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: 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 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 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 Investigating the problem of finite-time stability of non-autonomous neural networks with heterogeneous proportional delays Analizing the global dissipativity of non-autonomous neural networks with multiple proportional delays Establishing the existence, uniqueness and global attractivity of a positive periodic solution of a Nicholson model with nonlinear density-dependent mortality rate 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 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)) = − (t)xi (t) + j=1 (1) n cij (t)gj (xj (qij t)) + Ii (t), i ∈ [n], t > + 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 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 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 > 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, γ is the size at which the population reproduces at its maximum rate and τ ≥ 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 γ(t) Recently, Nicholson-type models with nonlinear density-dependent mortality terms have attracted considerable research attention In Chapter 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 Obtained results The thesis achieves the following main results: 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 Proved the global dissipativity of a class of neural networks with multi-proportional delays for both uniformly positive and singular self-feedback coefficients 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 Thesis organization Except the Introduction, Conclusion, List of Publications, and List of References, the remaining of the thesis is devided into four chapters Chapter 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 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 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 Chapter 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) = 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 = 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) > 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 = is uniformly stable if the aforementioned δ is independent of t0 Definition 1.2.2 The solution x = of (1.1) is said to be uniformly asymptotically stable if it is uniformly stable and there exists a δa > such that for any η > there exists a T (δa , η) such that φ C < δa implies x(t, φ) < η for all t ≥ t0 + T (δa , η) x = 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) = and u(s) > 0, v(s) > for s > If there exists a continuous positive definite functional V : R × C → R+ , u( φ(0) ) ≤ V (t, φ) ≤ v( φ 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 = of (1.1) is uniformly stable Moreover, if w(s) > for s > and lims→∞ u(s) = ∞ then the solution x = 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 ] 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 ≤ τ1 ≤ τ (t) ≤ τ2 , τ ′ (t) ≤ µ ≤ 1, ≤ 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) 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⊤ PA + ⊤ ατ1 e⊤ Qe v` ⊤ A⊤ P e1 − αe⊤ a Π2 = κ2 e1 Re1 − P e1 , Π1 = e1 Qe1 − (1 − µ)e ⊤ 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 ≤ τk (t) ≤ τ for all t ∈ [0, ∞), k ∈ [m], where τ > is a constant The function F : [0, ∞) × Rn × (C([−τ, ∞), Rn ))m → Rn Chapter 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 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, (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 > 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 (t) ≥ > 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 < 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 ≥ (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 ≺ • Compute n η = (−Mξ)l = ξj bij Fj + cij Gj ξi − i∈[n] (2.3) j=1 • The power convergence rate σmax can be iteratively computed as n max σ > s.t Hi (σ) = σξi + σ ln Gj cij ξj e qij − − 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 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, (t) ≥ > 0; and (ii) the self-feedback coefficients can be singular, that is, (t) > and inf t≥0 (t) = 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, (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) ≥ 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 ≥ (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 ≥ 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 > : 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 , i ∈ [n], such that (t) ≥ , ∀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 ≻ and γ = maxi∈[n] ; (2) System (3.1) is globally dissipative Corollary 3.2.2 Under assumptions (A3.1)-(A3.3), if n n > 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 = i∈[n] − 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) > and positive scalars a ˆi , i ∈ [n], such that t (t) ≥ a ˆi ϕ(t), sup t≥0 t ϕ(s)ds < ∞, lim t→∞ qij t ϕ(s)ds = ∞ (3.7) (A3.5): There are constants ˆbij ≥ 0, cˆij ≥ and Iˆi ≥ such that |bij (t)| ˆ |cij (t)| |Ii (t)| ≤ bij , ≤ cˆij , ≤ Iˆi , ∀i, j ∈ [n], t ≥ (t) (t) (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+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)|)}, = and Hη ≻ 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 ∞ ∞ ≤ x(0) − x∗ (0) η+ −λ0 ∞e t ϕ(s)ds , t≥0 (3.9) = and Hη ≻ 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 > 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) > and a t0 > such that a(t) ≥ las (t), t ≥ t0 t sup t≥t0 and t as (θ)dθ < ∞, lim qj t t→∞ as (θ)dθ = ∞ 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) − , e−λ − µ0 − µ0 t as (ζ)dζ , t ≥ (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 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 > 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) a(t) (C4.3) b− − a+ > 0, ̺ (C4.4) p + k=1 βk max a− − lim inf t→∞ = σ, − p β+ k=1 γ − k e − 1−γk r∗ , γ −r e2 e k ∗ < ̺b− , b+ b(t) a(t) > b− a+ σ > e > 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 > 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− − 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 = and assume that t b(t) sup = δ ∈ [0, 1), t≥0 a(t) a(s)ds → ∞, t → ∞ Then, ϕ(0)− N(t, t0 , ϕ) ≤ ln e t t0 a(s)ds − +δ 1−e t t0 a(s)ds → ln(δ) < 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− > and lim inf t→∞ b(t) ≥ eℓm > a(t) (4.5) Then, for any ϕ ∈ C0+ , lim inf N(t, t0 , ϕ) ≥ ℓm > 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 > 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, lim sup t→∞ a(t) p k=1 βk (t) = σ, γk (t) t ∈ [0, +∞), 1− σ > 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− > Assume that a − e p βk+ − ̺ > 0, (4.8a) b− − a+ > (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 − γk− r∗ , e2 eγk− r∗ max γk− Note that, by (4.8b), > and hence r∗ > In addition, since the condition r∗ < is not imposed, − can be positive, negative or zero For ≤ k ≤ p that − γk− r∗ ≤ 0, νk = 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 − 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 > 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 > are known coefficients, > The nonlinear density-dependent mortality term is given by D(N) = a − be−N , a > 0, b > 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 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− e p βk k=1 γk (4.15) Theorem 4.4.2 Assume that q βk k=1 where νˆk = max νˆk + eγk < a < b, (4.16) 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: 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 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 Established a new generalized exponential estimation for a type of Halanay inequalities with proportional delay (Corollary 3.2.5) 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) 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 This page is intentionally left blank 24 LIST OF PUBLICATIONS Le Van Hien, Doan Thai Son, Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays, Applied Mathematics and Compution 251 (2015) 14–23 (SCIE) Le Van Hien, Doan Thai Son, Hieu Trinh, On global dissipativity of nonautonomous neural networks with multiple proportional delays, IEEE Transactions on Neural Network and Learning Systems 29 (2018) 225–231 (SCI) Doan Thai Son, Le Van Hien, Trinh Tuan Anh, Global attractivity of positive periodic solution of a delayed Nicholson model with nonlinear density-dependent mortality term, Electronic Journal of Qualitative Theory of Differential Equations, No (2019) 1-21 (SCIE) The results of this thesis were reported at: • Seminar Differential and integral equations, Laboratory of Mathematical Analysis, Department of Mathematics, Hanoi National University of Education • Workshop of PhD students, Department of Mathematics, Hanoi National University of Education • Seminar Optimization and Control, Hanoi Institute of Mathematics, VAST ... density-dependent mortality terms have attracted considerable research attention In Chapter of this thesis, we study the problem of existence and global attractivity of positive periodic solution... Chapter 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... 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