MINISTRY OF EDUCATION AND TRAINING

HANOI NATIONAL UNIVERSITY OF EDUCATION

——————–o0o———————

DOAN THAI SON

STABILITY OF DIFFERENTIAL TIME-DELAY SYSTEMS

AND APPLICATIONS TO ECOLOGY MODELS

Major: 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. Trinh Tuan Anh

Referee 1: Professor Vu Ngoc Phat, Institute of Mathematics, VAST

Referee 2: Assoc.Prof. Do Duc Thuan, Hanoi University of Science and Technology

Referee 3: Assoc.Prof. Cung The Anh, Hanoi National University of Education

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........., 2019.

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 LyapunovKrasovskii 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. Purpose

This thesis is concerned with the problem of stability of time-delay systems which are

widely used in ecology models. Specifically, we consider the following problems

1. Investigating the problem of finite-time stability of non-autonomous neural networks with

heterogeneous proportional delays.

2. Analizing the global dissipativity of non-autonomous neural networks with multiple proportional delays.

3. Establishing the existence, uniqueness and global attractivity of a positive periodic solution

of a Nicholson model with nonlinear density-dependent mortality rate.

1

3. Objectives

3.1. Finite-time stability of non-autonomous neural networks with heterogeneous proportional delays

In recent years, dynamical neural networks have received a considerable attention due

to their potential applications in many fields such as image and signal processing, pattern

recognition, associative memory, parallel computing, solving optimization problems ect. In

most of the practical applications, it is of prime importance to ensure that the designed neural

networks be stable. On the other hand, time delays unavoidably exist in most application

networks and often become a source of oscillation, divergence, instability or bad performance.

A great deal of effort from researchers has been devoted to study the problems of stability

analysis, control and estimation for delayed neural networks during the past decade.

It is well-known that, in practical implementation of neural networks, time delays may

not be constants. They are not only time-varying but also proportional in many models.

Furthermore, a neural network usually has a spatial nature due to the presence of an amount

of parallel pathways of a variety of axon sizes and lengths, it is desirable to model them by

introducing continuously proportional delay over a certain duration of time. Proportional delay

is one of time-varying (monotonically increasing) and unbounded delays which is different from

most other types of delay such as time-varying bounded delays, bounded and/or unbounded

distributed delays. Its presence leads to an advantage is that the network’s running time can

be controlled based on the maximum delay allowed by the network. In addition, dealing with

the dynamic behavior of neural networks with proportional delays is an interesting problem

which is also much more complicated.

In Chapter 2 we consider the problem of finite-time stability of non-autonomous neural

networks with heterogeneous proportional delays described by the following system

n

x′i (t)

bij (t)fj (xj (t))

= − ai (t)xi (t) +

j=1

(1)

n

cij (t)gj (xj (qij t)) + Ii (t), i ∈ [n], t > 0.

+

j=1

By using novel comparison techniques, an explicit criterion is derived in terms of inequalities

for M-matrix ensuring that for each given bound on the initial conditions the state trajectories

of the system do not exceed a certain threshold over a pre-specified finite time interval. These

conditions are shown to be relaxed for the Lyapunov asymptotic stability through examples.

2

3.2. Global dissipativity of non-autonomous neural networks with multiple proportional delays

Dissipativity of dynamical systems, first introduced in the earlier of 1970s, has a meaningful physical concept and is an important characteristic of many mathematical models of physical

processes. In Chapter 3, we consider the problem of global dissipativity of the following neural

networks model

n

x′i (t) = −ai (t)xi (t) +

bij (t)fj (xj (t))

j=1

n

+

(2)

cij (t)gj (xj (qij t)) + Ii (t), i ∈ [n], t > 0.

j=1

Both the cases of uniform and non-uniform positive self-feedback coefficients −ai (t) are taken

into account simultaneously. Based on an extended comparison technique and M-matrix theory,

new unified delay-independent conditions are derived for both the existence of attracting sets

and global dissipativity of the system. On the basis of the obtained results, a generalized

exponential estimate for a class of Halanay-type inequalities with proportional delays, which

will be useful in the field of asymptotic behavior analysis of neural networks with delays, is also

established in this chapter.

3.3. Global attractivity of positive periodic solution of a delayed Nicholson model

with nonlinear density-dependent mortality term

Mathematical models are important for describing dynamics of phenomena in the real

world. For example, Nicholson used the following delay differential equation

N ′ (t) = −αN(t) + βN(t − τ )e−γN (t−τ ) ,

(3)

where α, β, γ are positive constants, to model the laboratory population of the Australian

sheep-blowfly. In the biology interpretation of equation (3), N(t) is the population size at

time t, α is the per capita daily adult mortality rate, β is the maximum per capita daily egg

production rate,

1

γ

is the size at which the population reproduces at its maximum rate and

τ ≥ 0 is the generation time (the time taken from birth to maturity). Model (3) is typically

referred to the Nicholson’s blowflies equation.

In the past few years, the qualitative theory for Nicholson model and its variants has been

extensively studied and developed. However, most of the existing works so far are devoted

to Nicholson-type models with linear mortality terms. Normally, a model of linear density3

dependent mortality rate will be most accurate for populations at low densities. According

to marine ecologists, many models in fishery such as marine protected areas or models of Bcell chronic lymphocytic leukemia dynamics are suitably described by Nicholson-type delay

differential equations with nonlinear density-dependent mortality rate of the form

N ′ (t) = −D(N(t)) + βN(t − τ )e−γN (t−τ ) ,

where the function D(N) might have one of the forms D(N) = a−be−N (type-I) or D(N) =

(4)

aN

b+N

(type-II) with positive constants a and b. A natural extension of (4) to the case of variable

coefficients and delays, which is more realistic in the theory of population dynamics is given by

N ′ (t) = −D(t, N(t)) + β(t)N(t − τ (t))e−γ(t)N (t−τ (t)),

where D(t, N) = a(t) − b(t)e−N or D(t, N) =

a(t)N

b(t)+N .

(5)

In model (5), D(t, N) is the death rate

of the population which depends on time t and the current population level N(t), B(t, N(t −

τ (t))) = β(t)N(t − τ (t))e−γ(t)N (t−τ (t)) is the time-dependent birth function which involves a

maturation delay τ (t) and gets its maximum

β(t)

γ(t)e

at rate

1

.

γ(t)

Recently, Nicholson-type models with nonlinear density-dependent mortality terms have

attracted considerable research attention. In Chapter 4 of this thesis, we study the problem of

existence and global attractivity of positive periodic solution of the following Nicholson model

p

βk (t)N(t − τk (t))e−γk (t)N (t−τk (t)) ,

′

N (t) = −D(t, N(t)) +

(6)

k=1

where D(t, N) = a(t) − b(t)e−N . Based on new comparison techniques via differential inequalities, we derive conditions for the existence and global attractivity of a unique positive periodic

solution of model (4.1). An application to Nicholson models with constant coefficients is also

presented.

4. Obtained results

The thesis achieves the following main results:

1. Established conditions in terms of M-matrices for finite stability and power-rate synchronization of Hopfiled neural networks with time-varying coefficients and heterogeneous proportional delays.

2. Proved the global dissipativity of a class of neural networks with multi-proportional delays

for both uniformly positive and singular self-feedback coefficients.

4

3. Derived conditions and proved the existence, uniqueness and global attractivity of a positive periodic solution to a Nicholson model with nonlinear density-dependent mortality

term.

The results presented in this thesis are based on three papers published on ISI indexed

international journals.

5. Thesis organization

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 concerning finite-time stability, dissipativity of certain classes of time-delay systems and

and some other auxiliary results, which will be useful for the presentation of the thesis. Chapter

2 investigates the problem of finite-time stability of Hopfield neuron networks with time-varying

connection weights and heterogeneous proportional delays. The global dissipativity of nonautonomous neural networks with multi-proportional delays is studied and presented in Chapter

3. Finally, the existence, uniqueness and global attractivity of a positive periodic solution to a

Nicholson model with nonlinear density-dependent mortality term is studied in Chapter 4.

5

Chapter 1

PREMILINARIES

In this chapter, we present some auxiliary results in matrix analysis, differential equations,

stability theory in the sense of Lyapunov and short time and the dissipativity of certain classes

of time-delay systems which will be used in the next chapters.

1.1. M-matrix

This section is concerned with basic concepts and properties of M-matrices.

1.2. Time-delay systems and the Lyapunov stability theory

Consider the following initial valued problem for functional differential equations

x′ (t) = f (t, xt ), t ≥ t0 ,

xt0 = φ,

(1.1)

where f : D = [t0 , ∞) × C → Rn and φ ∈ C = C([−r, 0], Rn ) is initial function. Assume that

f (t, 0) = 0 and the function f (t, φ) satisfies conditions that for any t0 ∈ [0, ∞) and φ ∈ C, the

problem (1.1) possesses a unique solution on [t0 , ∞).

Definition 1.2.1. The trival solution x = 0 of (1.1) is said to be stable (in the sense of Lyapunov) if for any t0 ∈ R+ , ǫ > 0, there esists a δ = δ(t0 , ǫ) > 0 such that for any solution

x(t, φ) of (1.1), if φ

C

< δ then x(t, φ) < ǫ for all t ≥ t0 . The solution x = 0 is uniformly

stable if the aforementioned δ is independent of t0 .

Definition 1.2.2. The solution x = 0 of (1.1) is said to be uniformly asymptotically stable if

it is uniformly stable and there exists a δa > 0 such that for any η > 0 there exists a T (δa , η)

such that φ

C

< δa implies x(t, φ) < η for all t ≥ t0 + T (δa , η). x = 0 is globally uniformly

asymptotically stable if δa can be arbitrarily selected.

Theorem 1.2.1 (Lyapunov-Krasovskii Theorem). Assume that f : R × C → Rn maps each set

R × Ω, where Ω is bounded set in C into a bounded set in Rn and u, v, w : R+ → R+ are

continuous non-decreasing functions, u(0) = 0, v(0) = 0 and u(s) > 0, v(s) > 0 for s > 0. If

there exists a continuous positive definite functional V : R × C → R+ ,

u( φ(0) ) ≤ V (t, φ) ≤ v( φ

6

C ),

∀φ ∈ C,

(1.2)

such that the derivative of V (t, φ) along trajectories of (1.1) is negative definite, that is,

V ′ (t, φ) ≤ −w( φ(0) ).

(1.3)

Then, the trivial solution x = 0 of (1.1) is uniformly stable. Moreover, if w(s) > 0 for s > 0

and lims→∞ u(s) = ∞ then the solution x = 0 is globally uniformly asymptotically stable.

1.3. Finite-time stability of dynamical systems

1.3.1. The concept of finite-time stability

The concept of finite-time stability (FTS) dates back to the 1950s, when it was introduced

in the Russian literature. Later, during the 1960s, this concept appeared in the western journals.

Roughly speaking, a system is said to be finite-time stable if, given a bound on the initial

conditions, its state does not exceed a certain threshold during a specified time interval. More

precisely, given the system

x′ (t) = f (t, x(t)),

x(t0 ) = x0 ,

(1.4)

where x(t) ∈ Rn is the system state vector, we can give the following formal definition.

Definition 1.3.1. Given an initial time t0 , a positive scaler T and two sets X0 , Xt . System

(1.4) is said to be finite-time stable with respect to (t0 , T, X0 , Xt ) if

x0 ∈ X0 =⇒ x(t, t0 , x0 ) ∈ Xt ,

∀t ∈ [t0 , t0 + T ].

Note that the trajectory set is allowed to vary in time. For well-posedness of the above

definition, it is required that X0 ⊂ Xt0 . However, in general, it is not required that X0 is

included in Xt for t > t0 . In addition, the sets X0 and Xt are typically given in the form of

ellipsoids ER (ρ) = {x⊤ Rx < ρ : x ∈ Rn }, where R ∈ Sn+ is a symmetric positive definite

matrices. The above definition can be stated as follows.

Definition 1.3.2. Given an initial time t0 , a scalar T > 0, a matrix R ∈ Sn+ and positive scalars

r1 < r2 . System (1.4) is said to be finite stable w.r.t (t0 , T, r1 , r2 , R) if for any x0 ∈ ER (r1 ),

the corresponding state trajectory x(t) = x(t; t0 , x0 ) of (1.4) satisfies x⊤ (t)Rx(t) < r2 for all

t ∈ [t0 , t0 + T ].

7

1.3.2. Finite-time stability of linear systems with mixed time-varying delays

Consider the following linear nonautonomous system with time-varying delays

t

′

x(s)ds, t ≥ 0,

x (t) = Ax(t) + Dx(t − τ (t)) + G

(1.5)

t−κ(t)

x(t) = φ(t),

t ∈ [−h, 0],

where x(t) ∈ Rn is the state vector, φ ∈ C([−h, 0], Rn ) is the initial function, A, D, G ∈ Rn×n

are known system matrices, τ (t), k(t) are time-varying delays which satisfy

0 ≤ τ1 ≤ τ (t) ≤ τ2 ,

τ ′ (t) ≤ µ ≤ 1,

0 ≤ κ1 ≤ κ(t) ≤ κ2 ,

where µ is a constant involving the rate of change of the discrete delay τ (t), τ1 , τ2 , κ1 , κ2 are

bounds of delays and h = max{τ2 , κ2 }.

Definition 1.3.3. Given T, r1 , r2 , where r1 < r2 . System (1.5) is said to be finite-time stable

w.r.t (r1 , r2 , T ) if for any φ ∈ C([−h, 0], Rn ), φ

∞

≤ r1 , one has x(t, φ)

∞

< r2 for all

t ∈ [0, T ].

Theorem 1.3.1. For given scalars T, r1 , r2 , r1 < r2 , system (1.5) is finite-time stable with

respect to (r1 , r2 , T ) if there exist positive scalars α, ρi , i = 1, 2, 3, 4, and symmetric positive

definite matrices P, Q, R ∈ Rn×n satisfying the following conditions

Π = Π0 + Π1 + Π2 < 0,

ρ1 In ≤ P ≤ ρ2 In ,

ρ2 + τ2 eατ2 ρ3 +

ρ1

(1.6a)

Q ≤ ρ3 In ,

eακ2 −1

ρ4

α

<

R ≤ ρ4 In ,

r2

r1

(1.6b)

2

e−αT ,

(1.6c)

where ei = 0n×(i−1)n In 0n×(3−i)n , i = 1, 2, 3, A = Ae1 + De2 + Ge3 , Π0 = e⊤

1 PA +

⊤

ατ1 e⊤ Qe v`

⊤

A⊤ P e1 − αe⊤

2 a Π2 = κ2 e1 Re1 −

1 P e1 , Π1 = e1 Qe1 − (1 − µ)e

2

1 ⊤

κ2 e3 Re3 .

1.4. Dissipativity of functional differential equations

In this section we introduce some preliminary results involving the dissipativity of certain

classes of time-delay systems. First, we consider the following system

x′ (t) = F (t, x(t), x(t − τ1 (t)), . . . , x(t − τm (t))),

x(t) = φ(t),

t ∈ [0, ∞),

(1.7)

t ∈ [−τ, 0],

where τk (.) are continuous time-delay functions satisfying 0 ≤ τk (t) ≤ τ for all t ∈ [0, ∞),

k ∈ [m], where τ > 0 is a constant. The function F : [0, ∞) × Rn × (C([−τ, ∞), Rn ))m → Rn

8

is continuous and satisfies

m

2 u, F (t, u, ψ1(.), . . . , ψm (.)) ≤ γ(t) + α(t) u

2

βk (t) ψk (t − τk (t))

+

2

(1.8)

k=1

for all t ∈ [0, ∞), u ∈ Rn v`a ψk (.) ∈ C([−τ, ∞), Rn ). The function F is also assumed to satisfy

conditions that for any φ(.) ∈ C([−τ, 0], Rn ), the problem (1.7) possesses a unique solution

x(t, φ) on [−τ, ∞).

Definition 1.4.1. System(1.7) is said to be globally dissipative if there exists a bounded set

B ⊂ Rn with the property that for any bounded set B ⊂ Rn , there exists a t∗ = t∗ (B) such

that for any φ(.) ∈ C([−τ, 0], Rn ), φ(t) ∈ B for all t ∈ [−τ, 0], one has x(t, φ) ∈ B for all

t ≥ t∗ (B). The set B is called an absorbing set of (1.7).

1.4.1. The Halanay inequality approach

Lemma 1.4.1 (Halanay inequality). Assume that the function u(t) ≥ 0, t ∈ (−∞, ∞), satisfies

u′ (t) ≤ γ(t) + α0 u(t) + β0

sup

t−τ ≤s≤t

u(s), t ≥ t0 , u(t) = θ(t), t ≤ t0 ,

(1.9)

where θ ∈ BC((−∞, t0 ], R+ ) is a continuous bounded function. If α0 + β0 < 0 then there exists

a scalar λ > 0 such that

u(t) ≤

where γ∗ = supt≥t0 γ(t) and θ

γ∗

+ θ

−(α0 + β0 )

∞

−λ(t−t0 )

,

∞e

t ≥ t0 ,

(1.10)

= supt≤t0 |θ(t)|.

Remark 1.4.1. Let α0 = supt≥0 α(t) and β0 = supt≥0 β(t). If α0 + β0 < 0 then, by Lemma

1.4.1, system (1.7) is globally dissipative. Specifically, for a given ǫ > 0, there exists a t∗ =

t∗ ( φ

∞ , ǫ)

> 0 such that

x(t)

2

<

γ∗

+ ǫ, t > t∗ .

−(α0 + β0 )

Therefore, system (1.7) is globally dissipative with the absorbing set B = B 0,

∗

− α0γ+β

+ǫ .

0

Moreover, the estimate (1.10) guarantees the exponential attraction of B.

Theorem 1.4.2. Let the function u(t) ≥ 0, t ∈ (−∞, ∞), satisfy

u′ (t) ≤ γ(t) + α(t)u(t) + β(t)

sup

u(s) (t ≥ t0 ),

u(t) = θ(t) (t ≤ t0 ),

(1.11)

t−τ (t)≤s≤t

where θ ∈ BC((−∞, t0 ], R+ ), α(t), β(t), γ(t) are continuous functions, β(t) ≥ 0, γ(t) ≥ 0 and

the delay τ (t) ≥ 0 satisfies t − τ (t) → ∞ as t → ∞. Assume that

α(t) + β(t) ≤ −σ < 0, t ≥ t0 ,

9

(1.12)

γ∗

+ θ ∞ , t ∈ [t0 , ∞). Additionally, if there

σ

exists a 0 < δ < 1 such that δα(t) + β(t) < 0, ∀t ≥ t0 , then for any ǫ > 0, there exists a

for some positive scalar σ. Then, u(t) ≤

t∗ = t∗ ( θ

∞ , ǫ)

> t0 by which

u(t) ≤

γ∗

+ ǫ,

σ

t ≥ t∗ .

1.4.2. Dissipativity of a class of nonlinear systems with proportional delay: A

changing of variable approach

Consider the following system

x′ (t) = g(x(t), x(qt)), t ≥ t0 > 0,

(1.13)

x(t) = ϕ(t), t ∈ [qt0 , t0 ],

where q is a constant, 0 < q < 1, and the function g satisfies

2 u, g(u, v) ≤ γ + α u

2

+β v

2

(1.14)

with α, β, γ are given constants. For the notational simplicity, we assume that t0 = 1. By the

change of variable y(t) = x(et ), system(1.14) can be stransformed to the following system with

a constant delay

y ′(t) = f (t, y(t), y(t − τ )), t > 0,

where τ = − ln(q) > 0 and

(1.15)

y(t) = ϕ(t), t ∈ [−τ, 0],

f (t, y(t), y(t − τ )) = et g(y(t), y(t − τ )).

(1.16)

From (1.14) and (1.16) we have

2 u, f (t, u, v) ≤ et γ + α u

2

+β v

2

.

(1.17)

Theorem 1.4.3. Let y(t) be a solution of (1.15)-(1.17) and assume that α + β < 0. Then, for

a given ǫ > 0, there exists a t∗ = t∗ ( ϕ

y(t)

2

∞ , ǫ)

<−

> 0 such that

γ

+ ǫ, ∀t > t∗ .

α+β

Therefore, system (1.15) is globally dissipative and B = B 0,

γ

+ ǫ is an absorbing set

− α+β

for a given ǫ > 0.

1.5. Auxiliary results

This section presents some technical lemmas and auxiliary results which will be used in

the next chapters.

10

Chapter 2

FINITE-TIME STABILITY OF NON-AUTONOMOUS NEURAL NETWORKS WITH

HETEROGENEOUS PROPORTIONAL DELAYS

In this chapter we study the problem of finite-time stability of non-autonomous neural networks with heterogeneous proportional delays. By introducing a novel constructive approach,

we derive explicit conditions in terms of matrix inequalities ensuring that the state trajectories

of the system do not exceed a certain threshold over a pre-specified finite time interval. As a

result, we also obtain conditions for the power-rate global stability of the system.

2.1. Model description

Consider the following neural networks model

n

x′i (t)

bij (t)fj (xj (t))

= −ai (t)xi (t) +

j=1

n

cij (t)gj (xj (qij t)) + Ii (t), t > 0,

+

(2.1)

j=1

xi (0) = x0i , i ∈ [n],

where xi (t) is the state variable (potential or voltage) of the ith neuron at time t, fj (.), gj (.),

j ∈ [n], are activation functions, ai (t) are self-inhibition terms, bij (t), cij (t) are time-varying

connection weights, Ii (t) are external inputs, qij ∈ (0, 1], i, j ∈ [n], are possibly heterogeneous

proportional delays, x0 = (x01 , . . . , x0n )T ∈ Rn is the initial state vector.

(A2.1) The neuron activation functions fi , gi , i ∈ [n], satisfy

−

li1

≤

fi (x) − fi (y)

+

≤ li1

,

x−y

−

li2

≤

gi (x) − gi (y)

+

≤ li2

, ∀x, y ∈ R, x = y,

x−y

− +

where lik

, lik , k = 1, 2, are known constants.

Remark 2.1.1. Let the function F : R+ × Rn × Rn×n → Rn be defined by F (t, u, v) =

(Fi (t, u, v)) where u = (ui ) ∈ Rn , v = (vij ) ∈ Rn×n and

n

n

cij (t)gj (vij ) + Ii (t).

bij (t)fj (uj ) +

Fi (t, u, v) = −ai (t)ui +

j=1

j=1

By (A2.1), F (t, u, v) is continuous and Lipschitz on R+ × Rn × Rn×n . Therefore, for a given

initial vector x0 ∈ Rn , there exists a unique solution x(t) = x(t, x0 ) of (2.1) on the interval

[0, ∞).

11

2.2. Finite-time stability of model (2.1)

Definition 2.2.1. For given a time T > 0 and positive numbers r1 < r2 , a solution x∗ (t) of

(2.1) is said to be finite-time stable with respect to (r1 , r2 , T ) if for any solution x(t) of (2.1),

x(0) − x∗ (0)

∞

≤ r1 =⇒ x(t) − x∗ (t)

∞

< r2 , ∀t ∈ [0, T ].

System (2.1) is said to be FTS with respect to (r1 , r2 , T ) if any solution x∗ (t) of (2.1) is FTS

with respect to (r1 , r2 , T ).

(A2.2) The matrices A(t) = diag{ai (t)}, B(t) = (bij (t)), C(t) = (cij (t)) satisfy

ai (t) ≥ ai > 0, |bij (t)| ≤ bij , |cij (t)| ≤ cij , ∀t ≥ 0, i, j ∈ [n].

+

−

+

−

Hereafter, let us denote for i ∈ [n] the constants Lfi = max{li1

, −li1

} and Lgi = max{li2

, −li2

}.

We also introduce the following matrix notations

A = diag{a1 , a2 , . . . , an },

B = (bij ),

Lf = diag{Lf1 , Lf2 , . . . , Lfn },

C = (cij ),

Lg = diag{Lg1 , Lg2 , . . . , Lgn },

M = BLf + CLg − A.

Theorem 2.2.1. Under assumptions (A2.1) and (A2.2), for given 0 < r1 < r2 and T > 0,

system (2.1) is finite-time stable with respect to (r1 , r2 , T ) if there exist a positive number γ and

a vector ξ ∈ Rn , ξ ≻ 0, satisfy the following conditions

(i) (M − γI) ξ ≺ 0,

(ii) C(ξ) <

r2 −γT

e

,

r1

where C(ξ) = ξ uξl−1 denotes the condition number of ξ.

Remark 2.2.1. Condition (i) in Theorem 2.2.1 does not guarantee the asymptotic stability of

system (2.1) in the sense of Lyapunov (LAS). Moreover, even conditions (i), (ii) are satisfied

for any T > 0, r2 > r1 > 0, system (2.1) may not be LAS.

2.3. Long-time behavior: Synchronization of model (2.1)

Theorem 2.3.1. Let assumptions (A2.1) and (A2.2) hold and assume that −M is a nonsingular M-matrix. Then, there exist positive constants β, σ, which are independent of solutions of

(2.1), such that for any two solutions x(t), x∗ (t) of (2.1), the following inequality holds

x(t) − x∗ (t)

∞

≤β

x(0) − x∗ (0)

(1 + t)σ

12

∞

, t ≥ 0.

(2.2)

Remark 2.3.1. Let x∗ (t) be a solution of (2.1). The estimate (2.2) shows that any state

trajectory x(t) of (2.1) will have similar behavior with x∗ (t) when the time is sufficiently large.

Thus, the family of solutions of (2.1) has the same behavior with x∗ (t) as the time t tends to

infinity. In the field the network control systems, this feature is referred to the synchronization.

Remark 2.3.2. The constant σ0 mentioned in the proof of Theorem 2.3.1 defines the powerrate synchronization of model (2.1). The power convergence rate σmax can be defined by the

following procedure

• Define a vector ξ ∈ Rn , ξ ≻ 0, such that Mξ ≺ 0.

• Compute

n

η = (−Mξ)l = min

ξj bij Fj + cij Gj

ai ξi −

i∈[n]

.

(2.3)

j=1

• The power convergence rate σmax can be iteratively computed as

n

max σ > 0 s.t. Hi (σ) = σξi +

σ ln

Gj cij ξj e

1

qij

− 1 − η ≤ 0, ∀i ∈ [n].

(2.4)

j=1

2.4. Numerical examples

This section presents some numerical examples and simulations to demonstrate the effectiveness of the obtained results in this chapter.

13

Chapter 3

GLOBAL DISSIPATIVITY OF NON-AUTONOMOUS NEURAL NETWORKS WITH

MULTIPLE PROPORTIONAL DELAYS

In this chapter we investigate the problem of dissipativity analysis of the following nonlinear differential system

n

x′i (t)

bij (t)fj (xj (t))

= −ai (t)xi (t) +

j=1

n

(3.1)

cij (t)gj (xj (qij t)) + Ii (t), t > 0, i ∈ [n],

+

j=1

in two cases: (i) the self-feedback coefficients are uniformly positive, that is, ai (t) ≥ ai > 0;

and (ii) the self-feedback coefficients can be singular, that is, ai (t) > 0 and inf t≥0 ai (t) = 0.

As discussed in Chapter 1, the proposed methods in the existing literate such as the use of

state transformation or Halanay inequalities are are now not appropriate for model (3.1) due

to the nature of its structure. Thus, to analize the dissipativity of model (3.1), we will develop

some new comparison techniques based on the theory of M-matrix to derive conditions for the

existence a generalized exponential attracting set. The content of this chapter is written based

on paper [2] in the publication list of this thesis.

3.1. Preliminaries

Consider a non-autonomous neural network model with multiple proportional delays presented in (3.1), where x(t) = (xi (t)) ∈ Rn is the state vector, ai (t) ∈ R+ , i ∈ [n], are selffeedback coefficients, bij (t) ∈ R and cij (t) ∈ R are neuron connection weights at time t, fj (.),

gj (.), j ∈ [n], are neuron activation functions, (Ii (t)) is the external input vector, qij ∈ (0, 1),

i, j ∈ [n], are heterogeneous proportional delay. The initial condition of (3.1) is specified as

xi (0) = x0i , i ∈ [n],

(3.2)

where x0 = (x0i ) ∈ Rn is a given vector.

(A3.1): There exist scalars Lfj ≥ 0, Lgj ≥ 0, j ∈ [n], such that

|fj (a) − fj (b)| ≤ Lfj |a − b|, |gj (a) − gj (b)| ≤ Lgj |a − b|,

∀a, b ∈ R.

(3.3)

(A3.2): The connection weights bij (t), cij (t), i, j ∈ [n], and external inputs Ii (t), i ∈ [n], are

14

bounded, i.e. there exist scalars bij , cij and I i , i, j ∈ [n], such that

|bij (t)| ≤ bij , |cij (t)| ≤ cij , |Ii (t)| ≤ I i , ∀t ≥ 0, i, j ∈ [n].

Definition 3.1.1. A compact set Ω ⊂ Rn is said to be a global attracting set of (3.1) if any

solution x(t) = x(t, x0 ) of (3.1) satisfies lim supt→∞ ρ(x(t), Ω) = 0, where ρ(x, Ω) = inf y∈Ω x −

y

∞

denotes the distance from x to Ω.

Definition 3.1.2. A compact set Ω ⊂ Rn is said to be a global generalized exponential attracting set of (3.1) if there exist a function κ( x0

∞)

≥ 0, a nondecreasing function σ(t) ≥ 0 such

that lim supt→∞ σ(t) = ∞ and any solution x(t) = x(t, x0 ) of (3.1) satisfies

ρ(x(t), Ω) ≤ κ( x0

−σ(t)

,

∞ )e

t ≥ 0.

(3.4)

If σ(t) = αt, where α is a positive scalar, then Ω is a global exponential attracting set of (3.1).

Definition 3.1.3. System (3.1) is said to be globally dissipative if there is a bounded set A ⊂

Rn such that for any bounded set B ⊂ Rn there is a time td = td (B) with the property that for

any initial condition x0 ∈ Φ, the corresponding solution x(t, x0 ) belongs to A for all t ≥ td (B).

Then A is called an absorbing set of (3.1).

Remark 3.1.1. If there exists a global generalized exponential attracting set Ω and the function

κ(.) defined in (3.4) is nondecreasing then system (3.1) is globally dissipative. Indeed, if Ω is

a global generalized exponential attracting set of (3.1) then for any bounded set B ⊂ Rn , let

r(B) = supx0 ∈B x0

∞,

we have

ρ(x(t), Ω) ≤ κ(r(B))e−σ(t), t ≥ 0.

For given any ǫ > 0, let Aǫ = {x ∈ Rn : ρ(x, Ω) ≤ ǫ} then Aǫ is a bounded set which contains

the set {x(t, x0 ) : t ≥ td (B), x0 ∈ B}, where td (B) = inf{t > 0 : σ(t) ≥ ln

κ(r(B))

}.

ǫ

Thus, Aǫ

is an absorbing set of (3.1) and system (3.1) is globally dissipative.

3.2. Global attractivity of the model (3.1)

3.2.1. Regular self-feedback coefficients

In this section, we derive disipativity conditions for system (3.1) under the following

assumption.

(A3.3): There exist positive scalars ai , i ∈ [n], such that

ai (t) ≥ ai , ∀t ≥ 0, i ∈ [n].

15

(3.5)

Let D = diag(a1 , a2 , . . . , an ) and M = D − BLf − CLg , where B = (bij ), C = (cij ),

Lf = diag(Lf1 , Lf2 , . . . , Lfn ) and Lg = diag(Lg1 , Lg2 , . . . , Lgn ). We have the following result.

Theorem 3.2.1. Under assumptions (A3.1)-(A3.3), if M is a nonsingular M-matrix then the

following assertions hold

(1) The set Ω defined by

Ω=

x ∈ Rn : x

∞

≤

γ

(Mχ)+

is a global generalized exponential attracting set of (3.1), where χ ∈ int(Rn+ ) satisfies

χ

∞

n

j=1 (bij |fj (0)| + cij |gj (0)|) + I i

= 1, Mχ ≻ 0 and γ = maxi∈[n]

;

(2) System (3.1) is globally dissipative.

Corollary 3.2.2. Under assumptions (A3.1)-(A3.3), if

n

n

ai >

bji + Lgi

Lfi

x

∞

≤

γ

σ∗

(3.6)

j=1

j=1

then Ω = x ∈ Rn :

cji , i ∈ [n]

is a global generalized exponential attracting set of (3.1) and

system (3.1) is globally dissipative, where

n

n

∗

σ = min

i∈[n]

ai − Lfi

bji −

j=1

Lgi

cji .

j=1

3.2.2. Singular self-feedback coefficients

In this section, we derive delay-independent conditions that ensure the global dissipativity

of system (3.1) without uniform positiveness of self-feedback coefficients.

For convenience, we introduce the following assumptions.

(A3.4): There exist a function ϕ(t) > 0 and positive scalars a

ˆi , i ∈ [n], such that

t

ai (t) ≥ a

ˆi ϕ(t), sup

t≥0

t

ϕ(s)ds < ∞, lim

t→∞

qij t

ϕ(s)ds = ∞.

(3.7)

0

(A3.5): There are constants ˆbij ≥ 0, cˆij ≥ 0 and Iˆi ≥ 0 such that

|bij (t)| ˆ |cij (t)|

|Ii (t)|

≤ bij ,

≤ cˆij ,

≤ Iˆi , ∀i, j ∈ [n], t ≥ 0.

ai (t)

ai (t)

ai (t)

(3.8)

ˆij =

Remark 3.2.1. Assumptions (A3.4) and (A3.5) are obviously satisfied with ˆbij = bij a−1

i , c

cij a−1

i , i, j ∈ [n], and ϕ(t) =

1

1+t

if assumptions (A3.2) and (A3.3) hold. Thus, (A3.4) and

(A3.5) can be regarded as extended conditions of (A3.2) and (A3.3).

Let us denote B = (ˆbij ), C = (ˆ

cij ) and H = En − (BLf + CLg ), where En denotes the

identity matrix in Rn×n .

16

Theorem 3.2.3. Assume assumptions (A3.1), (A3.4), (A3.5) are satisfied and H is a nonγˆ

singular M-matrix. Then system (3.1) is globally dissipative and the ball B 0, m

ˆ + ǫ is an

absorbing set of (3.1) for any ǫ > 0, where γˆ = maxi∈[n] {Iˆi +

m

ˆ = (Hη)+ and η ∈ int(Rn+ ) satisfying η

∞

n

ˆ

j=1 (bij |fj (0)|

+ cˆij |gj (0)|)},

= 1 and Hη ≻ 0.

Corollary 3.2.4. Under assumptions (A3.1), (A3.4) and (A3.5), if H is a nonsingular Mmatrix then system (3.1) is globally synchronous. More precisely, any two solutions x(t) and

x∗ (t) of (3.1) satisfy the following inequality

x(t) − x∗ (t)

where η ∈ int(Rn+ ) satisfies η

∞

∞

≤

1

x(0) − x∗ (0)

η+

−λ0

∞e

t

0 ϕ(s)ds

, t≥0

(3.9)

= 1 and Hη ≻ 0.

In the remaining of this section, let us consider the following Halanay-type inequality with

multiple proportional delays:

n

+

bj (t)u(qj t) + d(t), t > 0

D u(t) ≤ −a(t)u(t) +

(3.10)

j=1

where a(t) > 0, bj (t), j ∈ [n], and d(t) are continuous functions, qj ∈ (0, 1), j ∈ [n], are

proportional delays.

By some similar lines used in the proof of Theorem 3.2.3 we obtain the following result.

Corollary 3.2.5. Assume that there exist scalars l > 0, µ0 ∈ (0, 1), µ1 ≥ 0, a function as (t) > 0

and a t0 > 0 such that a(t) ≥ las (t), t ≥ t0

t

sup

t≥t0

and

t

as (θ)dθ < ∞, lim

qj t

t→∞

as (θ)dθ = ∞

0

n

|bj (t)| − µ0 a(t) ≤ 0, |d(t)| ≤ µ1 a(t), t ≥ t0 .

j=1

Then any solution u(t) of (3.10) converges exponentially within the bound

µ1

1−µ0 .

Specifically,

˜ such that any solution u(t) of (3.10) satisfies

there exists a positive constant λ

u(t) ≤

µ1

µ1

˜

+ max u(0) −

, 0 e−λ

1 − µ0

1 − µ0

t

0 as (ζ)dζ

, t ≥ 0.

(3.11)

3.3. Illustrative examples

This section presents some numerical examples to illustrate the effectiveness of the results

obtained in this chapter.

17

Chapter 4

GLOBAL ATTRACTIVITY OF POSITIVE PERIODIC SOLUTION OF A DELAYED

NICHOLSON MODEL WITH NONLINEAR MORTALITY TERM

In this chapter we study the problem of existence and global attractivity of positive

periodic solution of the following Nicholson model

p

βk (t)N(t − τk (t))e−γk (t)N (t−τk (t))

′

N (t) = −D(t, N(t)) +

(4.1)

k=1

nonlinear density-dependent mortality term D(t, N) = a(t) − b(t)e−N . Based on novel comparison techniques via differential and integral inequalities, we first derive conditions for the global

uniform permanence and dissipativity of the model (4.1). On the basis of the global uniform

permanence and dissipativity, the existence and global attractivity of a unique positive periodic solution of model (4.1) is then established. As an application to Nicholson models with

constant coefficients, improved results on the existence, uniqueness and global attractivity of a

positive equilibrium are also obtained. The content of this chapter is based upon the paper [3]

in the list of publication of this thesis.

4.1. Preliminaries

Consider a Nicholson model with delays and nonlinear density-dependent mortality term

of the form

p

βk (t)N(t − τk (t))e−γk (t)N (t−τk (t)) ,

′

N (t) = −D(t, N(t)) +

t ≥ t0 ,

(4.2)

k=1

N(t) = ϕ(t),

t ∈ [t0 − τM , t0 ],

(4.3)

where the density-dependent mortality term D(t, N) is of the form

D(t, N) = a(t) − b(t)e−N

(4.4)

and τM = max1≤k≤p τk+ represents the upper bound of delays.

Assumption (A):

(A4.1) a, b, γk : [0, ∞) → (0, ∞), βk : [0, ∞) → [0, ∞) and τk : [0, ∞) → [0, τM ] are continuous

bounded functions, where τM is some positive constant.

(A4.2) There exists an ω > 0 such that the functions a, b, βk , γk and τk belong to Pω (R+ ).

18

Condition (C):

(C4.1) a) b(t) ≥ a(t) ≥ a− > 0, b) θ

(C4.2) lim supt→∞

p

βk (t)

k=1 γk (t)

1

a(t)

(C4.3) b− − a+ > 0, ̺

(C4.4)

p

+

k=1 βk max

a− −

lim inf t→∞

= σ, 1 −

p

β+

k=1 γ −

k

1

e

−

1 1−γk r∗

, γ −r

e2

e k ∗

<

̺b−

,

b+

b(t)

a(t)

> 1.

b−

a+

.

σ

> 0.

e

> 0.

r∗ = ln

A preview of our main results is presented in the following table.

Conditions

(A4.1), (C4.1)

(A4.1), (C4.1a), (C4.2)

Results

Uniform permanence in C0+ , lim inf t→∞ N (t, t0 , ϕ) ≥ ln(θ).

b+

Uniform dissipativity in C0+ , lim supt→∞ N (t, t0 , ϕ) ≤ ln a− (1−

σ

) .

(A4.1), (C4.3)

(A4.1), (A4.2), (C4.3)

and (C4.4)

ln ab + ≤ lim inf t→∞ N (t, t0 , ϕ) ≤ lim supt→∞ N (t, t0 , ϕ) ≤ ln

There exists a unique positive ω-periodic solution N ∗ (t)

which is globally attractive in C0+ .

e

−

b+

̺

.

For a biological interpretation of the proposed conditions, it is reasonable that when the

population is absence the death rate is nonpositive (i.e. D(t, 0) ≤ 0) and D(t, N) is always

positive when N > 0. This gives rise to condition (C4.1). On the other hand, in most of

biological models, there typically exists a threshold related to the so-called carrying capacity.

When the population size is very large, over the carrying capacity, the death rate can be bigger

p

βk (t)

k=1 γk (t)e

than the maximum birth rate. The quantity

can be regarded as the maximum

birth rate of model (4.2). In addition, when N is large D(t, N) is approximate to a(t). By

this observation, we make an assumption to ensure that

p

βk (t)

k=1 γk (t)e

< a(t). This reveals the

imposing of condition (C4.2) when considering long-time behavior of the model. (C4.3) is a

testable condition derived from (C4.2) and (C4.1a) by taking into account the upper bound of

the associated rates. While condition (C4.3) only guarantees non-extinction and non-blowup

behavior, condition (C4.4) reveals that, by certain scaling coefficients, when maximum per

capita daily egg production rates are smaller than the gap between the maximum death rate

and birth rate (i.e. ̺ = a− −

1

e

p

β+

k=1 γ − ),

k

the population will be stable around a periodic

trajectory (in the case of periodic coefficients) or a positive equilibrium (for time-invariant

model).

19

4.2. Permanence of global positive solutions

4.2.1. Global existence of positive solutions

Theorem 4.2.1. Let assumption (A4.1) hold. Assume that b(t) ≥ a(t) for all t ∈ [0, ∞).

Then, for any initial condition ϕ ∈ C0+ , the solution N(t, t0 , ϕ) of system (4.2)-(4.4)satisfies

N(t, t0 , ϕ) > 0, t ∈ [t0 , η(ϕ)), and η(ϕ) = ∞.

Remark 4.2.1. To ensure the positiveness of solutions of (4.2)-(4.4) with initial conditions in

C0+ , condition b(t) ≥ a(t) cannot be relaxed. For a counterexample, let n = 1 and assume that

t

b(t)

sup

= δ ∈ [0, 1),

t≥0 a(t)

a(s)ds → ∞,

t → ∞.

0

Then,

ϕ(0)−

N(t, t0 , ϕ) ≤ ln e

t

t0

a(s)ds

−

+δ 1−e

t

t0

a(s)ds

→ ln(δ) < 0 as t → ∞.

4.2.2. Uniform permanence

In this section we derive conditions and prove the uniform permanence of model (4.2).

Theorem 4.2.2. Let assumption (A4.1) hold. Assume that b(t) ≥ a(t) ≥ a− > 0 and

lim inf

t→∞

b(t)

≥ eℓm > 1.

a(t)

(4.5)

Then, for any ϕ ∈ C0+ ,

lim inf N(t, t0 , ϕ) ≥ ℓm > 0.

t→∞

Remark 4.2.2. As a special case of (4.5), for bounded functions a(t) and b(t), if b− > a+ then

the scalar ℓm in (4.5) can be chosen as

ℓm = ln

b−

a+

.

The following result shows the uniform dissipativity of system (4.2)-(4.4) in C0+ in the

sense that there exists a constant ℓM > 0 such that lim supt→∞ N(t, t0 , ϕ) ≤ ℓM .

Theorem 4.2.3. Assume assumption (A4.1) and the following conditions hold

b+ ≥ b(t) ≥ a(t) ≥ a− > 0,

1

lim sup

t→∞ a(t)

p

k=1

βk (t)

= σ,

γk (t)

t ∈ [0, +∞),

1−

σ

> 0.

e

(4.6)

(4.7)

Then, system (4.2)–(4.4) is uniformly dissipative in C0+ . More precisely, for any initial condition

ϕ ∈ C0+ , the corresponding solution N(t, t0 , ϕ) of (4.2)-(4.4) satisfies

lim sup N(t, t0 , ϕ) ≤ ℓM

t→∞

20

ln

a−

b+

1−

σ

e

.

The following result is obtained as a consequence of Theorems 4.2.2 and 4.2.3.

Corollary 4.2.4. Let assumption (A4.1) hold, where a, b, βk , γk are bounded functions, γk− > 0.

Assume that

1

a −

e

p

βk+

−

̺

> 0,

(4.8a)

b− − a+ > 0.

(4.8b)

k=1

γk−

Then, for any ϕ ∈ C0+ , it holds that

ln

b−

a+

≤ lim inf N(t, t0 , ϕ) ≤ lim sup N(t, t0 , ϕ) ≤ ln

t→∞

t→∞

b+

̺

(4.9)

.

4.3. Global attractivity of positive periodic solution

In this section we assume that assumptions (A4.1), (A4.2) and conditions (4.8a)-(4.8b)

are satisfied. For convenience, we denote

b−

a+

r∗ = ln

b−

a+

γk− r∗

,

b+

̺

∗

r = ln

,

νk = max

1 1 − γk− r∗

,

e2 eγk− r∗

.

1

max γk−

Note that, by (4.8b),

> 1 and hence r∗ > 0. In addition, since the condition r∗ <

is not imposed, 1 −

can be positive, negative or zero. For 1 ≤ k ≤ p that 1 − γk− r∗ ≤ 0,

νk =

1

.

e2

We are now in a position to present the existence, uniqueness and global attractivity of a

positive periodic solution of system (4.2)-(4.4) as in the following theorem.

Theorem 4.3.1. Let assumptions (A4.1), (A4.2), conditions (4.8a), (4.8b) and the following

ones are satisfied

inf 1 − τk′ (t) = µ > 0,

t≥0

p

νk βk+ < µ

k=1

̺b−

,

b+

(4.10)

(4.11)

where ̺ is the constant defined in (4.8a). Then, system (4.2)-(4.4) has a unique positive ωperiodic solution N ∗ (t) which is globally attractive in C0+ .

Remark 4.3.1. Conditions (4.10) and (4.11) are involved a scalar µ > 0 related to the rate

of change of delay functions τk (t). However, this scalar can be relaxed and conditions (4.10),

(4.11) are reduced to the following one

p

νk βk+ <

k=1

̺b−

.

b+

More precisely, we state that in the following theorem.

21

(4.12)

Theorem 4.3.2. Under assumptions (A4.1) and (A4.2), assume that conditions (4.8a), (4.8b)

and (4.12) are satisfied. Then, system (4.2)-(4.4) has a unique positive ω-periodic solution

N ∗ (t) which is globally attractive in C0+ .

4.4. Attractivity of positive equilibrium

In this section, we apply our results presented in the preceding sections to the following

Nicholson model

p

βk N(t − τk (t))e−γk N (t−τk (t)) ,

′

N (t) = −D(N(t)) +

t ≥ t0 ≥ 0,

(4.13)

k=1

p

k=1 βk

where βk ≥ 0, γk > 0 are known coefficients,

> 0. The nonlinear density-dependent

mortality term is given by D(N) = a − be−N , a > 0, b > 0. Time-varying delays τk (t) are

continuous and bounded in the range [0, τM ].

For model (4.13), conditions (4.8a), (4.8b) are reduced to the following coupled condition

1

e

p

k=1

βk

< a < b.

γk

(4.14)

Proposition 4.4.1. Let condition (4.14) hold. Then, for any ϕ ∈ C0+ , it holds that

ln

b

a

≤ lim inf N(t, t0 , ϕ) ≤ lim sup N(t, t0 , ϕ) ≤ ln

t→∞

t→∞

b

a−

1

e

p

βk

k=1 γk

.

(4.15)

Theorem 4.4.2. Assume that

q

βk

k=1

where

νˆk = max

νˆk +

1

eγk

< a < b,

(4.16)

1 1 − γk ln( ab )

,

.

b

e2

eγk ln( a )

Then, model (4.13) has a unique positive equilibrium N ∗ which is globally attractive in C0+ .

4.5. Simulations

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

22

CONCLUSION

Main contributions

The main contributions of this thesis are as follows:

1. Established sufficient conditions in terms of M-matrix for finite-time stability (Theorem

2.2.1) and power-rate synchronization (Theorem 2.3.1) of Hopfiled neural networks with

time-varying connection weights and heterogeneous proportional delays.

2. Derived conditions and proved the global dissipativity both in the case of regular selffeedback terms (Theorem 3.2.1) and singular self-feedback terms (Theorem 3.2.3) for nonautonomous neural networks with multiple proportional delays.

3. Established a new generalized exponential estimation for a type of Halanay inequalities

with proportional delay (Corollary 3.2.5).

4. Proved the global existence, uniform permanence and dissipativity in C0+ of positive solutions to a delayed Nicholson model with nonlinear density-dependent mortality rate

(Theorems 4.2.1, 4.2.2, 4.2.3, Corollary 4.2.4).

5. Established the existence, uniqueness and global attractivity of a positive periodic solution

to a delayed Nicholson model (Theorem 4.3.1). An application concerning the existence

and attractivity of a unique positive equilibrium of Nicholson models with constant coefficients has also been obtained (Theorem 4.4.2).

Further topics

• Investigating the global dissipativity and synchronization of differential equations modeling

reaction-diffusion neural networks with time-varying connection weights and proportional

delays.

• The existence and asymptotic behavior of periodic, almost periodic positive solution to

delayed Nicholson models with fractional density-dependent mortality term of the form

a(t)N

D(t, N) =

.

N + b(t)

23

HANOI NATIONAL UNIVERSITY OF EDUCATION

——————–o0o———————

DOAN THAI SON

STABILITY OF DIFFERENTIAL TIME-DELAY SYSTEMS

AND APPLICATIONS TO ECOLOGY MODELS

Major: 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. Trinh Tuan Anh

Referee 1: Professor Vu Ngoc Phat, Institute of Mathematics, VAST

Referee 2: Assoc.Prof. Do Duc Thuan, Hanoi University of Science and Technology

Referee 3: Assoc.Prof. Cung The Anh, Hanoi National University of Education

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........., 2019.

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 LyapunovKrasovskii 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. Purpose

This thesis is concerned with the problem of stability of time-delay systems which are

widely used in ecology models. Specifically, we consider the following problems

1. Investigating the problem of finite-time stability of non-autonomous neural networks with

heterogeneous proportional delays.

2. Analizing the global dissipativity of non-autonomous neural networks with multiple proportional delays.

3. Establishing the existence, uniqueness and global attractivity of a positive periodic solution

of a Nicholson model with nonlinear density-dependent mortality rate.

1

3. Objectives

3.1. Finite-time stability of non-autonomous neural networks with heterogeneous proportional delays

In recent years, dynamical neural networks have received a considerable attention due

to their potential applications in many fields such as image and signal processing, pattern

recognition, associative memory, parallel computing, solving optimization problems ect. In

most of the practical applications, it is of prime importance to ensure that the designed neural

networks be stable. On the other hand, time delays unavoidably exist in most application

networks and often become a source of oscillation, divergence, instability or bad performance.

A great deal of effort from researchers has been devoted to study the problems of stability

analysis, control and estimation for delayed neural networks during the past decade.

It is well-known that, in practical implementation of neural networks, time delays may

not be constants. They are not only time-varying but also proportional in many models.

Furthermore, a neural network usually has a spatial nature due to the presence of an amount

of parallel pathways of a variety of axon sizes and lengths, it is desirable to model them by

introducing continuously proportional delay over a certain duration of time. Proportional delay

is one of time-varying (monotonically increasing) and unbounded delays which is different from

most other types of delay such as time-varying bounded delays, bounded and/or unbounded

distributed delays. Its presence leads to an advantage is that the network’s running time can

be controlled based on the maximum delay allowed by the network. In addition, dealing with

the dynamic behavior of neural networks with proportional delays is an interesting problem

which is also much more complicated.

In Chapter 2 we consider the problem of finite-time stability of non-autonomous neural

networks with heterogeneous proportional delays described by the following system

n

x′i (t)

bij (t)fj (xj (t))

= − ai (t)xi (t) +

j=1

(1)

n

cij (t)gj (xj (qij t)) + Ii (t), i ∈ [n], t > 0.

+

j=1

By using novel comparison techniques, an explicit criterion is derived in terms of inequalities

for M-matrix ensuring that for each given bound on the initial conditions the state trajectories

of the system do not exceed a certain threshold over a pre-specified finite time interval. These

conditions are shown to be relaxed for the Lyapunov asymptotic stability through examples.

2

3.2. Global dissipativity of non-autonomous neural networks with multiple proportional delays

Dissipativity of dynamical systems, first introduced in the earlier of 1970s, has a meaningful physical concept and is an important characteristic of many mathematical models of physical

processes. In Chapter 3, we consider the problem of global dissipativity of the following neural

networks model

n

x′i (t) = −ai (t)xi (t) +

bij (t)fj (xj (t))

j=1

n

+

(2)

cij (t)gj (xj (qij t)) + Ii (t), i ∈ [n], t > 0.

j=1

Both the cases of uniform and non-uniform positive self-feedback coefficients −ai (t) are taken

into account simultaneously. Based on an extended comparison technique and M-matrix theory,

new unified delay-independent conditions are derived for both the existence of attracting sets

and global dissipativity of the system. On the basis of the obtained results, a generalized

exponential estimate for a class of Halanay-type inequalities with proportional delays, which

will be useful in the field of asymptotic behavior analysis of neural networks with delays, is also

established in this chapter.

3.3. Global attractivity of positive periodic solution of a delayed Nicholson model

with nonlinear density-dependent mortality term

Mathematical models are important for describing dynamics of phenomena in the real

world. For example, Nicholson used the following delay differential equation

N ′ (t) = −αN(t) + βN(t − τ )e−γN (t−τ ) ,

(3)

where α, β, γ are positive constants, to model the laboratory population of the Australian

sheep-blowfly. In the biology interpretation of equation (3), N(t) is the population size at

time t, α is the per capita daily adult mortality rate, β is the maximum per capita daily egg

production rate,

1

γ

is the size at which the population reproduces at its maximum rate and

τ ≥ 0 is the generation time (the time taken from birth to maturity). Model (3) is typically

referred to the Nicholson’s blowflies equation.

In the past few years, the qualitative theory for Nicholson model and its variants has been

extensively studied and developed. However, most of the existing works so far are devoted

to Nicholson-type models with linear mortality terms. Normally, a model of linear density3

dependent mortality rate will be most accurate for populations at low densities. According

to marine ecologists, many models in fishery such as marine protected areas or models of Bcell chronic lymphocytic leukemia dynamics are suitably described by Nicholson-type delay

differential equations with nonlinear density-dependent mortality rate of the form

N ′ (t) = −D(N(t)) + βN(t − τ )e−γN (t−τ ) ,

where the function D(N) might have one of the forms D(N) = a−be−N (type-I) or D(N) =

(4)

aN

b+N

(type-II) with positive constants a and b. A natural extension of (4) to the case of variable

coefficients and delays, which is more realistic in the theory of population dynamics is given by

N ′ (t) = −D(t, N(t)) + β(t)N(t − τ (t))e−γ(t)N (t−τ (t)),

where D(t, N) = a(t) − b(t)e−N or D(t, N) =

a(t)N

b(t)+N .

(5)

In model (5), D(t, N) is the death rate

of the population which depends on time t and the current population level N(t), B(t, N(t −

τ (t))) = β(t)N(t − τ (t))e−γ(t)N (t−τ (t)) is the time-dependent birth function which involves a

maturation delay τ (t) and gets its maximum

β(t)

γ(t)e

at rate

1

.

γ(t)

Recently, Nicholson-type models with nonlinear density-dependent mortality terms have

attracted considerable research attention. In Chapter 4 of this thesis, we study the problem of

existence and global attractivity of positive periodic solution of the following Nicholson model

p

βk (t)N(t − τk (t))e−γk (t)N (t−τk (t)) ,

′

N (t) = −D(t, N(t)) +

(6)

k=1

where D(t, N) = a(t) − b(t)e−N . Based on new comparison techniques via differential inequalities, we derive conditions for the existence and global attractivity of a unique positive periodic

solution of model (4.1). An application to Nicholson models with constant coefficients is also

presented.

4. Obtained results

The thesis achieves the following main results:

1. Established conditions in terms of M-matrices for finite stability and power-rate synchronization of Hopfiled neural networks with time-varying coefficients and heterogeneous proportional delays.

2. Proved the global dissipativity of a class of neural networks with multi-proportional delays

for both uniformly positive and singular self-feedback coefficients.

4

3. Derived conditions and proved the existence, uniqueness and global attractivity of a positive periodic solution to a Nicholson model with nonlinear density-dependent mortality

term.

The results presented in this thesis are based on three papers published on ISI indexed

international journals.

5. Thesis organization

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 concerning finite-time stability, dissipativity of certain classes of time-delay systems and

and some other auxiliary results, which will be useful for the presentation of the thesis. Chapter

2 investigates the problem of finite-time stability of Hopfield neuron networks with time-varying

connection weights and heterogeneous proportional delays. The global dissipativity of nonautonomous neural networks with multi-proportional delays is studied and presented in Chapter

3. Finally, the existence, uniqueness and global attractivity of a positive periodic solution to a

Nicholson model with nonlinear density-dependent mortality term is studied in Chapter 4.

5

Chapter 1

PREMILINARIES

In this chapter, we present some auxiliary results in matrix analysis, differential equations,

stability theory in the sense of Lyapunov and short time and the dissipativity of certain classes

of time-delay systems which will be used in the next chapters.

1.1. M-matrix

This section is concerned with basic concepts and properties of M-matrices.

1.2. Time-delay systems and the Lyapunov stability theory

Consider the following initial valued problem for functional differential equations

x′ (t) = f (t, xt ), t ≥ t0 ,

xt0 = φ,

(1.1)

where f : D = [t0 , ∞) × C → Rn and φ ∈ C = C([−r, 0], Rn ) is initial function. Assume that

f (t, 0) = 0 and the function f (t, φ) satisfies conditions that for any t0 ∈ [0, ∞) and φ ∈ C, the

problem (1.1) possesses a unique solution on [t0 , ∞).

Definition 1.2.1. The trival solution x = 0 of (1.1) is said to be stable (in the sense of Lyapunov) if for any t0 ∈ R+ , ǫ > 0, there esists a δ = δ(t0 , ǫ) > 0 such that for any solution

x(t, φ) of (1.1), if φ

C

< δ then x(t, φ) < ǫ for all t ≥ t0 . The solution x = 0 is uniformly

stable if the aforementioned δ is independent of t0 .

Definition 1.2.2. The solution x = 0 of (1.1) is said to be uniformly asymptotically stable if

it is uniformly stable and there exists a δa > 0 such that for any η > 0 there exists a T (δa , η)

such that φ

C

< δa implies x(t, φ) < η for all t ≥ t0 + T (δa , η). x = 0 is globally uniformly

asymptotically stable if δa can be arbitrarily selected.

Theorem 1.2.1 (Lyapunov-Krasovskii Theorem). Assume that f : R × C → Rn maps each set

R × Ω, where Ω is bounded set in C into a bounded set in Rn and u, v, w : R+ → R+ are

continuous non-decreasing functions, u(0) = 0, v(0) = 0 and u(s) > 0, v(s) > 0 for s > 0. If

there exists a continuous positive definite functional V : R × C → R+ ,

u( φ(0) ) ≤ V (t, φ) ≤ v( φ

6

C ),

∀φ ∈ C,

(1.2)

such that the derivative of V (t, φ) along trajectories of (1.1) is negative definite, that is,

V ′ (t, φ) ≤ −w( φ(0) ).

(1.3)

Then, the trivial solution x = 0 of (1.1) is uniformly stable. Moreover, if w(s) > 0 for s > 0

and lims→∞ u(s) = ∞ then the solution x = 0 is globally uniformly asymptotically stable.

1.3. Finite-time stability of dynamical systems

1.3.1. The concept of finite-time stability

The concept of finite-time stability (FTS) dates back to the 1950s, when it was introduced

in the Russian literature. Later, during the 1960s, this concept appeared in the western journals.

Roughly speaking, a system is said to be finite-time stable if, given a bound on the initial

conditions, its state does not exceed a certain threshold during a specified time interval. More

precisely, given the system

x′ (t) = f (t, x(t)),

x(t0 ) = x0 ,

(1.4)

where x(t) ∈ Rn is the system state vector, we can give the following formal definition.

Definition 1.3.1. Given an initial time t0 , a positive scaler T and two sets X0 , Xt . System

(1.4) is said to be finite-time stable with respect to (t0 , T, X0 , Xt ) if

x0 ∈ X0 =⇒ x(t, t0 , x0 ) ∈ Xt ,

∀t ∈ [t0 , t0 + T ].

Note that the trajectory set is allowed to vary in time. For well-posedness of the above

definition, it is required that X0 ⊂ Xt0 . However, in general, it is not required that X0 is

included in Xt for t > t0 . In addition, the sets X0 and Xt are typically given in the form of

ellipsoids ER (ρ) = {x⊤ Rx < ρ : x ∈ Rn }, where R ∈ Sn+ is a symmetric positive definite

matrices. The above definition can be stated as follows.

Definition 1.3.2. Given an initial time t0 , a scalar T > 0, a matrix R ∈ Sn+ and positive scalars

r1 < r2 . System (1.4) is said to be finite stable w.r.t (t0 , T, r1 , r2 , R) if for any x0 ∈ ER (r1 ),

the corresponding state trajectory x(t) = x(t; t0 , x0 ) of (1.4) satisfies x⊤ (t)Rx(t) < r2 for all

t ∈ [t0 , t0 + T ].

7

1.3.2. Finite-time stability of linear systems with mixed time-varying delays

Consider the following linear nonautonomous system with time-varying delays

t

′

x(s)ds, t ≥ 0,

x (t) = Ax(t) + Dx(t − τ (t)) + G

(1.5)

t−κ(t)

x(t) = φ(t),

t ∈ [−h, 0],

where x(t) ∈ Rn is the state vector, φ ∈ C([−h, 0], Rn ) is the initial function, A, D, G ∈ Rn×n

are known system matrices, τ (t), k(t) are time-varying delays which satisfy

0 ≤ τ1 ≤ τ (t) ≤ τ2 ,

τ ′ (t) ≤ µ ≤ 1,

0 ≤ κ1 ≤ κ(t) ≤ κ2 ,

where µ is a constant involving the rate of change of the discrete delay τ (t), τ1 , τ2 , κ1 , κ2 are

bounds of delays and h = max{τ2 , κ2 }.

Definition 1.3.3. Given T, r1 , r2 , where r1 < r2 . System (1.5) is said to be finite-time stable

w.r.t (r1 , r2 , T ) if for any φ ∈ C([−h, 0], Rn ), φ

∞

≤ r1 , one has x(t, φ)

∞

< r2 for all

t ∈ [0, T ].

Theorem 1.3.1. For given scalars T, r1 , r2 , r1 < r2 , system (1.5) is finite-time stable with

respect to (r1 , r2 , T ) if there exist positive scalars α, ρi , i = 1, 2, 3, 4, and symmetric positive

definite matrices P, Q, R ∈ Rn×n satisfying the following conditions

Π = Π0 + Π1 + Π2 < 0,

ρ1 In ≤ P ≤ ρ2 In ,

ρ2 + τ2 eατ2 ρ3 +

ρ1

(1.6a)

Q ≤ ρ3 In ,

eακ2 −1

ρ4

α

<

R ≤ ρ4 In ,

r2

r1

(1.6b)

2

e−αT ,

(1.6c)

where ei = 0n×(i−1)n In 0n×(3−i)n , i = 1, 2, 3, A = Ae1 + De2 + Ge3 , Π0 = e⊤

1 PA +

⊤

ατ1 e⊤ Qe v`

⊤

A⊤ P e1 − αe⊤

2 a Π2 = κ2 e1 Re1 −

1 P e1 , Π1 = e1 Qe1 − (1 − µ)e

2

1 ⊤

κ2 e3 Re3 .

1.4. Dissipativity of functional differential equations

In this section we introduce some preliminary results involving the dissipativity of certain

classes of time-delay systems. First, we consider the following system

x′ (t) = F (t, x(t), x(t − τ1 (t)), . . . , x(t − τm (t))),

x(t) = φ(t),

t ∈ [0, ∞),

(1.7)

t ∈ [−τ, 0],

where τk (.) are continuous time-delay functions satisfying 0 ≤ τk (t) ≤ τ for all t ∈ [0, ∞),

k ∈ [m], where τ > 0 is a constant. The function F : [0, ∞) × Rn × (C([−τ, ∞), Rn ))m → Rn

8

is continuous and satisfies

m

2 u, F (t, u, ψ1(.), . . . , ψm (.)) ≤ γ(t) + α(t) u

2

βk (t) ψk (t − τk (t))

+

2

(1.8)

k=1

for all t ∈ [0, ∞), u ∈ Rn v`a ψk (.) ∈ C([−τ, ∞), Rn ). The function F is also assumed to satisfy

conditions that for any φ(.) ∈ C([−τ, 0], Rn ), the problem (1.7) possesses a unique solution

x(t, φ) on [−τ, ∞).

Definition 1.4.1. System(1.7) is said to be globally dissipative if there exists a bounded set

B ⊂ Rn with the property that for any bounded set B ⊂ Rn , there exists a t∗ = t∗ (B) such

that for any φ(.) ∈ C([−τ, 0], Rn ), φ(t) ∈ B for all t ∈ [−τ, 0], one has x(t, φ) ∈ B for all

t ≥ t∗ (B). The set B is called an absorbing set of (1.7).

1.4.1. The Halanay inequality approach

Lemma 1.4.1 (Halanay inequality). Assume that the function u(t) ≥ 0, t ∈ (−∞, ∞), satisfies

u′ (t) ≤ γ(t) + α0 u(t) + β0

sup

t−τ ≤s≤t

u(s), t ≥ t0 , u(t) = θ(t), t ≤ t0 ,

(1.9)

where θ ∈ BC((−∞, t0 ], R+ ) is a continuous bounded function. If α0 + β0 < 0 then there exists

a scalar λ > 0 such that

u(t) ≤

where γ∗ = supt≥t0 γ(t) and θ

γ∗

+ θ

−(α0 + β0 )

∞

−λ(t−t0 )

,

∞e

t ≥ t0 ,

(1.10)

= supt≤t0 |θ(t)|.

Remark 1.4.1. Let α0 = supt≥0 α(t) and β0 = supt≥0 β(t). If α0 + β0 < 0 then, by Lemma

1.4.1, system (1.7) is globally dissipative. Specifically, for a given ǫ > 0, there exists a t∗ =

t∗ ( φ

∞ , ǫ)

> 0 such that

x(t)

2

<

γ∗

+ ǫ, t > t∗ .

−(α0 + β0 )

Therefore, system (1.7) is globally dissipative with the absorbing set B = B 0,

∗

− α0γ+β

+ǫ .

0

Moreover, the estimate (1.10) guarantees the exponential attraction of B.

Theorem 1.4.2. Let the function u(t) ≥ 0, t ∈ (−∞, ∞), satisfy

u′ (t) ≤ γ(t) + α(t)u(t) + β(t)

sup

u(s) (t ≥ t0 ),

u(t) = θ(t) (t ≤ t0 ),

(1.11)

t−τ (t)≤s≤t

where θ ∈ BC((−∞, t0 ], R+ ), α(t), β(t), γ(t) are continuous functions, β(t) ≥ 0, γ(t) ≥ 0 and

the delay τ (t) ≥ 0 satisfies t − τ (t) → ∞ as t → ∞. Assume that

α(t) + β(t) ≤ −σ < 0, t ≥ t0 ,

9

(1.12)

γ∗

+ θ ∞ , t ∈ [t0 , ∞). Additionally, if there

σ

exists a 0 < δ < 1 such that δα(t) + β(t) < 0, ∀t ≥ t0 , then for any ǫ > 0, there exists a

for some positive scalar σ. Then, u(t) ≤

t∗ = t∗ ( θ

∞ , ǫ)

> t0 by which

u(t) ≤

γ∗

+ ǫ,

σ

t ≥ t∗ .

1.4.2. Dissipativity of a class of nonlinear systems with proportional delay: A

changing of variable approach

Consider the following system

x′ (t) = g(x(t), x(qt)), t ≥ t0 > 0,

(1.13)

x(t) = ϕ(t), t ∈ [qt0 , t0 ],

where q is a constant, 0 < q < 1, and the function g satisfies

2 u, g(u, v) ≤ γ + α u

2

+β v

2

(1.14)

with α, β, γ are given constants. For the notational simplicity, we assume that t0 = 1. By the

change of variable y(t) = x(et ), system(1.14) can be stransformed to the following system with

a constant delay

y ′(t) = f (t, y(t), y(t − τ )), t > 0,

where τ = − ln(q) > 0 and

(1.15)

y(t) = ϕ(t), t ∈ [−τ, 0],

f (t, y(t), y(t − τ )) = et g(y(t), y(t − τ )).

(1.16)

From (1.14) and (1.16) we have

2 u, f (t, u, v) ≤ et γ + α u

2

+β v

2

.

(1.17)

Theorem 1.4.3. Let y(t) be a solution of (1.15)-(1.17) and assume that α + β < 0. Then, for

a given ǫ > 0, there exists a t∗ = t∗ ( ϕ

y(t)

2

∞ , ǫ)

<−

> 0 such that

γ

+ ǫ, ∀t > t∗ .

α+β

Therefore, system (1.15) is globally dissipative and B = B 0,

γ

+ ǫ is an absorbing set

− α+β

for a given ǫ > 0.

1.5. Auxiliary results

This section presents some technical lemmas and auxiliary results which will be used in

the next chapters.

10

Chapter 2

FINITE-TIME STABILITY OF NON-AUTONOMOUS NEURAL NETWORKS WITH

HETEROGENEOUS PROPORTIONAL DELAYS

In this chapter we study the problem of finite-time stability of non-autonomous neural networks with heterogeneous proportional delays. By introducing a novel constructive approach,

we derive explicit conditions in terms of matrix inequalities ensuring that the state trajectories

of the system do not exceed a certain threshold over a pre-specified finite time interval. As a

result, we also obtain conditions for the power-rate global stability of the system.

2.1. Model description

Consider the following neural networks model

n

x′i (t)

bij (t)fj (xj (t))

= −ai (t)xi (t) +

j=1

n

cij (t)gj (xj (qij t)) + Ii (t), t > 0,

+

(2.1)

j=1

xi (0) = x0i , i ∈ [n],

where xi (t) is the state variable (potential or voltage) of the ith neuron at time t, fj (.), gj (.),

j ∈ [n], are activation functions, ai (t) are self-inhibition terms, bij (t), cij (t) are time-varying

connection weights, Ii (t) are external inputs, qij ∈ (0, 1], i, j ∈ [n], are possibly heterogeneous

proportional delays, x0 = (x01 , . . . , x0n )T ∈ Rn is the initial state vector.

(A2.1) The neuron activation functions fi , gi , i ∈ [n], satisfy

−

li1

≤

fi (x) − fi (y)

+

≤ li1

,

x−y

−

li2

≤

gi (x) − gi (y)

+

≤ li2

, ∀x, y ∈ R, x = y,

x−y

− +

where lik

, lik , k = 1, 2, are known constants.

Remark 2.1.1. Let the function F : R+ × Rn × Rn×n → Rn be defined by F (t, u, v) =

(Fi (t, u, v)) where u = (ui ) ∈ Rn , v = (vij ) ∈ Rn×n and

n

n

cij (t)gj (vij ) + Ii (t).

bij (t)fj (uj ) +

Fi (t, u, v) = −ai (t)ui +

j=1

j=1

By (A2.1), F (t, u, v) is continuous and Lipschitz on R+ × Rn × Rn×n . Therefore, for a given

initial vector x0 ∈ Rn , there exists a unique solution x(t) = x(t, x0 ) of (2.1) on the interval

[0, ∞).

11

2.2. Finite-time stability of model (2.1)

Definition 2.2.1. For given a time T > 0 and positive numbers r1 < r2 , a solution x∗ (t) of

(2.1) is said to be finite-time stable with respect to (r1 , r2 , T ) if for any solution x(t) of (2.1),

x(0) − x∗ (0)

∞

≤ r1 =⇒ x(t) − x∗ (t)

∞

< r2 , ∀t ∈ [0, T ].

System (2.1) is said to be FTS with respect to (r1 , r2 , T ) if any solution x∗ (t) of (2.1) is FTS

with respect to (r1 , r2 , T ).

(A2.2) The matrices A(t) = diag{ai (t)}, B(t) = (bij (t)), C(t) = (cij (t)) satisfy

ai (t) ≥ ai > 0, |bij (t)| ≤ bij , |cij (t)| ≤ cij , ∀t ≥ 0, i, j ∈ [n].

+

−

+

−

Hereafter, let us denote for i ∈ [n] the constants Lfi = max{li1

, −li1

} and Lgi = max{li2

, −li2

}.

We also introduce the following matrix notations

A = diag{a1 , a2 , . . . , an },

B = (bij ),

Lf = diag{Lf1 , Lf2 , . . . , Lfn },

C = (cij ),

Lg = diag{Lg1 , Lg2 , . . . , Lgn },

M = BLf + CLg − A.

Theorem 2.2.1. Under assumptions (A2.1) and (A2.2), for given 0 < r1 < r2 and T > 0,

system (2.1) is finite-time stable with respect to (r1 , r2 , T ) if there exist a positive number γ and

a vector ξ ∈ Rn , ξ ≻ 0, satisfy the following conditions

(i) (M − γI) ξ ≺ 0,

(ii) C(ξ) <

r2 −γT

e

,

r1

where C(ξ) = ξ uξl−1 denotes the condition number of ξ.

Remark 2.2.1. Condition (i) in Theorem 2.2.1 does not guarantee the asymptotic stability of

system (2.1) in the sense of Lyapunov (LAS). Moreover, even conditions (i), (ii) are satisfied

for any T > 0, r2 > r1 > 0, system (2.1) may not be LAS.

2.3. Long-time behavior: Synchronization of model (2.1)

Theorem 2.3.1. Let assumptions (A2.1) and (A2.2) hold and assume that −M is a nonsingular M-matrix. Then, there exist positive constants β, σ, which are independent of solutions of

(2.1), such that for any two solutions x(t), x∗ (t) of (2.1), the following inequality holds

x(t) − x∗ (t)

∞

≤β

x(0) − x∗ (0)

(1 + t)σ

12

∞

, t ≥ 0.

(2.2)

Remark 2.3.1. Let x∗ (t) be a solution of (2.1). The estimate (2.2) shows that any state

trajectory x(t) of (2.1) will have similar behavior with x∗ (t) when the time is sufficiently large.

Thus, the family of solutions of (2.1) has the same behavior with x∗ (t) as the time t tends to

infinity. In the field the network control systems, this feature is referred to the synchronization.

Remark 2.3.2. The constant σ0 mentioned in the proof of Theorem 2.3.1 defines the powerrate synchronization of model (2.1). The power convergence rate σmax can be defined by the

following procedure

• Define a vector ξ ∈ Rn , ξ ≻ 0, such that Mξ ≺ 0.

• Compute

n

η = (−Mξ)l = min

ξj bij Fj + cij Gj

ai ξi −

i∈[n]

.

(2.3)

j=1

• The power convergence rate σmax can be iteratively computed as

n

max σ > 0 s.t. Hi (σ) = σξi +

σ ln

Gj cij ξj e

1

qij

− 1 − η ≤ 0, ∀i ∈ [n].

(2.4)

j=1

2.4. Numerical examples

This section presents some numerical examples and simulations to demonstrate the effectiveness of the obtained results in this chapter.

13

Chapter 3

GLOBAL DISSIPATIVITY OF NON-AUTONOMOUS NEURAL NETWORKS WITH

MULTIPLE PROPORTIONAL DELAYS

In this chapter we investigate the problem of dissipativity analysis of the following nonlinear differential system

n

x′i (t)

bij (t)fj (xj (t))

= −ai (t)xi (t) +

j=1

n

(3.1)

cij (t)gj (xj (qij t)) + Ii (t), t > 0, i ∈ [n],

+

j=1

in two cases: (i) the self-feedback coefficients are uniformly positive, that is, ai (t) ≥ ai > 0;

and (ii) the self-feedback coefficients can be singular, that is, ai (t) > 0 and inf t≥0 ai (t) = 0.

As discussed in Chapter 1, the proposed methods in the existing literate such as the use of

state transformation or Halanay inequalities are are now not appropriate for model (3.1) due

to the nature of its structure. Thus, to analize the dissipativity of model (3.1), we will develop

some new comparison techniques based on the theory of M-matrix to derive conditions for the

existence a generalized exponential attracting set. The content of this chapter is written based

on paper [2] in the publication list of this thesis.

3.1. Preliminaries

Consider a non-autonomous neural network model with multiple proportional delays presented in (3.1), where x(t) = (xi (t)) ∈ Rn is the state vector, ai (t) ∈ R+ , i ∈ [n], are selffeedback coefficients, bij (t) ∈ R and cij (t) ∈ R are neuron connection weights at time t, fj (.),

gj (.), j ∈ [n], are neuron activation functions, (Ii (t)) is the external input vector, qij ∈ (0, 1),

i, j ∈ [n], are heterogeneous proportional delay. The initial condition of (3.1) is specified as

xi (0) = x0i , i ∈ [n],

(3.2)

where x0 = (x0i ) ∈ Rn is a given vector.

(A3.1): There exist scalars Lfj ≥ 0, Lgj ≥ 0, j ∈ [n], such that

|fj (a) − fj (b)| ≤ Lfj |a − b|, |gj (a) − gj (b)| ≤ Lgj |a − b|,

∀a, b ∈ R.

(3.3)

(A3.2): The connection weights bij (t), cij (t), i, j ∈ [n], and external inputs Ii (t), i ∈ [n], are

14

bounded, i.e. there exist scalars bij , cij and I i , i, j ∈ [n], such that

|bij (t)| ≤ bij , |cij (t)| ≤ cij , |Ii (t)| ≤ I i , ∀t ≥ 0, i, j ∈ [n].

Definition 3.1.1. A compact set Ω ⊂ Rn is said to be a global attracting set of (3.1) if any

solution x(t) = x(t, x0 ) of (3.1) satisfies lim supt→∞ ρ(x(t), Ω) = 0, where ρ(x, Ω) = inf y∈Ω x −

y

∞

denotes the distance from x to Ω.

Definition 3.1.2. A compact set Ω ⊂ Rn is said to be a global generalized exponential attracting set of (3.1) if there exist a function κ( x0

∞)

≥ 0, a nondecreasing function σ(t) ≥ 0 such

that lim supt→∞ σ(t) = ∞ and any solution x(t) = x(t, x0 ) of (3.1) satisfies

ρ(x(t), Ω) ≤ κ( x0

−σ(t)

,

∞ )e

t ≥ 0.

(3.4)

If σ(t) = αt, where α is a positive scalar, then Ω is a global exponential attracting set of (3.1).

Definition 3.1.3. System (3.1) is said to be globally dissipative if there is a bounded set A ⊂

Rn such that for any bounded set B ⊂ Rn there is a time td = td (B) with the property that for

any initial condition x0 ∈ Φ, the corresponding solution x(t, x0 ) belongs to A for all t ≥ td (B).

Then A is called an absorbing set of (3.1).

Remark 3.1.1. If there exists a global generalized exponential attracting set Ω and the function

κ(.) defined in (3.4) is nondecreasing then system (3.1) is globally dissipative. Indeed, if Ω is

a global generalized exponential attracting set of (3.1) then for any bounded set B ⊂ Rn , let

r(B) = supx0 ∈B x0

∞,

we have

ρ(x(t), Ω) ≤ κ(r(B))e−σ(t), t ≥ 0.

For given any ǫ > 0, let Aǫ = {x ∈ Rn : ρ(x, Ω) ≤ ǫ} then Aǫ is a bounded set which contains

the set {x(t, x0 ) : t ≥ td (B), x0 ∈ B}, where td (B) = inf{t > 0 : σ(t) ≥ ln

κ(r(B))

}.

ǫ

Thus, Aǫ

is an absorbing set of (3.1) and system (3.1) is globally dissipative.

3.2. Global attractivity of the model (3.1)

3.2.1. Regular self-feedback coefficients

In this section, we derive disipativity conditions for system (3.1) under the following

assumption.

(A3.3): There exist positive scalars ai , i ∈ [n], such that

ai (t) ≥ ai , ∀t ≥ 0, i ∈ [n].

15

(3.5)

Let D = diag(a1 , a2 , . . . , an ) and M = D − BLf − CLg , where B = (bij ), C = (cij ),

Lf = diag(Lf1 , Lf2 , . . . , Lfn ) and Lg = diag(Lg1 , Lg2 , . . . , Lgn ). We have the following result.

Theorem 3.2.1. Under assumptions (A3.1)-(A3.3), if M is a nonsingular M-matrix then the

following assertions hold

(1) The set Ω defined by

Ω=

x ∈ Rn : x

∞

≤

γ

(Mχ)+

is a global generalized exponential attracting set of (3.1), where χ ∈ int(Rn+ ) satisfies

χ

∞

n

j=1 (bij |fj (0)| + cij |gj (0)|) + I i

= 1, Mχ ≻ 0 and γ = maxi∈[n]

;

(2) System (3.1) is globally dissipative.

Corollary 3.2.2. Under assumptions (A3.1)-(A3.3), if

n

n

ai >

bji + Lgi

Lfi

x

∞

≤

γ

σ∗

(3.6)

j=1

j=1

then Ω = x ∈ Rn :

cji , i ∈ [n]

is a global generalized exponential attracting set of (3.1) and

system (3.1) is globally dissipative, where

n

n

∗

σ = min

i∈[n]

ai − Lfi

bji −

j=1

Lgi

cji .

j=1

3.2.2. Singular self-feedback coefficients

In this section, we derive delay-independent conditions that ensure the global dissipativity

of system (3.1) without uniform positiveness of self-feedback coefficients.

For convenience, we introduce the following assumptions.

(A3.4): There exist a function ϕ(t) > 0 and positive scalars a

ˆi , i ∈ [n], such that

t

ai (t) ≥ a

ˆi ϕ(t), sup

t≥0

t

ϕ(s)ds < ∞, lim

t→∞

qij t

ϕ(s)ds = ∞.

(3.7)

0

(A3.5): There are constants ˆbij ≥ 0, cˆij ≥ 0 and Iˆi ≥ 0 such that

|bij (t)| ˆ |cij (t)|

|Ii (t)|

≤ bij ,

≤ cˆij ,

≤ Iˆi , ∀i, j ∈ [n], t ≥ 0.

ai (t)

ai (t)

ai (t)

(3.8)

ˆij =

Remark 3.2.1. Assumptions (A3.4) and (A3.5) are obviously satisfied with ˆbij = bij a−1

i , c

cij a−1

i , i, j ∈ [n], and ϕ(t) =

1

1+t

if assumptions (A3.2) and (A3.3) hold. Thus, (A3.4) and

(A3.5) can be regarded as extended conditions of (A3.2) and (A3.3).

Let us denote B = (ˆbij ), C = (ˆ

cij ) and H = En − (BLf + CLg ), where En denotes the

identity matrix in Rn×n .

16

Theorem 3.2.3. Assume assumptions (A3.1), (A3.4), (A3.5) are satisfied and H is a nonγˆ

singular M-matrix. Then system (3.1) is globally dissipative and the ball B 0, m

ˆ + ǫ is an

absorbing set of (3.1) for any ǫ > 0, where γˆ = maxi∈[n] {Iˆi +

m

ˆ = (Hη)+ and η ∈ int(Rn+ ) satisfying η

∞

n

ˆ

j=1 (bij |fj (0)|

+ cˆij |gj (0)|)},

= 1 and Hη ≻ 0.

Corollary 3.2.4. Under assumptions (A3.1), (A3.4) and (A3.5), if H is a nonsingular Mmatrix then system (3.1) is globally synchronous. More precisely, any two solutions x(t) and

x∗ (t) of (3.1) satisfy the following inequality

x(t) − x∗ (t)

where η ∈ int(Rn+ ) satisfies η

∞

∞

≤

1

x(0) − x∗ (0)

η+

−λ0

∞e

t

0 ϕ(s)ds

, t≥0

(3.9)

= 1 and Hη ≻ 0.

In the remaining of this section, let us consider the following Halanay-type inequality with

multiple proportional delays:

n

+

bj (t)u(qj t) + d(t), t > 0

D u(t) ≤ −a(t)u(t) +

(3.10)

j=1

where a(t) > 0, bj (t), j ∈ [n], and d(t) are continuous functions, qj ∈ (0, 1), j ∈ [n], are

proportional delays.

By some similar lines used in the proof of Theorem 3.2.3 we obtain the following result.

Corollary 3.2.5. Assume that there exist scalars l > 0, µ0 ∈ (0, 1), µ1 ≥ 0, a function as (t) > 0

and a t0 > 0 such that a(t) ≥ las (t), t ≥ t0

t

sup

t≥t0

and

t

as (θ)dθ < ∞, lim

qj t

t→∞

as (θ)dθ = ∞

0

n

|bj (t)| − µ0 a(t) ≤ 0, |d(t)| ≤ µ1 a(t), t ≥ t0 .

j=1

Then any solution u(t) of (3.10) converges exponentially within the bound

µ1

1−µ0 .

Specifically,

˜ such that any solution u(t) of (3.10) satisfies

there exists a positive constant λ

u(t) ≤

µ1

µ1

˜

+ max u(0) −

, 0 e−λ

1 − µ0

1 − µ0

t

0 as (ζ)dζ

, t ≥ 0.

(3.11)

3.3. Illustrative examples

This section presents some numerical examples to illustrate the effectiveness of the results

obtained in this chapter.

17

Chapter 4

GLOBAL ATTRACTIVITY OF POSITIVE PERIODIC SOLUTION OF A DELAYED

NICHOLSON MODEL WITH NONLINEAR MORTALITY TERM

In this chapter we study the problem of existence and global attractivity of positive

periodic solution of the following Nicholson model

p

βk (t)N(t − τk (t))e−γk (t)N (t−τk (t))

′

N (t) = −D(t, N(t)) +

(4.1)

k=1

nonlinear density-dependent mortality term D(t, N) = a(t) − b(t)e−N . Based on novel comparison techniques via differential and integral inequalities, we first derive conditions for the global

uniform permanence and dissipativity of the model (4.1). On the basis of the global uniform

permanence and dissipativity, the existence and global attractivity of a unique positive periodic solution of model (4.1) is then established. As an application to Nicholson models with

constant coefficients, improved results on the existence, uniqueness and global attractivity of a

positive equilibrium are also obtained. The content of this chapter is based upon the paper [3]

in the list of publication of this thesis.

4.1. Preliminaries

Consider a Nicholson model with delays and nonlinear density-dependent mortality term

of the form

p

βk (t)N(t − τk (t))e−γk (t)N (t−τk (t)) ,

′

N (t) = −D(t, N(t)) +

t ≥ t0 ,

(4.2)

k=1

N(t) = ϕ(t),

t ∈ [t0 − τM , t0 ],

(4.3)

where the density-dependent mortality term D(t, N) is of the form

D(t, N) = a(t) − b(t)e−N

(4.4)

and τM = max1≤k≤p τk+ represents the upper bound of delays.

Assumption (A):

(A4.1) a, b, γk : [0, ∞) → (0, ∞), βk : [0, ∞) → [0, ∞) and τk : [0, ∞) → [0, τM ] are continuous

bounded functions, where τM is some positive constant.

(A4.2) There exists an ω > 0 such that the functions a, b, βk , γk and τk belong to Pω (R+ ).

18

Condition (C):

(C4.1) a) b(t) ≥ a(t) ≥ a− > 0, b) θ

(C4.2) lim supt→∞

p

βk (t)

k=1 γk (t)

1

a(t)

(C4.3) b− − a+ > 0, ̺

(C4.4)

p

+

k=1 βk max

a− −

lim inf t→∞

= σ, 1 −

p

β+

k=1 γ −

k

1

e

−

1 1−γk r∗

, γ −r

e2

e k ∗

<

̺b−

,

b+

b(t)

a(t)

> 1.

b−

a+

.

σ

> 0.

e

> 0.

r∗ = ln

A preview of our main results is presented in the following table.

Conditions

(A4.1), (C4.1)

(A4.1), (C4.1a), (C4.2)

Results

Uniform permanence in C0+ , lim inf t→∞ N (t, t0 , ϕ) ≥ ln(θ).

b+

Uniform dissipativity in C0+ , lim supt→∞ N (t, t0 , ϕ) ≤ ln a− (1−

σ

) .

(A4.1), (C4.3)

(A4.1), (A4.2), (C4.3)

and (C4.4)

ln ab + ≤ lim inf t→∞ N (t, t0 , ϕ) ≤ lim supt→∞ N (t, t0 , ϕ) ≤ ln

There exists a unique positive ω-periodic solution N ∗ (t)

which is globally attractive in C0+ .

e

−

b+

̺

.

For a biological interpretation of the proposed conditions, it is reasonable that when the

population is absence the death rate is nonpositive (i.e. D(t, 0) ≤ 0) and D(t, N) is always

positive when N > 0. This gives rise to condition (C4.1). On the other hand, in most of

biological models, there typically exists a threshold related to the so-called carrying capacity.

When the population size is very large, over the carrying capacity, the death rate can be bigger

p

βk (t)

k=1 γk (t)e

than the maximum birth rate. The quantity

can be regarded as the maximum

birth rate of model (4.2). In addition, when N is large D(t, N) is approximate to a(t). By

this observation, we make an assumption to ensure that

p

βk (t)

k=1 γk (t)e

< a(t). This reveals the

imposing of condition (C4.2) when considering long-time behavior of the model. (C4.3) is a

testable condition derived from (C4.2) and (C4.1a) by taking into account the upper bound of

the associated rates. While condition (C4.3) only guarantees non-extinction and non-blowup

behavior, condition (C4.4) reveals that, by certain scaling coefficients, when maximum per

capita daily egg production rates are smaller than the gap between the maximum death rate

and birth rate (i.e. ̺ = a− −

1

e

p

β+

k=1 γ − ),

k

the population will be stable around a periodic

trajectory (in the case of periodic coefficients) or a positive equilibrium (for time-invariant

model).

19

4.2. Permanence of global positive solutions

4.2.1. Global existence of positive solutions

Theorem 4.2.1. Let assumption (A4.1) hold. Assume that b(t) ≥ a(t) for all t ∈ [0, ∞).

Then, for any initial condition ϕ ∈ C0+ , the solution N(t, t0 , ϕ) of system (4.2)-(4.4)satisfies

N(t, t0 , ϕ) > 0, t ∈ [t0 , η(ϕ)), and η(ϕ) = ∞.

Remark 4.2.1. To ensure the positiveness of solutions of (4.2)-(4.4) with initial conditions in

C0+ , condition b(t) ≥ a(t) cannot be relaxed. For a counterexample, let n = 1 and assume that

t

b(t)

sup

= δ ∈ [0, 1),

t≥0 a(t)

a(s)ds → ∞,

t → ∞.

0

Then,

ϕ(0)−

N(t, t0 , ϕ) ≤ ln e

t

t0

a(s)ds

−

+δ 1−e

t

t0

a(s)ds

→ ln(δ) < 0 as t → ∞.

4.2.2. Uniform permanence

In this section we derive conditions and prove the uniform permanence of model (4.2).

Theorem 4.2.2. Let assumption (A4.1) hold. Assume that b(t) ≥ a(t) ≥ a− > 0 and

lim inf

t→∞

b(t)

≥ eℓm > 1.

a(t)

(4.5)

Then, for any ϕ ∈ C0+ ,

lim inf N(t, t0 , ϕ) ≥ ℓm > 0.

t→∞

Remark 4.2.2. As a special case of (4.5), for bounded functions a(t) and b(t), if b− > a+ then

the scalar ℓm in (4.5) can be chosen as

ℓm = ln

b−

a+

.

The following result shows the uniform dissipativity of system (4.2)-(4.4) in C0+ in the

sense that there exists a constant ℓM > 0 such that lim supt→∞ N(t, t0 , ϕ) ≤ ℓM .

Theorem 4.2.3. Assume assumption (A4.1) and the following conditions hold

b+ ≥ b(t) ≥ a(t) ≥ a− > 0,

1

lim sup

t→∞ a(t)

p

k=1

βk (t)

= σ,

γk (t)

t ∈ [0, +∞),

1−

σ

> 0.

e

(4.6)

(4.7)

Then, system (4.2)–(4.4) is uniformly dissipative in C0+ . More precisely, for any initial condition

ϕ ∈ C0+ , the corresponding solution N(t, t0 , ϕ) of (4.2)-(4.4) satisfies

lim sup N(t, t0 , ϕ) ≤ ℓM

t→∞

20

ln

a−

b+

1−

σ

e

.

The following result is obtained as a consequence of Theorems 4.2.2 and 4.2.3.

Corollary 4.2.4. Let assumption (A4.1) hold, where a, b, βk , γk are bounded functions, γk− > 0.

Assume that

1

a −

e

p

βk+

−

̺

> 0,

(4.8a)

b− − a+ > 0.

(4.8b)

k=1

γk−

Then, for any ϕ ∈ C0+ , it holds that

ln

b−

a+

≤ lim inf N(t, t0 , ϕ) ≤ lim sup N(t, t0 , ϕ) ≤ ln

t→∞

t→∞

b+

̺

(4.9)

.

4.3. Global attractivity of positive periodic solution

In this section we assume that assumptions (A4.1), (A4.2) and conditions (4.8a)-(4.8b)

are satisfied. For convenience, we denote

b−

a+

r∗ = ln

b−

a+

γk− r∗

,

b+

̺

∗

r = ln

,

νk = max

1 1 − γk− r∗

,

e2 eγk− r∗

.

1

max γk−

Note that, by (4.8b),

> 1 and hence r∗ > 0. In addition, since the condition r∗ <

is not imposed, 1 −

can be positive, negative or zero. For 1 ≤ k ≤ p that 1 − γk− r∗ ≤ 0,

νk =

1

.

e2

We are now in a position to present the existence, uniqueness and global attractivity of a

positive periodic solution of system (4.2)-(4.4) as in the following theorem.

Theorem 4.3.1. Let assumptions (A4.1), (A4.2), conditions (4.8a), (4.8b) and the following

ones are satisfied

inf 1 − τk′ (t) = µ > 0,

t≥0

p

νk βk+ < µ

k=1

̺b−

,

b+

(4.10)

(4.11)

where ̺ is the constant defined in (4.8a). Then, system (4.2)-(4.4) has a unique positive ωperiodic solution N ∗ (t) which is globally attractive in C0+ .

Remark 4.3.1. Conditions (4.10) and (4.11) are involved a scalar µ > 0 related to the rate

of change of delay functions τk (t). However, this scalar can be relaxed and conditions (4.10),

(4.11) are reduced to the following one

p

νk βk+ <

k=1

̺b−

.

b+

More precisely, we state that in the following theorem.

21

(4.12)

Theorem 4.3.2. Under assumptions (A4.1) and (A4.2), assume that conditions (4.8a), (4.8b)

and (4.12) are satisfied. Then, system (4.2)-(4.4) has a unique positive ω-periodic solution

N ∗ (t) which is globally attractive in C0+ .

4.4. Attractivity of positive equilibrium

In this section, we apply our results presented in the preceding sections to the following

Nicholson model

p

βk N(t − τk (t))e−γk N (t−τk (t)) ,

′

N (t) = −D(N(t)) +

t ≥ t0 ≥ 0,

(4.13)

k=1

p

k=1 βk

where βk ≥ 0, γk > 0 are known coefficients,

> 0. The nonlinear density-dependent

mortality term is given by D(N) = a − be−N , a > 0, b > 0. Time-varying delays τk (t) are

continuous and bounded in the range [0, τM ].

For model (4.13), conditions (4.8a), (4.8b) are reduced to the following coupled condition

1

e

p

k=1

βk

< a < b.

γk

(4.14)

Proposition 4.4.1. Let condition (4.14) hold. Then, for any ϕ ∈ C0+ , it holds that

ln

b

a

≤ lim inf N(t, t0 , ϕ) ≤ lim sup N(t, t0 , ϕ) ≤ ln

t→∞

t→∞

b

a−

1

e

p

βk

k=1 γk

.

(4.15)

Theorem 4.4.2. Assume that

q

βk

k=1

where

νˆk = max

νˆk +

1

eγk

< a < b,

(4.16)

1 1 − γk ln( ab )

,

.

b

e2

eγk ln( a )

Then, model (4.13) has a unique positive equilibrium N ∗ which is globally attractive in C0+ .

4.5. Simulations

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

22

CONCLUSION

Main contributions

The main contributions of this thesis are as follows:

1. Established sufficient conditions in terms of M-matrix for finite-time stability (Theorem

2.2.1) and power-rate synchronization (Theorem 2.3.1) of Hopfiled neural networks with

time-varying connection weights and heterogeneous proportional delays.

2. Derived conditions and proved the global dissipativity both in the case of regular selffeedback terms (Theorem 3.2.1) and singular self-feedback terms (Theorem 3.2.3) for nonautonomous neural networks with multiple proportional delays.

3. Established a new generalized exponential estimation for a type of Halanay inequalities

with proportional delay (Corollary 3.2.5).

4. Proved the global existence, uniform permanence and dissipativity in C0+ of positive solutions to a delayed Nicholson model with nonlinear density-dependent mortality rate

(Theorems 4.2.1, 4.2.2, 4.2.3, Corollary 4.2.4).

5. Established the existence, uniqueness and global attractivity of a positive periodic solution

to a delayed Nicholson model (Theorem 4.3.1). An application concerning the existence

and attractivity of a unique positive equilibrium of Nicholson models with constant coefficients has also been obtained (Theorem 4.4.2).

Further topics

• Investigating the global dissipativity and synchronization of differential equations modeling

reaction-diffusion neural networks with time-varying connection weights and proportional

delays.

• The existence and asymptotic behavior of periodic, almost periodic positive solution to

delayed Nicholson models with fractional density-dependent mortality term of the form

a(t)N

D(t, N) =

.

N + b(t)

23

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