Stability of positive solutions of nonlinear differential equations with delays

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION ——————–o0o——————— LE THI HONG DUNG STABILITY OF POSITIVE SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS WITH DELAYS IN NEURAL NETWORKS Speciality: Differential and Integral Equations Code: 9460103 SUMMARY OF DOCTORAL THESIS IN MATHEMATICS HA NOI-2024 This dissertation has been written on the basis of my research work carried at: Hanoi National University of Education Supervisor: Assoc. Le Van Hien Hanoi National University of Education Referee 1: Associate Professor Tran Dinh Ke Hanoi National University of Education Referee 2: Associate Professor Do Lan Water Resources University Referee 3: Associate Professor Duong Anh Tuan 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 .,2024 This dissertation is publicly available at: - HNUE Library Information Centre - The National Library of Vietnam INTRODUCTION A. Motivation Stability theory is one of the top priority research topics in the qualitative theory of differ- ential equations and, more general, systems and control theory. Date back to the pioneering work of Lyapunov, stability theory has been extensively developed.

Its intrinsic interest and relevance has been found in a variety of disciplines like mechanics, physics, chemistry, ecology or artificial intelligence. Appearing naturally in practice, many models in population, economic growth or labor migration are described by dynamic systems whose states are always nonnegative when the initial states and inputs are nonnegative. Such systems are called positive systems. Applica- tions of positive systems can be found in various disciplines, from physics, chemistry, ecology and epidemiology to economics, control engineering and telecommunications networks.

Re- search on positive systems shows that, besides a wide range of applications, positive systems also possess many properties that are not found in general systems. For example, based on the monotonicity and robustness induced by the positivity, positive systems are employed to design interval observers, state estimation or stability analysis of nonlinear systems. Thus, due to theoretical and practical features, the theory of positive systems has received ever- increasing interest in the past few years. While the theory of positive systems has been intensively studied for various kinds of linear systems, this area is still considerably less well-developed for nonlinear systems, in particular, for models arising in artificial and/or biological neural networks.

Typically, dynamics of a network is represented by a system of nonlinear differential equations with or without delay. In the past two decades, the study in- volving qualitative behavior of nonlinear systems describing various types of neural networks has attracted significant research attention due to a wide range of applications. The terminology of neural networks, appeared in the late 1800s, was mentioned by sci- entists while studying the function of human brain with the desire to be able to design computers that can work like the human brain, capable of learning through databases, re- membering experiences and use in appropriate situations. Over the history of more than 200 years, with the advent of computers, the study of neural networks has evolved extensive development and has obtained many important results in widen the ability to recognize and adapt to the industry computer technology.

Using ideas from experimental results in studies of human brain, many intelligent computers, which have components like neurons or sets of neurons and the connections between those components like the synapses of neurons, were invented. There are many kind of neural networks mentioned in the literature. Due to their specific structure and practical applicability, some popular models of neural networks such as Hop- field neural networks (HNNs), Cohen-Grossberg neural networks (CGNNs), inertial neural 1 networks (INNs), or bidirectional associative memory networks (BAMs) have been widely studied. However, there has been very little attempt devoted to the study of positive non- linear systems in neural networks.

For neural systems, the nonlinearity of neuron activation functions makes the study of positive neural networks more complicated and challenging, which requires in-depth knowledge and specific techniques. Although many results concerning systems and control theory for positive systems have been published in the past few decades, the field of qualitative research on long-term be- havior and stability of nonlinear neuronal systems with delays is still of great interest to mathematicians and engineers. Besides, many open issues related to the stability of posi- tive solutions in ecological models with multiple distinct delays are still being developed, especially for models with a structure that is more general and closer to reality. Developing qualitative research on this type of systems is generally more difficult and complicated due to the technical limitations and the capability of existing approaches.

This motives scholars for the research of positive solutions and stability of nonlinear differential equations with delays. Research aims The thesis focuses on problems of stability of several classes of nonlinear positive differ- ential systems with delay in neural network models. Specifically, the thesis researches the following issues. • Positive solutions and exponential stability of nonlinear time-delay systems in the model of BAM-Cohen-Grossberg neural networks.

• Exponential stability of positive conformable BAM neural networks with time-varying delays. • Exponential attractivity of positive inertial neural networks in bidirectional associative memory model with time-varying delays. The models considered are systems describing BAM-Hopfield networks with bounded delays. With assumptions related to connection weights and neural activation functions, we prove the systems are positive and aim to establish conditions in LP form to ensure exponential stability of system.

There are similarities in the analysis schemes for the systems considered, such as comparison techniques via inequalities, however, due to specific structure, it is necessary to have separate research methods and proof techniques for each individual system. Methodology • Comparison techniques via differential and integral inequalities. • M-matrix theory approach. 2 • The use of fixed point theorems and basic tools in nonlinear functional analysis.

Research topics This thesis is concerned with some issues of positive solutions and the stability of positive systems with delays. Specifically, the following topics will be researched and presented in this thesis. Stability of nonlinear time-delay systems in BAM-Cohen-Grossberg neural networks Consider a class of nonlinear systems describing BAM-Cohen-Grossberg neural networks with time-varying delays and nonlinear self-excitation rates of the form m m    X X  0 xi (t) = αi (xi (t)) − δi ϕi (xi (t)) + aij fj (yj (t)) + bij fj (yj (t − σj (t))) + Ii ,    j=1 j=1  n n  (1) X X 0  yj (t) = βj (yj (t)) − ρj ψj (yj (t)) + cji gi (xi (t)) + dji gi (xi (t − τi (t))) + Jj.    i=1 i=1 System (1) describes a BAM-Cohen-Grossberg neural network between n neurons in X -layer and m neurons in Y -layer.

xi (t) and yj (t) represent the state variables of the neurons; αi (xi ) and βj (yj ) are neural amplification functions, ϕi (xi ), ψj (yj ) are nonlinear decay rate functions and δi > 0, ρj > 0 are self-inhibition coefficients. fj , gi are neuron activation functions and aij , bij , cji , dji , i ∈ [n], j ∈ [m], are connection weights. τi (t) and σj (t) are the communication delays between neurons which satisfy 0 ≤ τi (t) ≤ τ , 0 ≤ σj (t) ≤ σ, (2) where τ and σ are known positive constants. Ii and Jj denote the external inputs to the ith neuron and j th neuron, respectively.

Initial conditions associated with system (1) are specified as follows x(t0 + ξ) = x0 (ξ), ξ ∈ [−τ , 0], y(t0 + θ) = y 0 (θ), θ ∈ [−σ, 0], (3) where x0 ∈ C([−τ , 0], Rn ) and y 0 ∈ C([−σ, 0], Rm ) are initial functions. The objective is to study the existence of global positive solutions and the existence, uniqueness and exponential stability of positive equilibrium point. Based on novel comparison techniques via differential inequalities, unified conditions for the existence and exponential stability of a unique positive equilibrium of model (1) are derived in terms of tractable LP-based conditions. Stability of positive conformable BAM neural networks with delays In Chapter 3, we consider a class of differential equations with delays described by con- formable fractional derivative (CFD).

This differential equation type can be used to describe 3 the dynamics of various practical models, including biological and artificial neural networks with heterogeneous time-varying delays. Consider the following system ! ! ! ! ! c α x(t) x(t) Af (y(t)) Bf (y σ (t)) I Dt0 = −Dβ,γ + + + , (4) y(t) y(t) Cg(x(t)) Dg(xτ (t)) J where c Dtα0 is the notation for the conformable fractional derivative. More details on CFD and the biology description of the system (4) will be presented in the next section. For a given t0 ≥ 0 and σ, τ are known positive constants, the initial condition of system (4) is specified as xt0 = x0 ∈ C([−τ , 0], Rn ), yt0 = y0 ∈ C([−σ, 0], Rm ), (5) that is, xt0 (s) = x(t0 + s) = x0 (s), s ∈ [−τ , 0], yt0 (θ) = y(t0 + θ) = y0 (θ), θ ∈ [−σ, 0].

By novel comparison techniques via fractional differential and integral inequalities, unified conditions in terms of tractable LP-based conditions for the existence and exponential sta- bility of a unique positive equilibrium of the CFD model (4) are derived. Stability of positive inertial BAM neural network model In Chapter 4, we consider a model of inertial BAM neural networks with delays described by the following second-order differential equations m X x00i (t) = −ai x0i (t) − ci xi (t) + rij fj (yj (t)) j=1 m X + sij fj (yj (t − σj (t))) + Ii , t ≥ t0 , i ∈ [n], (6a) j=1 n X yj00 (t) = −bj yj0 (t) − dj yj (t) + pji gi (xi (t)) i=1 n X + qji gi (xi (t − τi (t))) + Jj , t ≥ t0 , j ∈ [m]. (6b) i=1 The functions τi (t) and σj (t) represent the heterogeneous communication delays between neurons which are assumed to satisfy 0 ≤ τj (t) ≤ τ and 0 ≤ σi (t) ≤ σ for all t ≥ t0 , where τ , σ are known positive constants. 4 The initial condition associated with system (6) is defined by x(t0 + θ) = ϕ(θ), x0 (t0 + θ) = ϕd (θ), θ ∈ [−τ , 0], y(t0 + θ) = ψ(θ), y 0 (t0 + θ) = ψd (θ), θ ∈ [−σ, 0], where ϕ, ϕd ∈ C([−τ , 0], Rn ) and ψ , ψd ∈ C([−σ, 0], Rm ) are compatible initial functions.

Based on some novel comparison techniques developed from monotone dynamical systems theory, we then derive tractable conditions in terms of M-matrix involving self excitation coefficients and connection weights to ensure the positivity of solutions and the existence of a unique EP corresponding to an input vector of system (6). The derived conditions are then utilized to show that the unique EP is positive and globally attractive. Main contributions This dissertation is concerned with the positivity of solutions and exponential stability of positive equilibrium of nonlinear differential equations with delays in various types of neural networks. Main contribution are as follows.

Proposed new LP-based conditions for the positivity of solutions and exponential stabil- ity of positive equilibrium of BAM-Cohen-Grossberg neural networks with time-varying delays and nonlinear self-excitation rates. Proved the positivity and derived tractable conditions for the global exponential stability of a unique positive equilibrium of conformable BAM neural networks with communi- cation delays. Established LP-based conditions ensuring the positivity of solutions and global expo- nential stability of a unique positive equilibrium of inertial BAM neural networks with bounded delays.