Establishing the existence, uniqueness and global attractivity of a positive periodic solution of a Nicholson model with nonlinear density-dependent mortality rate... Finite-time stabili
Trang 1MINISTRY OF EDUCATION AND TRAININGHANOI 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
Trang 2This 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 ofEducation, 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
Trang 31 Motivation
Time delays are widely used in modeling practical modelsin control engineering, biologyand biological models, physical and chemical processes or artificial neural networks The pres-ence 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 itscomplete solution
A popular approach in stability analysis for time-delay systems is the use of the Krasovskii functional (LKF) method to derive sufficient conditions in terms of linear matrixinequalities (LMIs) However, it should be noted that finding effective LKF candidates fortime-delay systems is often connected with serious mathematical difficulties especially whendealing with nonlinear non-autonomous systems with bounded or unbounded time-varying de-lay In addition, extending the developed methodologies and existing results in the literature tononlinear time-delay systems proves to be a significant issue This research topic, however, hasnot been fully investigated, which gives much room for further development in particular fornonautonomous nonlinear systems with delays in the area of population dynamics and networkcontrol This motivates us for the present study in this thesis
pro-3 Establishing the existence, uniqueness and global attractivity of a positive periodic solution
of a Nicholson model with nonlinear density-dependent mortality rate
Trang 43 Objectives
3.1 Finite-time stability of non-autonomous neural networks with neous proportional delays
heteroge-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, patternrecognition, associative memory, parallel computing, solving optimization problems ect Inmost of the practical applications, it is of prime importance to ensure that the designed neuralnetworks be stable On the other hand, time delays unavoidably exist in most applicationnetworks 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 stabilityanalysis, control and estimation for delayed neural networks during the past decade
It is well-known that, in practical implementation of neural networks, time delays maynot 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 byintroducing continuously proportional delay over a certain duration of time Proportional delay
is one of time-varying (monotonically increasing) and unbounded delays which is different frommost other types of delay such as time-varying bounded delays, bounded and/or unboundeddistributed 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 withthe dynamic behavior of neural networks with proportional delays is an interesting problemwhich is also much more complicated
In Chapter 2 we consider the problem of finite-time stability of non-autonomous neuralnetworks with heterogeneous proportional delays described by the following system
Trang 53.2 Global dissipativity of non-autonomous neural networks with multiple portional delays
pro-Dissipativity of dynamical systems, first introduced in the earlier of 1970s, has a ful physical concept and is an important characteristic of many mathematical models of physicalprocesses In Chapter 3, we consider the problem of global dissipativity of the following neuralnetworks model
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 realworld 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 Australiansheep-blowfly In the biology interpretation of equation (3), N(t) is the population size attime t, α is the per capita daily adult mortality rate, β is the maximum per capita daily eggproduction rate, γ1 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 typicallyreferred to the Nicholson’s blowflies equation
In the past few years, the qualitative theory for Nicholson model and its variants has beenextensively 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
Trang 6density-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 cell chronic lymphocytic leukemia dynamics are suitably described by Nicholson-type delaydifferential equations with nonlinear density-dependent mortality rate of the form
B-N′(t) = −D(N(t)) + βN(t − τ )e−γN(t−τ ), (4)where the function D(N) might have one of the forms D(N) = a−be−N (type-I) or D(N) = b+NaN(type-II) with positive constants a and b A natural extension of (4) to the case of variablecoefficients 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)), (5)where D(t, N) = a(t) − b(t)e−N or D(t, N) = b(t)+Na(t)N 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 amaturation delay τ (t) and gets its maximum γ(t)eβ(t) at rate γ(t)1
Recently, Nicholson-type models with nonlinear density-dependent mortality terms haveattracted considerable research attention In Chapter 4 of this thesis, we study the problem ofexistence and global attractivity of positive periodic solution of the following Nicholson model
inequali-4 Obtained results
The thesis achieves the following main results:
1 Established conditions in terms of M-matrices for finite stability and power-rate nization of Hopfiled neural networks with time-varying coefficients and heterogeneous pro-portional delays
synchro-2 Proved the global dissipativity of a class of neural networks with multi-proportional delaysfor both uniformly positive and singular self-feedback coefficients
Trang 73 Derived conditions and proved the existence, uniqueness and global attractivity of a itive periodic solution to a Nicholson model with nonlinear density-dependent mortalityterm.
pos-The results presented in this thesis are based on three papers published on ISI indexedinternational journals
5 Thesis organization
Except the Introduction, Conclusion, List of Publications, and List of References, theremaining of the thesis is devided into four chapters Chapter 1 presents some preliminaryresults concerning finite-time stability, dissipativity of certain classes of time-delay systems andand 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-varyingconnection weights and heterogeneous proportional delays The global dissipativity of non-autonomous 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 aNicholson model with nonlinear density-dependent mortality term is studied in Chapter 4
Trang 8This 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, theproblem (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 punov) if for any t0 ∈ R+, ǫ > 0, there esists a δ = δ(t0, ǫ) > 0 such that for any solutionx(t, φ) of (1.1), if kφkC < δ then kx(t, φ)k < ǫ for all t ≥ t0 The solution x = 0 is uniformlystable if the aforementioned δ is independent of t0
Lya-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 kφkC < δa implies kx(t, φ)k < η for all t ≥ t0+ T (δa, η) x = 0 is globally uniformlyasymptotically 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+ arecontinuous non-decreasing functions, u(0) = 0, v(0) = 0 and u(s) > 0, v(s) > 0 for s > 0 Ifthere exists a continuous positive definite functional V :R× C → R+,
u(kφ(0)k) ≤ V (t, φ) ≤ v(kφkC), ∀φ ∈ C, (1.2)
Trang 9such that the derivative of V (t, φ) along trajectories of (1.1) is negative definite, that is,
Then, the trivial solution x = 0 of (1.1) is uniformly stable Moreover, if w(s) > 0 for s > 0and 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 initialconditions, its state does not exceed a certain threshold during a specified time interval Moreprecisely, 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 abovedefinition, it is required that X0 ⊂ Xt0 However, in general, it is not required that X0 isincluded in Xt for t > t0 In addition, the sets X0 and Xt are typically given in the form ofellipsoids ER(ρ) = {x⊤Rx < ρ : x ∈ Rn}, where R ∈ Sn
+ is a symmetric positive definitematrices 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 ]
Trang 101.3.2 Finite-time stability of linear systems with mixed time-varying delays
Consider the following linear nonautonomous system with time-varying delays
x′(t) = Ax(t) + Dx(t − τ (t)) + G
Z t t−κ(t)x(s)ds, t ≥ 0,x(t) = φ(t), t∈ [−h, 0],
(1.5)
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 arebounds 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 stablew.r.t (r1, r2, T) if for any φ ∈ C([−h, 0],Rn), kφk∞ ≤ r1, one has kx(t, φ)k∞ < 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 withrespect to (r1, r2, T) if there exist positive scalars α, ρi, i = 1, 2, 3, 4, and symmetric positivedefinite matrices P, Q, R ∈Rn×n satisfying the following conditions
1.4 Dissipativity of functional differential equations
In this section we introduce some preliminary results involving the dissipativity of certainclasses of time-delay systems First, we consider the following system
x′(t) = F (t, x(t), x(t − τ1(t)), , x(t − τm(t))), t∈ [0, ∞),x(t) = φ(t), t∈ [−τ, 0],
(1.7)
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
Trang 11is continuous and satisfies
2hu, F (t, u, ψ1(.), , ψm(.))i ≤ γ(t) + α(t)kuk2+
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) suchthat 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) + α0u(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) ≤ γ∗
−(α0+ β0) + kθk∞e
−λ(t−t 0 ), t ≥ t0, (1.10)where γ∗= supt≥t0γ(t) and kθk∞= supt≤t0|θ(t)|
Remark 1.4.1 Let α0 = supt≥0α(t) and β0 = supt≥0β(t) If α0+ β0 < 0 then, by Lemma1.4.1, system (1.7) is globally dissipative Specifically, for a given ǫ > 0, there exists a t∗ =
t∗(kφk∞, ǫ) > 0 such that
kx(t)k2 < γ∗
−(α0+ β0)+ ǫ, t > t∗.Therefore, system (1.7) is globally dissipative with the absorbing set B = B
0,q
− γ∗
α0+β 0 + ǫ
.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
t−τ(t)≤s≤t
u(s) (t ≥ t0), u(t) = θ(t) (t ≤ t0), (1.11)where θ ∈ BC((−∞, t0],R+), α(t), β(t), γ(t) are continuous functions, β(t) ≥ 0, γ(t) ≥ 0 andthe delay τ (t) ≥ 0 satisfies t − τ (t) → ∞ as t → ∞ Assume that
α(t) + β(t) ≤ −σ < 0, t ≥ t0, (1.12)
Trang 12for some positive scalar σ Then, u(t) ≤ γ∗
σ + kθk∞, t ∈ [t0,∞) Additionally, if thereexists a 0 < δ < 1 such that δα(t) + β(t) < 0, ∀t ≥ t0, then for any ǫ > 0, there exists a
t∗ = t∗(kθk∞, ǫ) > t0 by which
u(t) ≤ γ∗
σ + ǫ, 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,x(t) =ϕ(t), t ∈ [qtb 0, t0],
(1.13)
where q is a constant, 0 < q < 1, and the function g satisfies
2hu, g(u, v)i ≤ γ + αkuk2+ βkvk2 (1.14)with α, β, γ are given constants For the notational simplicity, we assume that t0 = 1 By thechange of variable y(t) = x(et), system(1.14) can be stransformed to the following system with
y′(t) = f (t, y(t), y(t − τ )), t > 0,y(t) = ϕ(t), t ∈ [−τ, 0],
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∗(kϕk∞, ǫ) > 0 such that
ky(t)k2<− γ
α+ β + ǫ, ∀t > t∗.Therefore, system (1.15) is globally dissipative and B = B
0,q
−α+βγ + ǫ
is an absorbing setfor a given ǫ > 0
1.5 Auxiliary results
This section presents some technical lemmas and auxiliary results which will be used inthe next chapters
Trang 13Chapter 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 works with heterogeneous proportional delays By introducing a novel constructive approach,
net-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 aresult, we also obtain conditions for the power-rate global stability of the system
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-varyingconnection weights, Ii(t) are external inputs, qij ∈ (0, 1], i, j ∈ [n], are possibly heterogeneousproportional 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)
x− y ≤ l
+ i1, l−i2≤ gi(x) − gi(y)
x− y ≤ l
+ i2, ∀x, y ∈R, x6= y,where l−ik, l+ik, 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