS
AND THEIR APPLICATIONS

Speciality: Differential and Integral Equations
Speciality code: 9 46 01 03

SUMMARY OF DOCTORAL THESIS IN MATHEMATICS

HANOI-2019


This dissertation has been written on the basis of my research work carried at:
Hanoi National University of Education

Supervisor:
Assoc. Prof. Le Van Hien
Dr. Tran Thi Loan

Referee 1: Professor Nguyen Minh Tri, Institute of Mathematics, Vietnam Academy
of Science and Technology
Referee 2: Professor Cung The Anh, Hanoi National University of Education
Referee 3: Associate Professor Nguyen Xuan Thao, Hanoi University of Science and
Technology

The thesis will be presented to the examining committee at Hanoi National University

of Education, 136 Xuan Thuy Road, Hanoi, Vietnam
At the time of ...., 2020

This dissertation is publicly available at:
- HNUE Library Information Centre
- The National Library of Vietnam


INTRODUCTION

1. Motivation
Time delays are widely used in modeling practical modelsin control engineering, biology and biological models, physical and chemical processes or artificial neural networks.
The presence of time-delay is often a source of poor performance, oscillation or instability.
Therefore, the stability of time-delay systems has been extensively studied during the past
decades. It is still one of the most burning problems in recent years due to the lack or the
absence of its complete solution.
A popular approach in stability analysis for time-delay systems is the use of the
Lyapunov-Krasovskii functional (LKF) method to derive sufficient conditions in terms of
linear matrix inequalities (LMIs). However, it should be noted that finding effective LKF
candidates for time-delay systems is often connected with serious mathematical difficulties
especially when dealing with nonlinear non-autonomous systems with bounded or unbounded
time-varying delay. In addition, extending the developed methodologies and existing results
in the literature to nonlinear time-delay systems proves to be a significant issue. This research topic, however, has not been fully investigated, which gives much room for further
development in particular for nonautonomous nonlinear systems with delays in the area of
population dynamics and network control. This motivates us for the present study in this
thesis.

2. Research aims
This thesis is concerned with the stability of some classes of nonlinear time-delay
systems in neural networks. Specifically, we consider the following problems

1. Investigating the problem of stability of non-autonomous neural networks with heterogeneous time-varying delays in the effect of destablizing impulses.
2. Stabilizing Hopfiled neural networks with proposition delays subject to stabilizing and
destablizing impulsive effects simultaneously.
3. Investigating the problem of exponential stability of positive equilibrium of inertial
neural networks with multiple time-varying delays.
4. Deriving conditions for the problem of exponential stability of a unique equilibrium of
positive BAM neural networks with multiple time-varying delays and nonlinear selfexcitation rates.

1


3. Objectives
3.1. Global exponential stability analysis of a class of non-autonomous neural
networks with heterogeneous delays and time-varying impulses
The states of various dynamical networks in the fields of artificial systems such as
mechanics, electronic and telecommunications networks, often suffer from instantaneous disturbances and undergo abrupt changes at certain instants. These may arise from switching
phenomena or frequency changes, and thus, they exhibit impulsive effects. With the effect of
impulses, stability of the networks may be destroyed. Therefore, delays and impulses heavily
affect the dynamical behaviors of the networks, and thus it is necessary to study both effects
of time-delay and impulses on the stability of neural networks. Up to now, considerable
effort of researchers has been devoted to investigating stability and asymptotic behavior of
neural networks with impulses.
However, the aforementioned works have been devoted to neural networks with constant
coefficients. As discussed in the many exsting literatures, non-autonomous phenomena often
occur in realistic systems, for instance, when considering a long-term dynamical behavior
of the system, the parameters of the system usually change along with time. Also, the
problem of stability analysis for non-autonomous systems usually requires specific and quite
different tools from the autonomous ones (systems with constant coefficients). There are
only few papers concerning stability of non-autonomous neural networks with heterogeneous
time-varying delays and impulsive effects.

In Chapter 2 we investigate the exponential stability of a class of non-autonomous
neural networks with heterogeneous delays and time-varying impulses
n

x′i (t)

= −di (t)xi (t) +

aij (t)fj (xj (t))
j=1

n

+
j=1

∆xi (tk )

bij (t)gj (xj (t − τij (t))) + Ii (t),

−
−
xi (t+
k ) − xi (tk ) = −σik xi (tk ),

t > 0, t = tk ,

(1)

k ∈ N.

Based on the comparison principle, an explicit criterion is derived in terms of inequalities for M-matrix ensuring the global exponential stability of the model under destabilizing
impulsive effects. The obtained results are shown improve some recent existing results. Finally, numerical examples are given to demonstrate the effectiveness the proposed conditions.

3.2. Exponential stability of impulsive neural networks with proportional delay
in the presence of periodic distribution impulses
Typically, a model of neural networks is composed of layers with a large number of cells
and connections. This fact reveals that NNs usually have a spatial nature due to the number
2


of parallel pathways, axon sizes and lengths. Thus, time delays encountered in the practical
implementation of NNs are usually time-varying. Proportional delays form a particular type
of unbounded time-varying delays, which are widely used in modeling various models in the
field of networking. It is realized that proportional delay provides most well-known quality of
service (QoS) models because of its controllable and predictable characteristics. Specifically,
when a network with proportional delays is utilized to represent an applied model, dynamics
of the system at time t is determined by its states x(t) and x(qt), where 0 < q < 1 is a
constant representing the ratio of time between current states and historical states. Thus,
the network’s running time can be controlled by the proportional factor q. Recently, the
problem of stability of various neural network models with proportional delays has attracted
considerably increasing research attention and, consequently, a large number of interesting
results have been reported in the literature.
On the other hand, impulsive dynamical systems (IDSs), in general, and impulsive
neural networks with delays (IDNNs) have received considerable research attention in recent
years. According to their strength, impulsive effects can be classified into two types named
as stabilizing impulses (SI) and destabilizing impulses (DI). An impulsive sequence is said
to be destabilizing if its effect can suppress the stability of dynamical systems while SI
can enhance the stability of dynamical systems. In most of the existing works concerning
stability of impulsive systems, SI and DI are considered separately.

In the second part of Chapter 2 we study the problem of exponential stability of the
following neural networks model
n

n

x′i (t)

= −di xi (t) +

bij gj (xj (qt)) + ui (t), t = tk ,

aij fj (xj (t)) +
j=1

j=1

(2)

−
−
∆xi (tk ) = xi (t+
k ) − xi (tk ) = −σik xi (tk ),

Both stabilizing and destabilizing impulsive effects are introduced in the model simultaneously. Based on the comparison principle, a unified stability criterion is first derived.
Then, on the basis of the derived stability conditions, the problem of designing a local state
feedback control law with bounded controller gains is addressed.

3.3. Positive solutions and global exponential stability of positive equilibrium
of inertial neural networks with multiple time-varying delays.

Conventional neural networks are typically described by first-order differential equations with or without delays. Recently, many authors focused on dynamics behaviors of
networks models called inertial neural networks (INNs). In state-space models, INNs are
described by systems of second-order differential equations, where the first-order derivative
terms are referred to as inertial terms. On one hand, there exist strong biological and engineering backgrounds for the introduction of inertial terms in neural systems, in particular
3


for those containing an inductance. On the other hand, the presence of inertial terms makes
it more difficult and challenging to analyze dynamic behaviors of INNs. Therefore, the investigation of INNs for both theoretical and practical reasons has attracted considerable
attention in the past few years.
Positive systems, in general, and positive neural networks (PNNs) can be used to
describe dynamics of various practical models where the associated state variables are subject
to positivity constraints according to the nature of phenomena. For instance, when ANNs
are designed for the purpose of identification or control of positive systems, state vectors of
the designed NNs are expected to inherit the positivity of the systems. Thus, as an essential
issue in applications, it is of interest and importance to study the stability problem of PNNs
with delays. To date, only a few results concerning stability of PNNs with delays have been
reported in the literature. In Chapter 3 we consider a class of INNs with delays described
by the following second-order differential system
d2 xi (t)
dxi (t)
= − ai
− bi xi (t) +
2
dt
dt

n

cij fj (xj (t))

j=1

n

+
j=1

dij fj (xj (t − τj (t))) + Ii , t ≥ 0, i ∈ [n].

(3)

By utilizing the comparison principle via differential inequalities, we first derive conditions
on damping coefficients, self excitation coefficients and connection weights under which all
state trajectories of the system starting from an admissible set of initial conditions are
always nonnegative. Then, based on the approach of using homeomorphisms in nonlinear
analysis, sufficient conditions for the existence, uniqueness and global exponential stability of
a positive equilibrium of the system are derived in the form of LP problems with M-matrices,
which can be effectively solved by various convex optimization algorithms.

2.4. Exponential stability of positive neural networks in bidirectional associative memory model with multiple time-varying delays and nonlinear selfexcitation rates
In 1987 Kosko introduced and studied stability and encoding properties of a class of
two-layer nonlinear feedback neural networks. Accordingly, the bidirectionality, referred to
forward and backward information flows, was introduced in neural nets to produce two-way
associative search for stored paired-data associations. This model was later called bidirectional associative memory (BAM) neural network. Roughly speaking, a BAM neural network
is composed of a number of neurons arranged into two layers namely X-layer and Y-layer.
The neurons in each layer are connected in the way that neurons in one layer are completely integrated to neurons in the other layer, whereas there is no interconnection among
neurons in the same layer. This structure performs a two-way associative search for stored
4

bipolar vector pairs and generalizes the single-layer autoassociative Hebbian correlation to
a two-layer pattern-matched heteroassociative circuits. Thus, BAM model possesses many
application prospects in the areas of pattern recognition, signal and image processing.
Typical applications of neural networks, for example, in optimization, control, or signal
processing, require that neural networks are designed to admit only one equilibrium point
which is globally asymptotically stable thereby avoiding the risks of having spurious equilibrium point and being trapped in local minima. Thus, it is relevant and very important to
study the stability problem of a unique equilibrium of a dynamic neural network. In Chapter 4 we investigate the problem of exponential stability of a unique equilibrium of positive
BAM neural networks with multiple time-varying delays and nonlinear self-excitation rates
m

m

x′i (t)

= −αi ϕi (xi (t)) +

yj′ (t) = −βj ψj (yj (t)) +

aij fj (yj (t)) +
j=1
n

j=1
n

cji gi (xi (t)) +
i=1

i=1

bij fj (yj (t − σj (t))) + Ii ,

(4)

dji gi (xi (t − τi (t))) + Jj .

(5)

A systematic approach involving extended comparison techniques via differential inequalities is presented. Combining with the use of Brouwer’s fixed point theorem and Mmatrix theory, tractable LP-based conditions are derived to ensure the existence and global
exponential stability of a unique positive equilibrium of the system. An extension to the
case of BAM neural networks with proportional delays is also presented.

4. Main contributions
1. Established conditions in terms of M-matrices forexponential stability of Hopfiled neural
networks with time-varying coefficients and destablizing impulses.
2. A unified stability criterion is first derived for the stability problem and application to
designing a local state feedback control law with bounded controller gains for Hopfiled
neural networks with proposition delays and periodic distribution impulses.
3. Based on the approach of using homeomorphisms in nonlinear analysis, sufficient conditions for the existence, uniqueness and global exponential stability of a positive equilibrium of inertial neural networks are derived in the form of LP problems with M-matrices.
4. Based on an extended comparison techniques via differential inequalities combining
with the use of Brouwer’s fixed point theorem and M-matrix theory, tractable LPbased conditions are derived to ensure the existence and global exponential stability of
a unique positive equilibrium of BAM neural networks with heterogeneous delays. An
extension to the case of BAM neural networks with proportional delays is also obtained.
The results presented in this thesis are based on 4 papers published on ISI/Scopus
international journals.
5


5. Thesis outline
Except the Introduction, Conclusion, List of Publications, and List of References, the

remaining of the thesis is devided into four chapters. Chapter 1 presents some preliminary results. Chapter 2 investigates the problem of exponential stability of Hopfield neuron
networks with time-varying connection weights and heterogeneous delays with impulses.
Positive solutions and the existence of a unique equilibrium of inertial neural networks with
time-varying delays is studied and presented in Chapter 3. Finally, the existence, uniqueness
and global exponential stability of a positive equilibrium to BAM neural networks is studied
in Chapter 4.

6


Chapter 1
PREMILINARIES

In this chapter, we briefly introduce state space model of biology neural networks and
present some auxiliary results in matrix analysis, differential equations, stability theory in
the sense of Lyapunov, impulsive differential systems.

1.1. Dynamic equation of biology neural networks
This section briefly introduces the history and general state space model of biology
neural networks.

1.2. Mathematical fundamentals
1.2.1. Time-delay systems and the Lyapunov stability theory
Fundamental results on the theory of functional differential equations are presented.

1.2.2. Functional impulsive differential equations
Basic theory of impulsive differential equations is presented.

1.2.3. Positive systems
This section briefly presents the positive linear system and introduces a result of the

stability of the nonlinear positive system

1.2.4. Auxiliary results
This section presents some technical lemmas and auxiliary results which will be used
in the next chapters.

7


Chapter 2
EXPONENTIAL STABILITY OF NON-AUTONOMOUS NEURAL NETWORKS
WITH HETEROGENEOUS TIME-VARYING DELAYS AND DESTABILIZING
IMPULSES

In this chapter we study the problem of exponential stability of non-autonomous neural
networks with heterogeneous delays. Based on the comparison principle, an explicit criterion
is derived in terms of inequalities for M-matrix ensuring the global exponential stability of
the model under destabilizing impulsive effects. The obtained results are shown improve
some recent existing results. Finally, numerical examples are given to demonstrate the
effectiveness the proposed conditions.

2.1. A motivation example
2.2. Nonautonomous neural networks with heterogeneous time-varying
delays and destablizing impulses
Consider a class of non-autonomous impulsive neural networks with heterogeneous
time-varying delays of the following form
n

x′i (t)

= −di (t)xi (t) +

aij (t)fj (xj (t))
j=1

n

+
j=1

bij (t)gj (xj (t − τij (t))) + Ii (t), t = tk ,

(2.1)

−
−
+
∆xi (tk ) = xi (t+
k ) − xi (tk ) = −σik xi (tk ), t = tk , k ∈ Z ,

xi (t) = φi (t),

t ∈ [−τ, 0], i ∈ [n].

(A2.1) The matrices D(t) = diag(d1 (t), d2 (t), . . . , dn (t)), A(t) = (aij (t)) and B(t) = (bij (t))
are continuous on each interval (tk , tk+1 ), k ≥ 0, and there exist scalars dˆi , a+ , b+ such
ij

ij

that

di (t) ≥ dˆi > 0,

|aij (t)| ≤ a+
ij ,

|bij (t)| ≤ b+
ij ,

∀t ≥ 0, i, j ∈ [n].

(A2.2) The neural activation functions fi , gi , i ∈ [n], satisfy
−
≤
ljf

fj (a) − fj (b)
+
≤ ljf
,
a−b

−
ljg
≤

gj (a) − gj (b)
+
≤ ljg

,
a−b

∀a, b ∈ R, a = b,

−
+
−
+
where ljf
, ljf
, ljg
and ljg
are known constants.

(A2.3) There exists positive sequence (γk )k∈Z+ such that 1 − γk ≤ σik ≤ 1 + γk , ∀i ∈ [n],
k ∈ N.

8


− + − +
Remark 2.2.1. The constants ljf
, ljf , ljg , ljg , j ∈ [n] in assumption (A2.2) are allowed

to be positive, negative or zero. As discussed in the existing literature, for autonomous
neural networks, assumption (A2.2) can lead to less conservative stability conditions than the
descriptions on the Lipschitz-type activation functions or the sigmoid activation functions.
However, in order to establish stability conditions for non-autonomous impulsive neural
network (2.1) we ultilize the following estimations which can be easily derived from (A2.2)

+
−
, −ljf
}|a − b|,
|fj (a) − fj (b)| ≤ max{ljf

+
−
|gj (a) − gj (b)| ≤ max{ljg
, −ljg
}|a − b|.

+
−
+
−
, −ljg
}.
, −ljf
} and ljg = max{ljg
Hereafter, let us denote for j ∈ [n] the constants ljf = max{ljf

Remark 2.2.2. Under assumptions (A2.1), (A2.2), for each initial function φ ∈ C([−τ, 0], Rn ),
there exists a unique solution x(t, φ) of (2.1) which is piecewise continuous on R+ with possible discontinuities at t = tk , k ∈ N.
Remark 2.2.3. When γk > 1, the absolute value of the state can be enlarged and the
impulses can potentially destroy the stability of system (2.1). We refer this type of impulses
to as destabilizing impulses. When γk ≤ 1, the impulsive effects are inactive or stabilizing.

In this paper, and as mentioned in the preceding section, we assume that the impulses are
destabilizing and taking values in a finite set {µ1 , µ2 , . . . , µq }, where µi > 1, i ∈ [q].

Let us denote tjk , j ∈ [q], the activation times of the destabilizing impulses with

impulsive strength µj , that means tjk = tk if γk = µj .

Remark 2.2.4. It is well known that when the network dynamics are stable but the impulsive effects are destabilizing, the impulses should not occur too frequently in order to
guarantee stability. In this chapter, we derive conditions ensuring that the non-autonomous
neural network (2.1) is globally exponentially stable under destabilizing impulsive effects. In
regard to this observation, we assume that
(A2.4) There are positive numbers ρj such that
tj(k+1) − tjk ≥ ρj ,

∀j ∈ [q], k ∈ N.

Remark 2.2.5. In the case of constant impulse, q = 1 and assumption (A2.4) can be replaced
by the average impulsive interval condition, that is there exist positive integer N0 and positive
number Ta such that
t−s
t−s
− N0 ≤ Nζ (t, s) ≤
+ N0 ,
Ta
Ta

∀t > s ≥ 0,

(2.2)

where Nζ (t, s) denotes the number of impulsive times of the impulsive sequence ζ = {t1 , t2 , . . .}
on interval (s, t).

Definition 2.2.1. The impulsive neural network (2.1) is said to be globally exponentially
stable if there exist positive constants α, β such that, for any two solutions x(t), xˆ(t) of (2.1)
with respectively initial function φ, ψ ∈ C([−τ, 0], Rn ), the following inequality holds
x(t) − xˆ(t)

∞

≤β φ−ψ
9

−αt
,
∞e

t ≥ 0.

(2.3)


The main objective of this section is to derive new conditions in terms of testable matrix
inequalities ensuring the global exponential stability of the neural network (2.1) based on
M-matrix theory and some efficient techniques which have been developed for time-varying
systems with bounded delays.

2.3. Stability conditions
To facilitate in presenting our results, let us introduce the following matrix notations
q

ˆ = diag(dˆ1 , dˆ2 , . . . , dˆn ),
D

Lf = diag(l1f , l2f , . . . , lnf ),

Aˆ =

(a+
ij ),

ˆ=
B

+
eσ0 τij b+
ij

,

σ0 =
j=1

ln µj
,
ρj

Lg = diag(l1g , l2g , . . . , lng ),

ˆ f + BL
ˆ g + σ0 En − D.
ˆ
M = AL
Theorem 2.3.1. Let Assumptions (A2.1)-(A2.4) hold. Then the impulsive neural network

(2.1) is globally exponentially stable if there exists a vector χ ∈ Rn+ such that
ˆ f + BL
ˆ g + σ0 En − D
ˆ χ ≺ 0.
AL

(2.4)

Remark 2.3.1. It can be found in many existing works which deal with time-varying impulses, the impulsive strength sequence (γk ) is usually assumed to be bounded that is there
exists a constant µ > 0 such that γk ≤ µ, ∀k ∈ N.
Corollary 2.3.2. Assume that Assumptions (A2.1)-(A2.3) hold and there exists a Ta > 0
satisfying (2.2). Then the neural network (2.1) is globally exponentially stable if there exist
a constant µ ≥ 1 and a vector χ˜ ∈ Rn+ such that γk ≤ µ, k ∈ N, and
ˆ f + BL
˜ g +σ
ˆ χ
AL
˜0 En − D
˜ ≺ 0,
where σ
˜0 =

ln µ
Ta

(2.5)

+

˜ = eσ˜0 τij b+ .

and B
ij

Remark 2.3.2. It should be pointed out that Theorem 2.3.1 and Corollary 2.3.2 are devoted
to non-autonomous neural networks with bounded impulses. However, it can be seen from
the proof of Theorem 2.3.1 that our approach can also be used for non-autonomous neural
networks with unbounded impulses. In that case, the following condition which is widely
used in the literature can be employed
∃γ0 ≥ 0 :

ln γk
≤ γ0 , ∀k ≥ 1.
tk − tk−1

(2.6)

Then stability conditions of the network (2.1) incorporating (2.6) are formulated in the
following corollary.
Corollary 2.3.3. Let Assumptions (A2.1)-(A2.3) and condition (2.6) hold. Then the neural
network (2.1) is globally exponentially stable if there exists a vector χˆ ∈ Rn+ satisfying
ˆ f + BL
ˇ g + γ0 En − D
ˆ χˆ ≺ 0
AL
10

(2.7)


+

ˇ = eγ0 τij b+ .
where B
ij
As a special case, when σik = 0, i ∈ [n], k ≥ 1, and I(t) = 0, system (2.1) becomes the

following nonlinear non-autonomous system without impulses
n

n

x′i (t)

= −di (t)xi (t) +

xi (t) = φi (t),

aij (t)fj (xj (t)) +
j=1

j=1

bij (t)gj (xj (t − τij (t))),

t ≥ 0,

(2.8)

t ∈ [−τ, 0].

From the proof of Theorem 2.3.1 we get the following result.
Corollary 2.3.4. Under Assumptions (A2.1), (A2.2), assume that there exists a vector υ ∈

Rn+ such that

ˆ f + B + Lg − D
ˆ υ ≺ 0,
AL

(2.9)

where B + = (b+
ij ). Then system (2.8) is globally exponentially stable. Furthermore, every
solution x(t, φ) of (2.8) satisfies
x(t, φ)

∞

≤ Cυ φ

−η0 t
,
∞e

t ≥ 0,

υi
where Cυ = max1≤i≤n ( min1≤j≤n
υj ), 0 < η0 ≤ min1≤i≤n ηi and ηi is the unique positive

solution of the scalar equation

1
−dˆi +
υi

n

+

f
+ ηi τij g
a+
lj υj + ηi = 0.
ij lj + bij e

(2.10)

j=1

2.4. Stabilization of Hopfield neural networks with proportional delays and periodic distribution impulsive effects
Consider a neural network model with a proportional delay described as
n

n

x′i (t)

= −di xi (t) +

bij gj (xj (qt)) + ui (t), t = tk ,

aij fj (xj (t)) +
j=1

j=1

−
−
∆xi (tk ) = xi (t+
k ) − xi (tk ) = −σik xi (tk ),

(2.11)

xi (t) = x0i , t ∈ [qt0 , t0 ], i ∈ [n],

where n is the number of neurons, xi (t) and ui (t) are the state variable and control input of
ith neuron at time t, respectively. The factor q ∈ (0, 1) is a constant involving history time.

More specifically, in the interpretation of model (2.11), dynamics of ith neuron at time t is
determined by the current states xj (t), j ∈ [n], and the states xj (qt) at history time qt which

is proportional to current time t with a constant rate q. In this meaning, the constant q is
referred to as proportional delay. Since qt = t − τ (t), where τ (t) = (1 − q)t → ∞ as t → ∞,
proportional delays form a class of unbounded time-varying delays.

The strengths of stabilizing impulses and destablizing impulses are assumed to take
values in finite sets Is = {ρ1 , ρ2 , . . . , ρM } and Iu = {µ1 , µ2 , . . . , µN }, where 0 < ρi < 1 for
11

i ∈ [M] and µj > 1 for j ∈ [N]. In addition, we denote as tsik and tujk the impulsive instances
of stabilizing impulses with strength ρi and the impulsive instances of destabilizing impulses
with strength µj , respectively. That means, for any i ∈ [M] and j ∈ [N], tsik = tk if γk = ρi
and tujk = tk if γk = µj .

(A2.5) There exist positive numbers τis , τju , and integers qi ∈ N0 , rj ∈ N0 , i ∈ [M], j ∈ [N],
satisfying the following conditions for any t > s ≥ t0

t−s
t−s
− qi ≤ Nρi (t, s) ≤ s + qi ,
s
τi
τi
t−s
t−s
− rj ≤ Nµj (t, s) ≤ u + rj ,
u
τj
τj

(2.12)

where Nρi (t, s) and Nµj (t, s) present the frequencies of impulsive strengths ρi and µj on
interval (s, t), respectively.
We will design a local state feedback control law (LSFCL) of the form
ui (t) = −ki xi (t), i ∈ [n],

(2.13)

to stabilize system (2.11), where ki , i ∈ [n], are controller gains. Due to practical config-

urations of the inputs, we assume the controller gains ki , i ∈ [n], are confined in intervals

[kil , kiu ], where kil , kiu , i ∈ [n], are known constants. Under the LSFCL (2.13), the closed-loop

system of (2.11) can be written as


x′ (t) = −D x(t) + Af (x(t)) + Bg(x(qt)), t = t ,
c
k
x(t+ ) = Jk x(t− ), k ∈ N,
k

(2.14)

k

where Dc = diag(di + ki ), A = (aij ), B = (bij ), f (x(t)) = (fj (xj (t))), g(x(qt)) = (gj (xj (qt)))
and Jk = diag(1 − σik ).

2.4.1. Stability of the closed-loop system (2.14)
Definition 2.4.1. System (2.14) is said to be generalized globally exponentially stable (GGES)
if there exist a positive scalar κ and an increasing function σ(t) > 0, σ(t) → ∞ as t → ∞,

such that any solution x(t) = x(t, x0 ) of (2.14) satisfies the following estimation
x(t) ≤ κ x0 e−σ(t) , t ≥ t0 .
We denote the matrices |A| = (|aij |), |B| = (|bij |) and

⊤
M = −2Dc + sym(|A|Lf ) + θ−1 |B|Lg L⊤
g |B| , θ > 0,

where Lf = diag(Lfi ) and Lg = diag(Lgi ).
Theorem 2.4.1. Let assumptions (A2.2), (A2.3) and (A2.5) hold. Assume that there exist
positive scalars α and θ satisfying the following conditions
M + αEn < 0,
12

(2.15a)


(2.15b)

α > pθ,
M

ln(ρi )
+
τis

i=1

where p =

M
i=1

2rj −2qi

N
.
j=1 µj ρi

N

j=1

ln(µj )
= 0,
τju

(2.15c)

Then, system (2.14) is GGES. More precisely, there exists

a constant σ > 0 such that any solution x(t) = x(t, x0 ) of (2.14) satisfies
√
p
σ
x0 e− 2 ln(1+t) , t ≥ t0 .
x(t) ≤
(1 + qt0 )σ

(2.16)

By the Schur complement lemma, conditions (2.15a) and (2.15b) can be recast into the
following linear matrix inequalities (LMIs)
−2Dc + sym(|A|Lf ) + αEn |B|Lg
Lg |B|⊤

< 0,

(2.17a)

pθ − α < 0.

(2.17b)

−θEn

As a special case of model (2.11), let us consider the following neural network model
with alternatively impulsive effects


x(t)
˙
= −Dx(t) + Af (x(t)) + Bg(x(qt)), t = kTs ,
x(t+ ) = γk x(t− ),
k

(2.18)

k

where Ts > 0 is a sampling time. Assume that there exists a scalar γ∗ , 0 < |γ∗ | = 1, such

that γ2k+1 = γ∗ and γ2k+2 = γ∗−1 for any k ∈ N0 . It is clear that
t−s
t−s

− 1 ≤ Nγ∗ (t, s) ≤
+ 1,
2Ts
2Ts
t−s
t−s
− 1 ≤ Nγ −1 (t, s) ≤
+ 1.
∗
2Ts
2Ts

By Theorem 2.4.1, we have the following result.
Corollary 2.4.2. Under assumptions (A2.2), (A2.3) and (A2.5), system (2.18) is GGES if
there exists a scalar θ > 0 satisfying the following condition
θ max γ∗2 ,

1
γ∗2

+ m < 0,

(2.19)

⊤ .
where m = λmax −2D + sym(|A|Lf ) + θ−1 |B|Lg L⊤
g |B|

2.4.2. Stabilization coditions
Based on condition (2.17), the problem of designing a LSFCL (2.13) that makes the

closed-loop system (2.14) GGES is presented in the following theorem.
Theorem 2.4.3. Under assumptions (A2.2), (A2.3) and (A2.5), assume that condition (2.15c)
is satisfied. Then, system (2.11) is exponentially stabilizable under LSFCL (2.13) if the following LMIs are feasible for scalar α > 0, θ > 0, and a diagonal matrix Z = diag(zi ) ∈ Rn×n
13


−2D + sym(|A|Lf ) + αEn + Z |B|Lg

< 0,

(2.20a)

Z + 2diag(kil ) ≤ 0,

(2.20b)

α > pθ.

(2.20c)

Lg |B|⊤

−θEn

Z + 2diag(kiu ) ≥ 0,

Moreover, the controller gain matrix is given by
1
Kc = − Z.
2

14

(2.21)


Chapter 3
POSITIVE SOLUTIONS AND EXPONENTIAL STABILITY OF INNERTIAL
NEURAL NETWORKS WITH TIME-VARYING DELAYS

In this chapter we consider a class of INNs with delays. By utilizing the comparison
principle via differential inequalities, we first derive conditions on damping coefficients, self
excitation coefficients and connection weights under which all state trajectories of the system
starting from an admissible set of initial conditions are always nonnegative. Then, based on
the approach of using homeomorphisms in nonlinear analysis, sufficient conditions for the
existence, uniqueness and global exponential stability of a positive equilibrium of the system
are derived in the form of LP problems with M-matrices, which can be effectively solved by
various convex optimization algorithms.

3.1. Model description
Consider a class of INNs with delays described by the following second-order differential
system
d2 xi (t)
dxi (t)
= − ai
− bi xi (t) +
2
dt
dt

n

cij fj (xj (t))
j=1

n

+
j=1

dij fj (xj (t − τj (t))) + Ii , t ≥ 0, i ∈ [n].

(3.1)

Denote x(t) = (x1 (t), x2 (t), . . . , xn (t))⊤ as the state vector, A = diag{a1 , . . . , an },

B = diag{b1 , . . . , bn }, C = (cij )n×n , D = (dij )n×n and I = (I1 , I2 , . . . , In )⊤ . System (3.1)
can be written in the following matrix-vector form

x′′ (t) = −Ax′ (t) − Bx(t) + Cf (x(t)) + Df (xτ (t)) + I,

(3.2)

where f (x(t)) = (fj (xj (t))) and f (xτ (t)) = (fj (xj (t − τj (t)))). In regard to (3.2), each initial
condition of (3.1) is defined by compatible functions φ = (φi ) and φˆ = (φˆi ) in C([−τ + , 0], Rn )
as
xi (s) = φi (s), x′i (s) = φˆi (s), s ∈ [−τ + , 0], i ∈ [n].

(3.3)

By using the following state transformation
dxi (t)
ηi yi (t) =
+ ξi xi (t), i ∈ [n],
(3.4)
dt
where ηi = 0 and ξi , i ∈ [n], are constants, system (3.1) can be written in the following
vector form


x′ (t) = −D x(t) + D y(t),
η
ξ
y ′(t) = −Dα y(t) + Dβ x(t) + D−1 [Cf (x(t)) + Df (xτ (t)) + I] ,
η

15

(3.5)


where y(t) = (y1 (t), y2 (t), . . . , yn(t))⊤ , Dξ = diag{ξ1 , . . . , ξn }, Dη = diag{η1 , . . . , ηn }, Dα =

diag{α1 , . . . , αn }, Dβ = diag{β1 , . . . , βn } and αi = ai − ξi , βi = ηi−1 (αi ξi − bi ), i ∈ [n].

Assumption (A3): The activation functions fj (.), j ∈ [n], are continuous and there

exist positive constants ljf , j ∈ [n], satisfying the following condition

fj (a) − fj (b)

≤ ljf , a = b.
(3.6)
a−b
Proposition 3.1.1. Under assumption (A3), for any initial condition (3.3), there exists a
0≤

unique solution of system (3.1) defined on the interval [0, ∞), which is absolutely continuous

in t.

Let x(t) be a solution of (3.1). If its trajectory is confined within the first orthant
(i.e. x(t) ∈ Rn+ for all t ≥ 0), then x(t) is said to be a positive solution. For a given

transformation (3.4), where ηi > 0 and ξi > 0, i ∈ [n], we define the following set of
admissible initial functions for system (3.1)
AT =

(φi ), (φˆi ) ∈ C([−τ + , 0], Rn) : φi (s) ≥ 0,
ψi = ηi−1 (ξi φi (s) + φˆi (s)) ≥ 0, s ∈ [−τ + , 0], i ∈ [n] .

(3.7)

Clearly, AT contains all nonnegative nondecreasing initial functions φi with φˆi (s) = φ′i (s).
Definition 3.1.1. System (3.1) is said to be positive if for any initial condition that belongs
to AT and any input vector I = (Ii ) ∈ Rn+ , the corresponding solution x(t) of (3.1) is
positive.

Definition 3.1.2. Given an input vector I ∈ Rn+ . A vector x∗ ∈ Rn+ is said to be a positive

equilibrium point (EP) of system (3.1) if it satisfies the following algebraic system

−Bx∗ + Cf (x∗ ) + Df (x∗ ) + I = 0.

(3.8)

The following auxiliary result will be used to issue the existence of such positive EP
for system (3.1).
Lemma 3.1.2. Assume that Dη ≻ 0 and Dξ ≻ 0. Then, a vector x∗ ∈ Rn+ is a positive

EP of system (3.1) if and only if (x∗ , y∗ ), where y∗ = Dη−1 Dξ x∗ , is a positive EP of system
(3.5), that is,


−D x + D y = 0,
η ∗
ξ ∗
Dη (−Dα y∗ + Dβ x∗ ) + Cf (x∗ ) + Df (x∗ ) + I = 0.

(3.9)

3.2. Positive solutions of INNs with delays

Lemma 3.2.1. For given coefficients ai > 0, bi > 0, there exists a transformation (3.4) with
ηi > 0 and ξi > 0, i ∈ [n], such that Dα ≻ 0 and Dβ ≻ 0 if and only if the following
condition holds

a2i − 4bi > 0, i ∈ [n].
16

(3.10)

Theorem 3.2.2. Let assumption (A3) and condition (3.10) hold. Assume that C = (cij )
and D = (dij )

0

0. Then, system (3.1) is positive for any bounded delays.

3.3. Existence of an equilibrium for INNs
In this section, we derive conditions for the existence of a unique EP for INNs in the
form of (3.1). By Lemma 3.1.2, system (3.1) possesses an EP if and only if system (3.5)
does. As revealed from (3.9), for a given input vector I ∈ Rn , an EP of (3.1) exists if and
x
, x, y ∈ Rn , and the
only if the equation H(χ) = 0 has a solution χ∗ ∈ R2n , where χ =
y
mapping H : R2n → R2n is defined by
H(χ) =

−Dξ x + Dη y

(Dαξ − B)x − Dαη y + Cf (x) + Df (x) + I

(3.11)

,

where Dαξ = Dα Dξ and Dαη = Dα Dη .
Theorem 3.3.1. Assume the assumptions of Theorem 3.2.2 hold and there exists a vector
χ0 ∈ R2n , χ0 ≻ 0, such that

⊤

−Dξ

Dη

Dαξ − B + (C + D)Lf

−Dαη

χ0 ≺ 0.

Then, for any given input vector I ∈ Rn , there exists a unique EP χ∗ =

(3.12)
x∗
y∗

of system

(3.5).

3.4. Exponential stability of positive equilibrium
In this section, we will prove that under the assumptions of Theorem 3.2.2, the unique
EP χ∗ of (3.5) is positive and GES for any delays τj (t) ∈ [0, τ + ].

Theorem 3.4.1. Under the assumptions of Theorem 3.2.2, if there exists a vector χ
ˆ0 ∈ R2n ,
χˆ0 ≻ 0, such that

−Dξ

Dη

Dαξ − B + (C + D)Lf

−Dαη

χˆ0 ≺ 0,

then, for any input vector I ∈ Rn+ , system (3.5) has a unique positive EP χ∗ =
which is GES for any delays τj (t) ∈ [0, τ + ].

17

(3.13)
x∗
y∗

∈ R2n
+,


Chapter 4
EXPONENTIAL STABILITY OF POSITIVE NEURAL NETWORKS IN
BIDIRECTIONAL ASSOCIATIVE MEMORY MODEL WITH DELAYS

In this chapter we consider the problem of exponential stability of positive neural networks in bidirectional associative memory (BAM) model with multiple time-varying delays
and nonlinear self-excitation rates. Based on a systematic approach involving extended
comparison techniques via differential inequalities, we first prove the positivity of state trajectories initializing from a positive cone called the admissible set of initial conditions. In

combination with the use of Brouwer’s fixed point theorem and M-matrix theory, we then
derive conditions for the existence and global exponential stability of a unique positive equilibrium of the model. An extension to the case of BAM neural networks with proportional
delays is also presented.

4.1. Model description
Consider the following BAM neural networks model with delays
m

m

x′i (t)

= −αi ϕi (xi (t)) +

yj′ (t) = −βj ψj (yj (t)) +

aij fj (yj (t)) +
j=1
n

j=1
n

cji gi (xi (t)) +
i=1

i=1

bij fj (yj (t − σj (t))) + Ii ,

(4.1)

dji gi (xi (t − τi (t))) + Jj , t ≥ t0 . (4.2)

We denote the matrices Dα = diag{α1 , α2 , . . . , αn }, Dβ = diag{β1 , β2 , . . . , βm }, A =

(aij ) ∈ Rn×m , B = (bij ) ∈ Rn×m , C = (cji ) ∈ Rm×n , D = (dji ) ∈ Rm×n and vectors

I = (Ii ) ∈ Rn , J = (Jj ) ∈ Rm , then system (4.1)-(4.2) can be written in the following

matrix-vector form


x′ (t) = −D Φ(x(t)) + Af (y(t)) + Bf (y(t − σ(t))) + I,
α
y ′(t) = −Dβ Ψ(y(t)) + Cg(x(t)) + Dg(x(t − τ (t))) + J.

(4.3)

Let D be the set of continuous functions ϕ : R → R satisfying ϕ(0) = 0 and there exist
positive scalars lϕ , l˜ϕ such that the following condition
lϕ ≤

ϕ(a) − ϕ(b) ˜
≤ lϕ
a−b

(4.4)

holds for all a, b ∈ R, a = b. Clearly, the function class D includes all linear functions

ϕ(a) = γϕ a where γϕ is some positive scalar.

Assumption (A4.1): The decay rate functions ϕi , ψj , i ∈ [n], j ∈ [m], are assumed to belong

the function class D.

18


Assumption (A4.2): The neuron activation functions fj (.), gi (.), i ∈ [n], j ∈ [m], are

continuous, fj (0) = 0, gi (0) = 0, and there exist positive constants Lfj , Lgi such that
0≤

fj (a) − fj (b)
≤ Lfj ,
a−b

0≤

gi (a) − gi (b)
≤ Lgi ,
a−b

a = b.

(4.5)

Proposition 4.1.1. Under Assumptions (A4.1) and (A4.2), for any initial functions x0 ∈

C([−τ , 0], Rn ), y 0 ∈ C([−σ, 0], Rm ), there exists a unique solution vec(x(t), y(t)) of system
(4.3) defined on [t0 , ∞), which is absolutely continuous in t.

Let χ(t) = vec(x(t), y(t)) be a solution of system (4.3). If the trajectory of χ(t) is
confined within the first orthant, that is, χ(t) ∈ Rn+m
for all t ≥ t0 , then χ(t) is said to be
+

a positive solution of (4.3). We define the following admissible set of initial conditions for
system (4.3)
A=

φ=

x0
y0

: x0 (ξ)

0, ∀ξ ∈ [−τ , 0], y 0(θ)

0, ∀θ ∈ [−σ, 0]

.

(4.6)

Definition 4.1.1. System (4.3) is said to be positive if for any initial condition φ ∈ A and
nonnegative input vector vec(I, J) ∈ Rn+m

, the corresponding solution χ(t) = vec(x(t), y(t))
+
of (4.3) is positive.

Definition 4.1.2. For a given input vector J = vec(I, J) ∈ Rn+m , a point χ∗ = vec(x∗ , y ∗ ),

x∗ ∈ Rn , y ∗ ∈ Rm , is said to be an equilibrium point (EP) of system (4.3) if it satisfies the
following algebraic system


−D Φ(x∗ ) + (A + B) f (y ∗) + I = 0
α
−Dβ Ψ(y ∗ ) + (C + D) g(x∗ ) + J = 0.

Moreover, χ∗ is a positive EP if it is an EP and χ∗

(4.7)

0.

Definition 4.1.3. A positive EP χ∗ = vec(x∗ , y ∗ ) of system (4.3) is said to be globally
exponentially stable (GES) if there exist positive scalars κ, γ such that any solution χ(t) =
vec(x(t), y(t)) of (4.3) satisfies the following inequality
x(t) − x∗ + y(t) − y ∗ ≤ κ

x0 − x∗

∞

+ y0 − y∗

∞

e−γ(t−t0 ) , t ≥ t0 .

4.2. Positive solutions of BAM neural networks with delays
Theorem 4.2.1. Under assumptions (A4.1) and (A4.2), if the connection weight matrices
A
B
A, B, C, and D are nonnegative (equivalently, M =
0), then system (4.3)
C ⊤ D⊤
is positive subject to bounded delays and the admissible set of initial conditions A.

4.3. Existence of an equilibrium
First, it can be verified from system (4.7) that, for a given input vector J = vec(I, J) ∈

Rn+m , a vector χ∗ = vec(x∗ , y ∗) ∈ Rn+m is an EP of model (4.3) if and only if it satisfies
19


the following algebraic system


D−1 ((A + B)f (y ∗) + I) = Φ(x∗ )
α

(4.8)

D−1 ((C + D)g(x∗ ) + J) = Ψ(y ∗).

β

Revealed by system (4.8), we define a mapping H : Rn+m → Rn+m by
Φ−1 Dα−1 ((A + B)f (y) + I)

H(χ) =

Ψ−1 Dβ−1 ((C + D)g(x) + J)

(4.9)

,

where χ = vec(x, y), x ∈ Rn and y ∈ Rm . More specifically, the mapping H(χ) defined in
(4.9) can be written as

˜ 1 (x) · · · h
˜ m (x)
H(χ) = h1 (y) · · · hn (y) h

⊤

,

where
hi (y) =

ϕ−1
i

˜ j (x) = ψ −1
h
j

1
αi
1
βj

m

(aij + bij ) fj (yj ) + Ii

, i ∈ [n],

(cji + dji ) gi (xi ) + Jj

, j ∈ [m],

j=1
n

i=1

−1
and ϕ−1
i (.), ψj (.) denote the inverse functions of ϕi (.) and ψj (.), respectively. In regards

to equations (4.8) and (4.9), a vector χ∗ ∈ Rn+m is an EP of model (4.3) if and only if it is

a fixed point of the mapping H(χ), that is, H(χ∗ ) = χ∗ . Based on the Brouwer’s fixed point

theorem, we have the following result.

Theorem 4.3.1. Let assumptions (A4.1) and (A4.2) hold. Assume that
0n×n

ρ

K2

K1

0m×m

< 1,

(4.10)

1 ) ∈ Rn×m , K = (k 2 ) ∈ Rm×n and
where K1 = (kij
2
ji

1 −1
l (|aij | + |bij |) Lfj ,
αi ϕi
1
2
(|cji | + |dji |) Lgi .

kji
= lψ−1
j
βj

1
kij
=

Then, system (4.3) has at least one EP.
Remark 4.3.1. In the proof of Theorem 4.3.1, if the connection weight matrices A, B, C,
D, and input vectors I, J are nonnegative, then
ui

1
αi

and
vj

1
βj

m

(aij + bij ) fj (yj ) + Ii

≥ 0, i ∈ [n],

(cji + dji ) gi (xi ) + Jj

≥ 0, j ∈ [m],

j=1
n

i=1

20


for any x

˜ j (x) = ψ −1 (vj ) ≥ 0 for all
0. Thus, hi (y) = ϕ−1 (ui ) ≥ 0 and h
j

0 and y

i ∈ [n], j ∈ [m], and χ = vec(x, y)

⊂ Rn+m
. This shows that
0. Therefore, H Rn+m
+
+

H : B+ → B+ , where B+ = B ∩ Rn+m
. Since
+

B+ =

χ ∈ Rn+m

0

δ

χ

̺

is also a convex compact subset of Rn+m , by the Brouwer’s fixed point theorem, the continuous mapping H has at least one fixed point χ∗+ ∈ B+ , which is a positive EP of system

(4.3). We summarize this result in the following corollary.

Corollary 4.3.2. With the assumptions of Theorem 4.3.1, if the connection weight matrices
A, B, C and D are nonnegative, then for any nonnegative input vector J = vec(I, J),
system (4.3) has at least one positive EP χ∗ ∈ Rn+m
.
+

˜ be an (n + m) × (n + m) block matrix of the form
Remark 4.3.2. Let Ω
˜=
Ω

˜ V˜
U
˜ Z˜

W

˜ ∈ Rm×n and Z˜ ∈ Rm×m . If Z˜ is nonsingular, then we
where U˜ ∈ Rn×n , V˜ ∈ Rn×m , W
have

˜ V˜
U˜ − V˜ Z˜ −1 W
0m×n
Z˜

=

U˜ V˜
˜ Z˜
W

En
0n×m
˜
−Z˜ −1 W
Em

.

Therefore,
˜ = det(U˜ − V˜ Z˜ −1 W
˜ ) det(Z).
˜
det(Ω)

(4.11)

By utilizing equality (4.11), for any λ = 0, we have
det(λEn+m − K) = det

λEn −K1

−K2 λEm

1
= λm det λEn − K1 K2
λ
= λm−n det(λ2 En − K1 K2 ).
The aforementioned identity shows that λ ∈ σ(K)\{0} if and only if µ = λ2 ∈ σ(K1 K2 )\{0}.
Consequently,

ρ(K) < 1 ⇐⇒ ρ(K1 K2 ) < 1.
Based on the above observation, we have the following results.
Proposition 4.3.3. Condition (4.10) holds if and only if one of the two following conditions
does
ρ

1 −1
l
αi ϕi

m

j=1

1 −1
l (|aij | + |bij |) |cjk | + |djk | Lfj Lgk
βj ψj
21

< 1;
n×n

(4.12)


1 −1
l
βj ψj

ρ

n

i=1

1 −1
l (|cji | + |dji |) (|aik | + |bik |) Lfk Lgi
αi ϕi

< 1.

(4.13)

m×m

Corollary 4.3.4. Let assumptions (A4.1) and (A4.2) hold. Assume that the connection
weight matrices are nonnegative and either condition (4.12) or condition (4.13) is satisfied.
Then, for any nonnegative input vector J = vec(I, J) ∈ Rn+m
, system (4.3) has at least
+
one positive EP χ∗ ∈ Rn+m
.
+

4.4. Exponential stability of positive EP
The results presented in the preceeding section only guarantee the existence of at least
one positive EP. In this section, we will prove that under the assumptions of Theorems 4.2.1
and 4.3.1 system (4.3) has a unique positive EP χ∗ , which is globally exponentially stable.
Theorem 4.4.1. Let assumptions (A4.1) and (A4.2) hold. Assume that the connection
weight matrices are nonnegative and one of the three conditions (4.10), (4.12), or (4.13)
is satisfied. Then, for any input vector J = vec(I, J) ∈ Rn+m
, system (4.3) has a unique
+
positive EP χ∗ ∈ Rn+m
, which is GES for any delays τi (t) ∈ [0, τ ] and σj (t) ∈ [0, σ].
+

4.5. Positive BAM neural networks with multi-proportional delays
Proportional delays belong to a type of unbounded time-varying delays, which are different from most other types of delays such as bounded time-varying delays or distributed
delays. Typically, in analysis of neural networks with proportional delays, the use of comparison techniques via certain types of differential inequalities proves to be an effective approach.
As an application, in this section we extend the result of Theorem 4.4.1 to the following BAM
neural networks model with proportional delays
m

m

x′i (t)

= −αi ϕi (xi (t)) +

yj′ (t) = −βj ψj (yj (t)) +

bij fj (yj (qij t)) + Ii ,

aij fj (yj (t)) +

(4.14)

j=1
n

j=1
n

cji gi (xi (t)) +
i=1

i=1

dji gi (xi (pji t)) + Jj , t ≥ t0 > 0, (4.15)

where 0 < pji < 1, 0 < qij < 1, i ∈ [n], j ∈ [m], are proportional delay factors. Other

coefficients and functions of the model (4.14)-(4.15) are similarly described as in system
(4.1)-(4.2). The initial condition of system (4.14)-(4.15) is specified as follows
xi (ξ) = x0i (ξ), ξ ∈ [p∗ t0 , t0 ],

yj (θ) = yj0(θ), θ ∈ [q∗ t0 , t0 ],

(4.16)

where p∗ = mini,j pji , q∗ = mini,j qij and x0i ∈ C([p∗ t0 , t0 ], R), yj0 ∈ C([q∗ t0 , t0 ], R). Similar
to Proposition 4.1.1, it can be verified that under assumptions (A4.1) and (A4.2) system

(4.14)-(4.16) has a unique solution χ(t) = vec(x(t), y(t)) defined on [t0 , ∞). Moreover, if
the connection weight matrices A, B, C, D are nonnegative and x0 (ξ) = (x0i (ξ))

y 0 (θ) = (yj0 (θ))

0 for all ξ ∈ [p∗ t0 , t0 ], θ ∈ [q∗ t0 , t0 ] then χ(t)
22

0 for all t ≥ t0 .

0,


Theorem 4.5.1. Under the assumptions of Theorem 4.4.1, for any input vector J = vec(I, J) ∈

Rn+m
, system (4.14)-(4.16) has a unique positive EP χ∗ ∈ Rn+m
, which is generalized glob+
+

ally exponentially stable (GGES). More precisely, there exist positive scalars κ and ̟ such
that any solution χ(t) = vec(x(t), y(t)) of system (4.14)-(4.15) satisfies
x0 − x∗

x(t) − x∗ + y(t) − y ∗ ≤ κ

0
∗
∞+ y −y

−̟ ln

∞

e

1+t
1+t0

, t ≥ t0 .

(4.17)

Remark 4.5.1. As a special case of system (4.14)-(4.15), we consider the following recurrent
neural networks (RNNs) model with proportional delays
m

m

x′i (t)

= −αi ϕi (xi (t)) +

aij fj (yj (t)) +
j=1

j=1

For system (4.18), with A = (aij )

bij gj (yj (qij t)) + Ii , t ≥ t0 > 0.

0 and B = (bij )

(4.18)

0, conditions (4.10), (4.12) and

(4.13) are reduced to the following one
ρ

1 −1
l
αi ϕi

n

aij Lfj + bij Lgj

< 1.

(4.19)

n×n

j=1

Similar to Theorem 4.5.1, if assumptions (A4.1), (A4.2) and condition (4.19) are satisfied
then, for any input vector I = (Ii )

0, model (4.18) has a unique positive EP χ∗ ∈ Rn+

which is GGES. Moreover, if I ≻ 0 then χ∗ ≻ 0. This result extends Theorems 3.1 and 3.2

in Yang (2019).

4.6. Simulations
In this section we give two examples to illustrate the effectiveness of the obtained
results.

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