MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION ——————–o0o——————— LE DAO HAI AN STABILITY OF NONLINEAR TIME-DELAY SYSTEMS AND THEIR APPLICATIONS Speciality: Differential and Integral Equations Speciality code: 46 01 03 SUMMARY OF DOCTORAL THESIS IN MATHEMATICS HANOI-2019 This dissertation has been written on the basis of my research work carried at: Hanoi National University of Education Supervisor: Assoc Prof Le Van Hien Dr Tran Thi Loan Referee 1: Professor Nguyen Minh Tri, Institute of Mathematics, Vietnam Academy of Science and Technology Referee 2: Professor Cung The Anh, Hanoi National University of Education Referee 3: Associate Professor Nguyen Xuan Thao, Hanoi University of Science and Technology The thesis will be presented to the examining committee at Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Vietnam At the time of , 2020 This dissertation is publicly available at: - HNUE Library Information Centre - The National Library of Vietnam INTRODUCTION Motivation Time delays are widely used in modeling practical modelsin control engineering, biology and biological models, physical and chemical processes or artificial neural networks The presence of time-delay is often a source of poor performance, oscillation or instability Therefore, the stability of time-delay systems has been extensively studied during the past decades It is still one of the most burning problems in recent years due to the lack or the absence of its complete solution A popular approach in stability analysis for time-delay systems is the use of the Lyapunov-Krasovskii functional (LKF) method to derive sufficient conditions in terms of linear matrix inequalities (LMIs) However, it should be noted that finding effective LKF candidates for time-delay systems is often connected with serious mathematical difficulties especially when dealing with nonlinear non-autonomous systems with bounded or unbounded time-varying delay In addition, extending the developed methodologies and existing results in the literature to nonlinear time-delay systems proves to be a significant issue This research topic, however, has not been fully investigated, which gives much room for further development in particular for nonautonomous nonlinear systems with delays in the area of population dynamics and network control This motivates us for the present study in this thesis Research aims This thesis is concerned with the stability of some classes of nonlinear time-delay systems in neural networks Specifically, we consider the following problems Investigating the problem of stability of non-autonomous neural networks with heterogeneous time-varying delays in the effect of destablizing impulses Stabilizing Hopfiled neural networks with proposition delays subject to stabilizing and destablizing impulsive effects simultaneously Investigating the problem of exponential stability of positive equilibrium of inertial neural networks with multiple time-varying delays Deriving conditions for the problem of exponential stability of a unique equilibrium of positive BAM neural networks with multiple time-varying delays and nonlinear selfexcitation rates Objectives 3.1 Global exponential stability analysis of a class of non-autonomous neural networks with heterogeneous delays and time-varying impulses The states of various dynamical networks in the fields of artificial systems such as mechanics, electronic and telecommunications networks, often suffer from instantaneous disturbances and undergo abrupt changes at certain instants These may arise from switching phenomena or frequency changes, and thus, they exhibit impulsive effects With the effect of impulses, stability of the networks may be destroyed Therefore, delays and impulses heavily affect the dynamical behaviors of the networks, and thus it is necessary to study both effects of time-delay and impulses on the stability of neural networks Up to now, considerable effort of researchers has been devoted to investigating stability and asymptotic behavior of neural networks with impulses However, the aforementioned works have been devoted to neural networks with constant coefficients As discussed in the many exsting literatures, non-autonomous phenomena often occur in realistic systems, for instance, when considering a long-term dynamical behavior of the system, the parameters of the system usually change along with time Also, the problem of stability analysis for non-autonomous systems usually requires specific and quite different tools from the autonomous ones (systems with constant coefficients) There are only few papers concerning stability of non-autonomous neural networks with heterogeneous time-varying delays and impulsive effects In Chapter we investigate the exponential stability of a class of non-autonomous neural networks with heterogeneous delays and time-varying impulses n x′i (t) = −di (t)xi (t) + aij (t)fj (xj (t)) j=1 n + j=1 ∆xi (tk ) bij (t)gj (xj (t − τij (t))) + Ii (t), − − xi (t+ k ) − xi (tk ) = −σik xi (tk ), t > 0, t = tk , (1) k ∈ N Based on the comparison principle, an explicit criterion is derived in terms of inequalities for M-matrix ensuring the global exponential stability of the model under destabilizing impulsive effects The obtained results are shown improve some recent existing results Finally, numerical examples are given to demonstrate the effectiveness the proposed conditions 3.2 Exponential stability of impulsive neural networks with proportional delay in the presence of periodic distribution impulses Typically, a model of neural networks is composed of layers with a large number of cells and connections This fact reveals that NNs usually have a spatial nature due to the number of parallel pathways, axon sizes and lengths Thus, time delays encountered in the practical implementation of NNs are usually time-varying Proportional delays form a particular type of unbounded time-varying delays, which are widely used in modeling various models in the field of networking It is realized that proportional delay provides most well-known quality of service (QoS) models because of its controllable and predictable characteristics Specifically, when a network with proportional delays is utilized to represent an applied model, dynamics of the system at time t is determined by its states x(t) and x(qt), where < q < is a constant representing the ratio of time between current states and historical states Thus, the network’s running time can be controlled by the proportional factor q Recently, the problem of stability of various neural network models with proportional delays has attracted considerably increasing research attention and, consequently, a large number of interesting results have been reported in the literature On the other hand, impulsive dynamical systems (IDSs), in general, and impulsive neural networks with delays (IDNNs) have received considerable research attention in recent years According to their strength, impulsive effects can be classified into two types named as stabilizing impulses (SI) and destabilizing impulses (DI) An impulsive sequence is said to be destabilizing if its effect can suppress the stability of dynamical systems while SI can enhance the stability of dynamical systems In most of the existing works concerning stability of impulsive systems, SI and DI are considered separately In the second part of Chapter we study the problem of exponential stability of the following neural networks model n n x′i (t) = −di xi (t) + bij gj (xj (qt)) + ui (t), t = tk , aij fj (xj (t)) + j=1 j=1 (2) − − ∆xi (tk ) = xi (t+ k ) − xi (tk ) = −σik xi (tk ), Both stabilizing and destabilizing impulsive effects are introduced in the model simultaneously Based on the comparison principle, a unified stability criterion is first derived Then, on the basis of the derived stability conditions, the problem of designing a local state feedback control law with bounded controller gains is addressed 3.3 Positive solutions and global exponential stability of positive equilibrium of inertial neural networks with multiple time-varying delays Conventional neural networks are typically described by first-order differential equations with or without delays Recently, many authors focused on dynamics behaviors of networks models called inertial neural networks (INNs) In state-space models, INNs are described by systems of second-order differential equations, where the first-order derivative terms are referred to as inertial terms On one hand, there exist strong biological and engineering backgrounds for the introduction of inertial terms in neural systems, in particular for those containing an inductance On the other hand, the presence of inertial terms makes it more difficult and challenging to analyze dynamic behaviors of INNs Therefore, the investigation of INNs for both theoretical and practical reasons has attracted considerable attention in the past few years Positive systems, in general, and positive neural networks (PNNs) can be used to describe dynamics of various practical models where the associated state variables are subject to positivity constraints according to the nature of phenomena For instance, when ANNs are designed for the purpose of identification or control of positive systems, state vectors of the designed NNs are expected to inherit the positivity of the systems Thus, as an essential issue in applications, it is of interest and importance to study the stability problem of PNNs with delays To date, only a few results concerning stability of PNNs with delays have been reported in the literature In Chapter we consider a class of INNs with delays described by the following second-order differential system d2 xi (t) dxi (t) = − − bi xi (t) + dt dt n cij fj (xj (t)) j=1 n + j=1 dij fj (xj (t − τj (t))) + Ii , t ≥ 0, i ∈ [n] (3) By utilizing the comparison principle via differential inequalities, we first derive conditions on damping coefficients, self excitation coefficients and connection weights under which all state trajectories of the system starting from an admissible set of initial conditions are always nonnegative Then, based on the approach of using homeomorphisms in nonlinear analysis, sufficient conditions for the existence, uniqueness and global exponential stability of a positive equilibrium of the system are derived in the form of LP problems with M-matrices, which can be effectively solved by various convex optimization algorithms 2.4 Exponential stability of positive neural networks in bidirectional associative memory model with multiple time-varying delays and nonlinear selfexcitation rates In 1987 Kosko introduced and studied stability and encoding properties of a class of two-layer nonlinear feedback neural networks Accordingly, the bidirectionality, referred to forward and backward information flows, was introduced in neural nets to produce two-way associative search for stored paired-data associations This model was later called bidirectional associative memory (BAM) neural network Roughly speaking, a BAM neural network is composed of a number of neurons arranged into two layers namely X-layer and Y-layer The neurons in each layer are connected in the way that neurons in one layer are completely integrated to neurons in the other layer, whereas there is no interconnection among neurons in the same layer This structure performs a two-way associative search for stored bipolar vector pairs and generalizes the single-layer autoassociative Hebbian correlation to a two-layer pattern-matched heteroassociative circuits Thus, BAM model possesses many application prospects in the areas of pattern recognition, signal and image processing Typical applications of neural networks, for example, in optimization, control, or signal processing, require that neural networks are designed to admit only one equilibrium point which is globally asymptotically stable thereby avoiding the risks of having spurious equilibrium point and being trapped in local minima Thus, it is relevant and very important to study the stability problem of a unique equilibrium of a dynamic neural network In Chapter we investigate the problem of exponential stability of a unique equilibrium of positive BAM neural networks with multiple time-varying delays and nonlinear self-excitation rates m m x′i (t) = −αi ϕi (xi (t)) + yj′ (t) = −βj ψj (yj (t)) + aij fj (yj (t)) + j=1 n j=1 n cji gi (xi (t)) + i=1 i=1 bij fj (yj (t − σj (t))) + Ii , (4) dji gi (xi (t − τi (t))) + Jj (5) A systematic approach involving extended comparison techniques via differential inequalities is presented Combining with the use of Brouwer’s fixed point theorem and Mmatrix theory, tractable LP-based conditions are derived to ensure the existence and global exponential stability of a unique positive equilibrium of the system An extension to the case of BAM neural networks with proportional delays is also presented Main contributions Established conditions in terms of M-matrices forexponential stability of Hopfiled neural networks with time-varying coefficients and destablizing impulses A unified stability criterion is first derived for the stability problem and application to designing a local state feedback control law with bounded controller gains for Hopfiled neural networks with proposition delays and periodic distribution impulses Based on the approach of using homeomorphisms in nonlinear analysis, sufficient conditions for the existence, uniqueness and global exponential stability of a positive equilibrium of inertial neural networks are derived in the form of LP problems with M-matrices Based on an extended comparison techniques via differential inequalities combining with the use of Brouwer’s fixed point theorem and M-matrix theory, tractable LPbased conditions are derived to ensure the existence and global exponential stability of a unique positive equilibrium of BAM neural networks with heterogeneous delays An extension to the case of BAM neural networks with proportional delays is also obtained The results presented in this thesis are based on papers published on ISI/Scopus international journals 5 Thesis outline Except the Introduction, Conclusion, List of Publications, and List of References, the remaining of the thesis is devided into four chapters Chapter presents some preliminary results Chapter investigates the problem of exponential stability of Hopfield neuron networks with time-varying connection weights and heterogeneous delays with impulses Positive solutions and the existence of a unique equilibrium of inertial neural networks with time-varying delays is studied and presented in Chapter Finally, the existence, uniqueness and global exponential stability of a positive equilibrium to BAM neural networks is studied in Chapter Chapter PREMILINARIES In this chapter, we briefly introduce state space model of biology neural networks and present some auxiliary results in matrix analysis, differential equations, stability theory in the sense of Lyapunov, impulsive differential systems 1.1 Dynamic equation of biology neural networks This section briefly introduces the history and general state space model of biology neural networks 1.2 Mathematical fundamentals 1.2.1 Time-delay systems and the Lyapunov stability theory Fundamental results on the theory of functional differential equations are presented 1.2.2 Functional impulsive differential equations Basic theory of impulsive differential equations is presented 1.2.3 Positive systems This section briefly presents the positive linear system and introduces a result of the stability of the nonlinear positive system 1.2.4 Auxiliary results This section presents some technical lemmas and auxiliary results which will be used in the next chapters Chapter EXPONENTIAL STABILITY OF NON-AUTONOMOUS NEURAL NETWORKS WITH HETEROGENEOUS TIME-VARYING DELAYS AND DESTABILIZING IMPULSES In this chapter we study the problem of exponential stability of non-autonomous neural networks with heterogeneous delays Based on the comparison principle, an explicit criterion is derived in terms of inequalities for M-matrix ensuring the global exponential stability of the model under destabilizing impulsive effects The obtained results are shown improve some recent existing results Finally, numerical examples are given to demonstrate the effectiveness the proposed conditions 2.1 A motivation example 2.2 Nonautonomous neural networks with heterogeneous time-varying delays and destablizing impulses Consider a class of non-autonomous impulsive neural networks with heterogeneous time-varying delays of the following form n x′i (t) = −di (t)xi (t) + aij (t)fj (xj (t)) j=1 n + j=1 bij (t)gj (xj (t − τij (t))) + Ii (t), t = tk , (2.1) − − + ∆xi (tk ) = xi (t+ k ) − xi (tk ) = −σik xi (tk ), t = tk , k ∈ Z , xi (t) = φi (t), t ∈ [−τ, 0], i ∈ [n] (A2.1) The matrices D(t) = diag(d1 (t), d2 (t), , dn (t)), A(t) = (aij (t)) and B(t) = (bij (t)) are continuous on each interval (tk , tk+1 ), k ≥ 0, and there exist scalars dˆi , a+ , b+ such ij ij that di (t) ≥ dˆi > 0, |aij (t)| ≤ a+ ij , |bij (t)| ≤ b+ ij , ∀t ≥ 0, i, j ∈ [n] (A2.2) The neural activation functions fi , gi , i ∈ [n], satisfy − ≤ ljf fj (a) − fj (b) + ≤ ljf , a−b − ljg ≤ gj (a) − gj (b) + ≤ ljg , a−b ∀a, b ∈ R, a = b, − + − + where ljf , ljf , ljg and ljg are known constants (A2.3) There exists positive sequence (γk )k∈Z+ such that − γk ≤ σik ≤ + γk , ∀i ∈ [n], k ∈ N + ˇ = eγ0 τij b+ where B ij As a special case, when σik = 0, i ∈ [n], k ≥ 1, and I(t) = 0, system (2.1) becomes the following nonlinear non-autonomous system without impulses n n x′i (t) = −di (t)xi (t) + xi (t) = φi (t), aij (t)fj (xj (t)) + j=1 j=1 bij (t)gj (xj (t − τij (t))), t ≥ 0, (2.8) t ∈ [−τ, 0] From the proof of Theorem 2.3.1 we get the following result Corollary 2.3.4 Under Assumptions (A2.1), (A2.2), assume that there exists a vector υ ∈ Rn+ such that ˆ f + B + Lg − D ˆ υ ≺ 0, AL (2.9) where B + = (b+ ij ) Then system (2.8) is globally exponentially stable Furthermore, every solution x(t, φ) of (2.8) satisfies x(t, φ) ∞ ≤ Cυ φ −η0 t , ∞e t ≥ 0, υi where Cυ = max1≤i≤n ( min1≤j≤n υj ), < η0 ≤ min1≤i≤n ηi and ηi is the unique positive solution of the scalar equation −dˆi + υi n + f + ηi τij g a+ lj υj + ηi = ij lj + bij e (2.10) j=1 2.4 Stabilization of Hopfield neural networks with proportional delays and periodic distribution impulsive effects Consider a neural network model with a proportional delay described as n n x′i (t) = −di xi (t) + bij gj (xj (qt)) + ui (t), t = tk , aij fj (xj (t)) + j=1 j=1 − − ∆xi (tk ) = xi (t+ k ) − xi (tk ) = −σik xi (tk ), (2.11) xi (t) = x0i , t ∈ [qt0 , t0 ], i ∈ [n], where n is the number of neurons, xi (t) and ui (t) are the state variable and control input of ith neuron at time t, respectively The factor q ∈ (0, 1) is a constant involving history time More specifically, in the interpretation of model (2.11), dynamics of ith neuron at time t is determined by the current states xj (t), j ∈ [n], and the states xj (qt) at history time qt which is proportional to current time t with a constant rate q In this meaning, the constant q is referred to as proportional delay Since qt = t − τ (t), where τ (t) = (1 − q)t → ∞ as t → ∞, proportional delays form a class of unbounded time-varying delays The strengths of stabilizing impulses and destablizing impulses are assumed to take values in finite sets Is = {ρ1 , ρ2 , , ρM } and Iu = {µ1 , µ2 , , µN }, where < ρi < for 11 i ∈ [M] and µj > for j ∈ [N] In addition, we denote as tsik and tujk the impulsive instances of stabilizing impulses with strength ρi and the impulsive instances of destabilizing impulses with strength µj , respectively That means, for any i ∈ [M] and j ∈ [N], tsik = tk if γk = ρi and tujk = tk if γk = µj (A2.5) There exist positive numbers τis , τju , and integers qi ∈ N0 , rj ∈ N0 , i ∈ [M], j ∈ [N], satisfying the following conditions for any t > s ≥ t0 t−s t−s − qi ≤ Nρi (t, s) ≤ s + qi , s τi τi t−s t−s − rj ≤ Nµj (t, s) ≤ u + rj , u τj τj (2.12) where Nρi (t, s) and Nµj (t, s) present the frequencies of impulsive strengths ρi and µj on interval (s, t), respectively We will design a local state feedback control law (LSFCL) of the form ui (t) = −ki xi (t), i ∈ [n], (2.13) to stabilize system (2.11), where ki , i ∈ [n], are controller gains Due to practical config- urations of the inputs, we assume the controller gains ki , i ∈ [n], are confined in intervals [kil , kiu ], where kil , kiu , i ∈ [n], are known constants Under the LSFCL (2.13), the closed-loop system of (2.11) can be written as  x′ (t) = −D x(t) + Af (x(t)) + Bg(x(qt)), t = t , c k x(t+ ) = Jk x(t− ), k ∈ N, k (2.14) k where Dc = diag(di + ki ), A = (aij ), B = (bij ), f (x(t)) = (fj (xj (t))), g(x(qt)) = (gj (xj (qt))) and Jk = diag(1 − σik ) 2.4.1 Stability of the closed-loop system (2.14) Definition 2.4.1 System (2.14) is said to be generalized globally exponentially stable (GGES) if there exist a positive scalar κ and an increasing function σ(t) > 0, σ(t) → ∞ as t → ∞, such that any solution x(t) = x(t, x0 ) of (2.14) satisfies the following estimation x(t) ≤ κ x0 e−σ(t) , t ≥ t0 We denote the matrices |A| = (|aij |), |B| = (|bij |) and ⊤ M = −2Dc + sym(|A|Lf ) + θ−1 |B|Lg L⊤ g |B| , θ > 0, where Lf = diag(Lfi ) and Lg = diag(Lgi ) Theorem 2.4.1 Let assumptions (A2.2), (A2.3) and (A2.5) hold Assume that there exist positive scalars α and θ satisfying the following conditions M + αEn < 0, 12 (2.15a) (2.15b) α > pθ, M ln(ρi ) + τis i=1 where p = M i=1 2rj −2qi N j=1 µj ρi N j=1 ln(µj ) = 0, τju (2.15c) Then, system (2.14) is GGES More precisely, there exists a constant σ > such that any solution x(t) = x(t, x0 ) of (2.14) satisfies √ p σ x0 e− ln(1+t) , t ≥ t0 x(t) ≤ (1 + qt0 )σ (2.16) By the Schur complement lemma, conditions (2.15a) and (2.15b) can be recast into the following linear matrix inequalities (LMIs) −2Dc + sym(|A|Lf ) + αEn |B|Lg Lg |B|⊤ < 0, (2.17a) pθ − α < (2.17b) −θEn As a special case of model (2.11), let us consider the following neural network model with alternatively impulsive effects  x(t) ˙ = −Dx(t) + Af (x(t)) + Bg(x(qt)), t = kTs , x(t+ ) = γk x(t− ), k (2.18) k where Ts > is a sampling time Assume that there exists a scalar γ∗ , < |γ∗ | = 1, such that γ2k+1 = γ∗ and γ2k+2 = γ∗−1 for any k ∈ N0 It is clear that t−s t−s − ≤ Nγ∗ (t, s) ≤ + 1, 2Ts 2Ts t−s t−s − ≤ Nγ −1 (t, s) ≤ + ∗ 2Ts 2Ts By Theorem 2.4.1, we have the following result Corollary 2.4.2 Under assumptions (A2.2), (A2.3) and (A2.5), system (2.18) is GGES if there exists a scalar θ > satisfying the following condition θ max γ∗2 , γ∗2 + m < 0, (2.19) ⊤ where m = λmax −2D + sym(|A|Lf ) + θ−1 |B|Lg L⊤ g |B| 2.4.2 Stabilization coditions Based on condition (2.17), the problem of designing a LSFCL (2.13) that makes the closed-loop system (2.14) GGES is presented in the following theorem Theorem 2.4.3 Under assumptions (A2.2), (A2.3) and (A2.5), assume that condition (2.15c) is satisfied Then, system (2.11) is exponentially stabilizable under LSFCL (2.13) if the following LMIs are feasible for scalar α > 0, θ > 0, and a diagonal matrix Z = diag(zi ) ∈ Rn×n 13 −2D + sym(|A|Lf ) + αEn + Z |B|Lg < 0, (2.20a) Z + 2diag(kil ) ≤ 0, (2.20b) α > pθ (2.20c) Lg |B|⊤ −θEn Z + 2diag(kiu ) ≥ 0, Moreover, the controller gain matrix is given by Kc = − Z 14 (2.21) Chapter POSITIVE SOLUTIONS AND EXPONENTIAL STABILITY OF INNERTIAL NEURAL NETWORKS WITH TIME-VARYING DELAYS In this chapter we consider a class of INNs with delays By utilizing the comparison principle via differential inequalities, we first derive conditions on damping coefficients, self excitation coefficients and connection weights under which all state trajectories of the system starting from an admissible set of initial conditions are always nonnegative Then, based on the approach of using homeomorphisms in nonlinear analysis, sufficient conditions for the existence, uniqueness and global exponential stability of a positive equilibrium of the system are derived in the form of LP problems with M-matrices, which can be effectively solved by various convex optimization algorithms 3.1 Model description Consider a class of INNs with delays described by the following second-order differential system d2 xi (t) dxi (t) = − − bi xi (t) + dt dt n cij fj (xj (t)) j=1 n + j=1 dij fj (xj (t − τj (t))) + Ii , t ≥ 0, i ∈ [n] (3.1) Denote x(t) = (x1 (t), x2 (t), , xn (t))⊤ as the state vector, A = diag{a1 , , an }, B = diag{b1 , , bn }, C = (cij )n×n , D = (dij )n×n and I = (I1 , I2 , , In )⊤ System (3.1) can be written in the following matrix-vector form x′′ (t) = −Ax′ (t) − Bx(t) + Cf (x(t)) + Df (xτ (t)) + I, (3.2) where f (x(t)) = (fj (xj (t))) and f (xτ (t)) = (fj (xj (t − τj (t)))) In regard to (3.2), each initial condition of (3.1) is defined by compatible functions φ = (φi ) and φˆ = (φˆi ) in C([−τ + , 0], Rn ) as xi (s) = φi (s), x′i (s) = φˆi (s), s ∈ [−τ + , 0], i ∈ [n] (3.3) By using the following state transformation dxi (t) ηi yi (t) = + ξi xi (t), i ∈ [n], (3.4) dt where ηi = and ξi , i ∈ [n], are constants, system (3.1) can be written in the following vector form  x′ (t) = −D x(t) + D y(t), η ξ y ′(t) = −Dα y(t) + Dβ x(t) + D−1 [Cf (x(t)) + Df (xτ (t)) + I] , η 15 (3.5) where y(t) = (y1 (t), y2 (t), , yn(t))⊤ , Dξ = diag{ξ1 , , ξn }, Dη = diag{η1 , , ηn }, Dα = diag{α1 , , αn }, Dβ = diag{β1 , , βn } and αi = − ξi , βi = ηi−1 (αi ξi − bi ), i ∈ [n] Assumption (A3): The activation functions fj (.), j ∈ [n], are continuous and there exist positive constants ljf , j ∈ [n], satisfying the following condition fj (a) − fj (b) ≤ ljf , a = b (3.6) a−b Proposition 3.1.1 Under assumption (A3), for any initial condition (3.3), there exists a 0≤ unique solution of system (3.1) defined on the interval [0, ∞), which is absolutely continuous in t Let x(t) be a solution of (3.1) If its trajectory is confined within the first orthant (i.e x(t) ∈ Rn+ for all t ≥ 0), then x(t) is said to be a positive solution For a given transformation (3.4), where ηi > and ξi > 0, i ∈ [n], we define the following set of admissible initial functions for system (3.1) AT = (φi ), (φˆi ) ∈ C([−τ + , 0], Rn) : φi (s) ≥ 0, ψi = ηi−1 (ξi φi (s) + φˆi (s)) ≥ 0, s ∈ [−τ + , 0], i ∈ [n] (3.7) Clearly, AT contains all nonnegative nondecreasing initial functions φi with φˆi (s) = φ′i (s) Definition 3.1.1 System (3.1) is said to be positive if for any initial condition that belongs to AT and any input vector I = (Ii ) ∈ Rn+ , the corresponding solution x(t) of (3.1) is positive Definition 3.1.2 Given an input vector I ∈ Rn+ A vector x∗ ∈ Rn+ is said to be a positive equilibrium point (EP) of system (3.1) if it satisfies the following algebraic system −Bx∗ + Cf (x∗ ) + Df (x∗ ) + I = (3.8) The following auxiliary result will be used to issue the existence of such positive EP for system (3.1) Lemma 3.1.2 Assume that Dη ≻ and Dξ ≻ Then, a vector x∗ ∈ Rn+ is a positive EP of system (3.1) if and only if (x∗ , y∗ ), where y∗ = Dη−1 Dξ x∗ , is a positive EP of system (3.5), that is,  −D x + D y = 0, η ∗ ξ ∗ Dη (−Dα y∗ + Dβ x∗ ) + Cf (x∗ ) + Df (x∗ ) + I = (3.9) 3.2 Positive solutions of INNs with delays Lemma 3.2.1 For given coefficients > 0, bi > 0, there exists a transformation (3.4) with ηi > and ξi > 0, i ∈ [n], such that Dα ≻ and Dβ ≻ if and only if the following condition holds a2i − 4bi > 0, i ∈ [n] 16 (3.10) Theorem 3.2.2 Let assumption (A3) and condition (3.10) hold Assume that C = (cij ) and D = (dij ) 0 Then, system (3.1) is positive for any bounded delays 3.3 Existence of an equilibrium for INNs In this section, we derive conditions for the existence of a unique EP for INNs in the form of (3.1) By Lemma 3.1.2, system (3.1) possesses an EP if and only if system (3.5) does As revealed from (3.9), for a given input vector I ∈ Rn , an EP of (3.1) exists if and x , x, y ∈ Rn , and the only if the equation H(χ) = has a solution χ∗ ∈ R2n , where χ = y mapping H : R2n → R2n is defined by H(χ) = −Dξ x + Dη y (Dαξ − B)x − Dαη y + Cf (x) + Df (x) + I (3.11) , where Dαξ = Dα Dξ and Dαη = Dα Dη Theorem 3.3.1 Assume the assumptions of Theorem 3.2.2 hold and there exists a vector χ0 ∈ R2n , χ0 ≻ 0, such that ⊤ −Dξ Dη Dαξ − B + (C + D)Lf −Dαη χ0 ≺ Then, for any given input vector I ∈ Rn , there exists a unique EP χ∗ = (3.12) x∗ y∗ of system (3.5) 3.4 Exponential stability of positive equilibrium In this section, we will prove that under the assumptions of Theorem 3.2.2, the unique EP χ∗ of (3.5) is positive and GES for any delays τj (t) ∈ [0, τ + ] Theorem 3.4.1 Under the assumptions of Theorem 3.2.2, if there exists a vector χ ˆ0 ∈ R2n , χˆ0 ≻ 0, such that −Dξ Dη Dαξ − B + (C + D)Lf −Dαη χˆ0 ≺ 0, then, for any input vector I ∈ Rn+ , system (3.5) has a unique positive EP χ∗ = which is GES for any delays τj (t) ∈ [0, τ + ] 17 (3.13) x∗ y∗ ∈ R2n +, Chapter EXPONENTIAL STABILITY OF POSITIVE NEURAL NETWORKS IN BIDIRECTIONAL ASSOCIATIVE MEMORY MODEL WITH DELAYS In this chapter we consider the problem of exponential stability of positive neural networks in bidirectional associative memory (BAM) model with multiple time-varying delays and nonlinear self-excitation rates Based on a systematic approach involving extended comparison techniques via differential inequalities, we first prove the positivity of state trajectories initializing from a positive cone called the admissible set of initial conditions In combination with the use of Brouwer’s fixed point theorem and M-matrix theory, we then derive conditions for the existence and global exponential stability of a unique positive equilibrium of the model An extension to the case of BAM neural networks with proportional delays is also presented 4.1 Model description Consider the following BAM neural networks model with delays m m x′i (t) = −αi ϕi (xi (t)) + yj′ (t) = −βj ψj (yj (t)) + aij fj (yj (t)) + j=1 n j=1 n cji gi (xi (t)) + i=1 i=1 bij fj (yj (t − σj (t))) + Ii , (4.1) dji gi (xi (t − τi (t))) + Jj , t ≥ t0 (4.2) We denote the matrices Dα = diag{α1 , α2 , , αn }, Dβ = diag{β1 , β2 , , βm }, A = (aij ) ∈ Rn×m , B = (bij ) ∈ Rn×m , C = (cji ) ∈ Rm×n , D = (dji ) ∈ Rm×n and vectors I = (Ii ) ∈ Rn , J = (Jj ) ∈ Rm , then system (4.1)-(4.2) can be written in the following matrix-vector form  x′ (t) = −D Φ(x(t)) + Af (y(t)) + Bf (y(t − σ(t))) + I, α y ′(t) = −Dβ Ψ(y(t)) + Cg(x(t)) + Dg(x(t − τ (t))) + J (4.3) Let D be the set of continuous functions ϕ : R → R satisfying ϕ(0) = and there exist positive scalars lϕ , l˜ϕ such that the following condition lϕ ≤ ϕ(a) − ϕ(b) ˜ ≤ lϕ a−b (4.4) holds for all a, b ∈ R, a = b Clearly, the function class D includes all linear functions ϕ(a) = γϕ a where γϕ is some positive scalar Assumption (A4.1): The decay rate functions ϕi , ψj , i ∈ [n], j ∈ [m], are assumed to belong the function class D 18 Assumption (A4.2): The neuron activation functions fj (.), gi (.), i ∈ [n], j ∈ [m], are continuous, fj (0) = 0, gi (0) = 0, and there exist positive constants Lfj , Lgi such that 0≤ fj (a) − fj (b) ≤ Lfj , a−b 0≤ gi (a) − gi (b) ≤ Lgi , a−b a = b (4.5) Proposition 4.1.1 Under Assumptions (A4.1) and (A4.2), for any initial functions x0 ∈ C([−τ , 0], Rn ), y ∈ C([−σ, 0], Rm ), there exists a unique solution vec(x(t), y(t)) of system (4.3) defined on [t0 , ∞), which is absolutely continuous in t Let χ(t) = vec(x(t), y(t)) be a solution of system (4.3) If the trajectory of χ(t) is confined within the first orthant, that is, χ(t) ∈ Rn+m for all t ≥ t0 , then χ(t) is said to be + a positive solution of (4.3) We define the following admissible set of initial conditions for system (4.3) A= φ= x0 y0 : x0 (ξ) 0, ∀ξ ∈ [−τ , 0], y 0(θ) 0, ∀θ ∈ [−σ, 0] (4.6) Definition 4.1.1 System (4.3) is said to be positive if for any initial condition φ ∈ A and nonnegative input vector vec(I, J) ∈ Rn+m , the corresponding solution χ(t) = vec(x(t), y(t)) + of (4.3) is positive Definition 4.1.2 For a given input vector J = vec(I, J) ∈ Rn+m , a point χ∗ = vec(x∗ , y ∗ ), x∗ ∈ Rn , y ∗ ∈ Rm , is said to be an equilibrium point (EP) of system (4.3) if it satisfies the following algebraic system  −D Φ(x∗ ) + (A + B) f (y ∗) + I = α −Dβ Ψ(y ∗ ) + (C + D) g(x∗ ) + J = Moreover, χ∗ is a positive EP if it is an EP and χ∗ (4.7) Definition 4.1.3 A positive EP χ∗ = vec(x∗ , y ∗ ) of system (4.3) is said to be globally exponentially stable (GES) if there exist positive scalars κ, γ such that any solution χ(t) = vec(x(t), y(t)) of (4.3) satisfies the following inequality x(t) − x∗ + y(t) − y ∗ ≤ κ x0 − x∗ ∞ + y0 − y∗ ∞ e−γ(t−t0 ) , t ≥ t0 4.2 Positive solutions of BAM neural networks with delays Theorem 4.2.1 Under assumptions (A4.1) and (A4.2), if the connection weight matrices A B A, B, C, and D are nonnegative (equivalently, M = 0), then system (4.3) C ⊤ D⊤ is positive subject to bounded delays and the admissible set of initial conditions A 4.3 Existence of an equilibrium First, it can be verified from system (4.7) that, for a given input vector J = vec(I, J) ∈ Rn+m , a vector χ∗ = vec(x∗ , y ∗) ∈ Rn+m is an EP of model (4.3) if and only if it satisfies 19 the following algebraic system  D−1 ((A + B)f (y ∗) + I) = Φ(x∗ ) α (4.8) D−1 ((C + D)g(x∗ ) + J) = Ψ(y ∗) β Revealed by system (4.8), we define a mapping H : Rn+m → Rn+m by Φ−1 Dα−1 ((A + B)f (y) + I) H(χ) = Ψ−1 Dβ−1 ((C + D)g(x) + J) (4.9) , where χ = vec(x, y), x ∈ Rn and y ∈ Rm More specifically, the mapping H(χ) defined in (4.9) can be written as ˜ (x) · · · h ˜ m (x) H(χ) = h1 (y) · · · hn (y) h ⊤ , where hi (y) = ϕ−1 i ˜ j (x) = ψ −1 h j αi βj m (aij + bij ) fj (yj ) + Ii , i ∈ [n], (cji + dji ) gi (xi ) + Jj , j ∈ [m], j=1 n i=1 −1 and ϕ−1 i (.), ψj (.) denote the inverse functions of ϕi (.) and ψj (.), respectively In regards to equations (4.8) and (4.9), a vector χ∗ ∈ Rn+m is an EP of model (4.3) if and only if it is a fixed point of the mapping H(χ), that is, H(χ∗ ) = χ∗ Based on the Brouwer’s fixed point theorem, we have the following result Theorem 4.3.1 Let assumptions (A4.1) and (A4.2) hold Assume that 0n×n ρ K2 K1 0m×m < 1, (4.10) ) ∈ Rn×m , K = (k ) ∈ Rm×n and where K1 = (kij ji −1 l (|aij | + |bij |) Lfj , αi ϕi (|cji | + |dji |) Lgi kji = lψ−1 j βj kij = Then, system (4.3) has at least one EP Remark 4.3.1 In the proof of Theorem 4.3.1, if the connection weight matrices A, B, C, D, and input vectors I, J are nonnegative, then ui αi and vj βj m (aij + bij ) fj (yj ) + Ii ≥ 0, i ∈ [n], (cji + dji ) gi (xi ) + Jj ≥ 0, j ∈ [m], j=1 n i=1 20 for any x ˜ j (x) = ψ −1 (vj ) ≥ for all Thus, hi (y) = ϕ−1 (ui ) ≥ and h j and y i ∈ [n], j ∈ [m], and χ = vec(x, y) ⊂ Rn+m This shows that Therefore, H Rn+m + + H : B+ → B+ , where B+ = B ∩ Rn+m Since + B+ = χ ∈ Rn+m δ χ ̺ is also a convex compact subset of Rn+m , by the Brouwer’s fixed point theorem, the continuous mapping H has at least one fixed point χ∗+ ∈ B+ , which is a positive EP of system (4.3) We summarize this result in the following corollary Corollary 4.3.2 With the assumptions of Theorem 4.3.1, if the connection weight matrices A, B, C and D are nonnegative, then for any nonnegative input vector J = vec(I, J), system (4.3) has at least one positive EP χ∗ ∈ Rn+m + ˜ be an (n + m) × (n + m) block matrix of the form Remark 4.3.2 Let Ω ˜= Ω ˜ V˜ U ˜ Z˜ W ˜ ∈ Rm×n and Z˜ ∈ Rm×m If Z˜ is nonsingular, then we where U˜ ∈ Rn×n , V˜ ∈ Rn×m , W have ˜ V˜ U˜ − V˜ Z˜ −1 W 0m×n Z˜ = U˜ V˜ ˜ Z˜ W En 0n×m ˜ −Z˜ −1 W Em Therefore, ˜ = det(U˜ − V˜ Z˜ −1 W ˜ ) det(Z) ˜ det(Ω) (4.11) By utilizing equality (4.11), for any λ = 0, we have det(λEn+m − K) = det λEn −K1 −K2 λEm = λm det λEn − K1 K2 λ = λm−n det(λ2 En − K1 K2 ) The aforementioned identity shows that λ ∈ σ(K)\{0} if and only if µ = λ2 ∈ σ(K1 K2 )\{0} Consequently, ρ(K) < ⇐⇒ ρ(K1 K2 ) < Based on the above observation, we have the following results Proposition 4.3.3 Condition (4.10) holds if and only if one of the two following conditions does ρ −1 l αi ϕi m j=1 −1 l (|aij | + |bij |) |cjk | + |djk | Lfj Lgk βj ψj 21 < 1; n×n (4.12) −1 l βj ψj ρ n i=1 −1 l (|cji | + |dji |) (|aik | + |bik |) Lfk Lgi αi ϕi < (4.13) m×m Corollary 4.3.4 Let assumptions (A4.1) and (A4.2) hold Assume that the connection weight matrices are nonnegative and either condition (4.12) or condition (4.13) is satisfied Then, for any nonnegative input vector J = vec(I, J) ∈ Rn+m , system (4.3) has at least + one positive EP χ∗ ∈ Rn+m + 4.4 Exponential stability of positive EP The results presented in the preceeding section only guarantee the existence of at least one positive EP In this section, we will prove that under the assumptions of Theorems 4.2.1 and 4.3.1 system (4.3) has a unique positive EP χ∗ , which is globally exponentially stable Theorem 4.4.1 Let assumptions (A4.1) and (A4.2) hold Assume that the connection weight matrices are nonnegative and one of the three conditions (4.10), (4.12), or (4.13) is satisfied Then, for any input vector J = vec(I, J) ∈ Rn+m , system (4.3) has a unique + positive EP χ∗ ∈ Rn+m , which is GES for any delays τi (t) ∈ [0, τ ] and σj (t) ∈ [0, σ] + 4.5 Positive BAM neural networks with multi-proportional delays Proportional delays belong to a type of unbounded time-varying delays, which are different from most other types of delays such as bounded time-varying delays or distributed delays Typically, in analysis of neural networks with proportional delays, the use of comparison techniques via certain types of differential inequalities proves to be an effective approach As an application, in this section we extend the result of Theorem 4.4.1 to the following BAM neural networks model with proportional delays m m x′i (t) = −αi ϕi (xi (t)) + yj′ (t) = −βj ψj (yj (t)) + bij fj (yj (qij t)) + Ii , aij fj (yj (t)) + (4.14) j=1 n j=1 n cji gi (xi (t)) + i=1 i=1 dji gi (xi (pji t)) + Jj , t ≥ t0 > 0, (4.15) where < pji < 1, < qij < 1, i ∈ [n], j ∈ [m], are proportional delay factors Other coefficients and functions of the model (4.14)-(4.15) are similarly described as in system (4.1)-(4.2) The initial condition of system (4.14)-(4.15) is specified as follows xi (ξ) = x0i (ξ), ξ ∈ [p∗ t0 , t0 ], yj (θ) = yj0(θ), θ ∈ [q∗ t0 , t0 ], (4.16) where p∗ = mini,j pji , q∗ = mini,j qij and x0i ∈ C([p∗ t0 , t0 ], R), yj0 ∈ C([q∗ t0 , t0 ], R) Similar to Proposition 4.1.1, it can be verified that under assumptions (A4.1) and (A4.2) system (4.14)-(4.16) has a unique solution χ(t) = vec(x(t), y(t)) defined on [t0 , ∞) Moreover, if the connection weight matrices A, B, C, D are nonnegative and x0 (ξ) = (x0i (ξ)) y (θ) = (yj0 (θ)) for all ξ ∈ [p∗ t0 , t0 ], θ ∈ [q∗ t0 , t0 ] then χ(t) 22 for all t ≥ t0 0, Theorem 4.5.1 Under the assumptions of Theorem 4.4.1, for any input vector J = vec(I, J) ∈ Rn+m , system (4.14)-(4.16) has a unique positive EP χ∗ ∈ Rn+m , which is generalized glob+ + ally exponentially stable (GGES) More precisely, there exist positive scalars κ and ̟ such that any solution χ(t) = vec(x(t), y(t)) of system (4.14)-(4.15) satisfies x0 − x∗ x(t) − x∗ + y(t) − y ∗ ≤ κ ∗ ∞+ y −y −̟ ln ∞ e 1+t 1+t0 , t ≥ t0 (4.17) Remark 4.5.1 As a special case of system (4.14)-(4.15), we consider the following recurrent neural networks (RNNs) model with proportional delays m m x′i (t) = −αi ϕi (xi (t)) + aij fj (yj (t)) + j=1 j=1 For system (4.18), with A = (aij ) bij gj (yj (qij t)) + Ii , t ≥ t0 > 0 and B = (bij ) (4.18) 0, conditions (4.10), (4.12) and (4.13) are reduced to the following one ρ −1 l αi ϕi n aij Lfj + bij Lgj < (4.19) n×n j=1 Similar to Theorem 4.5.1, if assumptions (A4.1), (A4.2) and condition (4.19) are satisfied then, for any input vector I = (Ii ) 0, model (4.18) has a unique positive EP χ∗ ∈ Rn+ which is GGES Moreover, if I ≻ then χ∗ ≻ This result extends Theorems 3.1 and 3.2 in Yang (2019) 4.6 Simulations In this section we give two examples to illustrate the effectiveness of the obtained results 23 CONCLUSION Main contributions The main contributions of this thesis are as follows: Established conditions in terms of M-matrices forexponential stability of Hopfiled neural networks with time-varying coefficients and destablizing impulses (Theorem 2.3.1) A unified stability criterion is first derived for the stability problem and application to designing a local state feedback control law with bounded controller gains for Hopfiled neural networks with proposition delays and periodic distribution impulses (Theorem 2.4.1) Based on the approach of using homeomorphisms in nonlinear analysis, sufficient conditions for the existence, uniqueness and global exponential stability of a positive equilibrium of inertial neural networks are derived in the form of LP problems with M-matrices (Theorems 3.2.2-3.4.1) Based on an extended comparison techniques via differential inequalities combining with the use of Brouwer’s fixed point theorem and M-matrix theory, tractable LPbased conditions are derived to ensure the existence and global exponential stability of a unique positive equilibrium of BAM neural networks with heterogeneous delays An extension to the case of BAM neural networks with proportional delays is also obtained (Theorems 4.2.1-4.5.1) Further topics • Investigating the problem of stability/synchronization of nonautonomous neural net- work with mixed impusive effects For this model, conventional methods based on Lyapunov or Razumikhin functions are no longer available Moreover, the idea of using varying-rate of the model parameters to provide stability conditions is an interesting and challenging issue • Studying important problems in the systems and control theory for delayed neural network models or models with interval parameters 24 LIST OF PUBLICATIONS [CT1] L.D Hai An, L.V Hien, T.T Loan, Exponential stability of non-autonomous neural networks with heterogeneous time-varying delays and destabilizing impulses, Vietnam Journal of Mathematics 45 (2017) 425–440 (ESCI/Scopus) [CT2] L.D Hai-An, L.V Hien, T.T Loan, On exponential stability of neural networks with proportional delays and periodic distribution impulsive effects, Differential Equations and Dynamical Systems (2019) DOI: 10.1007/s12591-019-00459-x (ESCI/Scopus) [CT3] L.V Hien, L.D Hai-An, Positive solutions and exponential stability of positive equilibrium of inertial neural networks with multiple time-varying delays, Neural Computing and Applications 31 (2019) 6933–6943 (SCIE) [CT4] L.V Hien, L.D Hai-An, Exponential stability of positive neural networks in bidirectional associative memory model with delays, Mathematical Methods in the Applied Sciences 42 (2019) 6339–6357 (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 • Department of Optimization and Control, Hanoi Institute of Mathematics, Vietnam Academy of Sciences and Technology ... challenging to analyze dynamic behaviors of INNs Therefore, the investigation of INNs for both theoretical and practical reasons has attracted considerable attention in the past few years Positive... self-excitation rates Based on a systematic approach involving extended comparison techniques via differential inequalities, we first prove the positivity of state trajectories initializing from a positive... Technology The thesis will be presented to the examining committee at Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Vietnam At the time of , 2020 This dissertation is publicly