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.

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



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

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)) +

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


(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

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 ,



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

+ 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


t∗ .
−(α0 + β0 )

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)



(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∗ .


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


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

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


(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],

+


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 ⊂

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



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.

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

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

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))


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)



̺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+

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



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+ ,


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.


≤ 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+
̺


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 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→∞


< 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).