Asymptotic Stability Theorem Research Articles

Critical waves of a stage-structured epidemic model with latent period

We study the epidemic waves connecting the disease-free equilibrium and endemic equilibrium for the critical speed of a stage-structured epidemic model with latent period. The method is limiting approaches combined with priori estimates, Lyapunov functional method, Lyapunov-LaSalle asymptotic stability theorem and Barbalat’s theorem. The conclusion illustrates in the degenerate case, the epidemic waves still exist. Simulations are given as well to demonstrate the evolution of wave profiles.

Active impedance control based adaptive locomotion for a bionic hexapod robot

AbstractIn recent years, with the continuous development of human exploration of the natural world, there has been a growing demand across various fields for robots capable of free movement in diverse environments. In this study, we address the issue of compliant control for a hexapod robot in diverse environments and propose a novel control method based on an adaptive impedance model for position control. Our approach enables the hexapod robot to stabilize foot force on complex terrains while preserving balance and body height. Specifically, we analyze the algorithm's parameters and stability by establishing the hexapod robot's structural and impedance control models. To tackle this challenge, we introduce an adaptive impedance control algorithm that estimates environmental parameters using Lyapunov's asymptotic stability theorem and achieves tracking of actual foot‐end forces to desired foot forces. Furthermore, to ensure body stability and height, we incorporate attitude feedback and body feedback. Experimental results from foot force control experiments conducted on a multilegged robot demonstrate that our proposed algorithm enhances the adaptability and robustness of the robot. This research holds significant implications for the stable control of hexapod robots in complex environments and has promising practical applications.

NOVEL UNIFIED STABILITY CRITERION FOR FRACTIONAL-ORDER TIME DELAY SYSTEMS WITH STRONG RESISTANCE TO FRACTIONAL ORDERS

In this study, a novel unified stability criterion is first proposed for general fractional-order systems with time delay when the fractional order is from 0 to 1. Such a new unified criterion has the advantage of having an initiative link with the fractional orders. A further advantage is that the corresponding asymptotic stability theorem, derived from the proposed criterion used to analyze the asymptotic stability, is only slightly affected by the change of the fractional order. In addition, the unified stability criterion is applied to general multi-dimensional nonlinear fractional-order systems with time delays, the corresponding asymptotic stability criterion is applied by combining the vector Lyapunov function with the M-matrix method. Compared with the traditional stability criterion, the unified stability criterion is slightly influenced by the changing fractional order and large time delays. The reliability and effectiveness of the novel uniform stability criterion were verified through three representative examples.

Modeling the co-infection of HTLV-2 and HIV-1 in vivo

<p>Human T-lymphotropic virus type 2 (HTLV-2) and human immunodeficiency virus type 1 (HIV-1) are two infectious retroviruses that infect immune cells, CD8<sup>+</sup> T cells and CD4<sup>+</sup> T cells, respectively. Multiple studies have revealed co-infected patients with HTLV-2 and HIV-1. In this paper, we formulated a new mathematical model for the co-infection of HTLV-2 and HIV-1 in vivo. The HIV-1-specific B-cell response is included. Six ordinary differential equations made up the model, which depicted the interactions between uninfected CD4<sup>+</sup> T cells, HIV-1-infected CD4<sup>+</sup> T cells, HIV-1 particles, uninfected CD8<sup>+</sup> T cells, HTLV-2-infected CD8<sup>+</sup> T cells, and HIV-1-specific B cells. We carried out a thorough study of the model, demonstrating the boundedness and nonnegativity of the solutions. Additionally, we determined the equilibrium points and demonstrated, under specific conditions, their global stability. The global asymptotic stability of all equilibria was established by constructing appropriate Lyapunov functions and applying the Lyapunov-LaSalle asymptotic stability theorem. We provide numerical simulations to corroborate the theoretical findings. We investigated how the B-cell response affects the dynamics of HIV-1 and HTLV-2 co-infection. The results suggested that the B-cell response regulates and inhibits the spread of HIV-1. We present a comparison between HTLV-2 or HIV-1 mono-infections and co-infections with HTLV-2 and HIV-1. Our findings support earlier research, suggesting that co-infection with HTLV-2 may be able to maintain the behavior dynamics of the CD4<sup>+</sup> T cells, inhibit HIV-1 replication, and postpone the onset of AIDS. However, co-infected patients with HTLV-2 and HIV-1 may experience a greater occurrence of HTLV-2-related T-cell malignant diseases.</p>

Global Dynamics of a Diffusive Within-Host HTLV/HIV Co-Infection Model with Latency

In several publications, the dynamical system of HIV and HTLV mono-infections taking into account diffusion, as well as latently infected cells in cellular transmission has been mathematically analyzed. However, no work has been conducted on HTLV/HIV co-infection dynamics taking both factors into consideration. In this paper, a partial differential equations (PDEs) model of HTLV/HIV dual infection was developed and analyzed, considering the cells’ and viruses’ spatial mobility. CD4+T cells are the primary target of both HTLV and HIV. For HIV, there are three routes of transmission: free-to-cell (FTC), latent infected-to-cell (ITC), and active ITC. In contrast, HTLV transmits horizontally through ITC contact and vertically through the mitosis of active HTLV-infected cells. In the beginning, the well-posedness of the model was investigated by proving the existence of global solutions and the boundedness. Eight threshold parameters that determine the existence and stability of the eight equilibria of the model were obtained. Lyapunov functions together with the Lyapunov–LaSalle asymptotic stability theorem were used to investigate the global stability of all equilibria. Finally, the theoretical results were verified utilizing numerical simulations.

Exponential Stability of Nonlinear Time-Varying Delay Differential Equations via Lyapunov–Razumikhin Technique

In this article, some new sufficient conditions for the exponential stability of nonlinear time-varying delay differential equations are given. An extension of the classical asymptotical stability theorem in terms of a Lyapunov–Razumikhin function is obtained. The condition of non-positivity of the time derivative of a Razumikhin function is weakened. Additionally, the resulting sufficient asymptotic stability conditions allow us to guarantee uniform exponential stability and evaluate the exponential convergence rate of the system solutions. The effectiveness of the results is demonstrated by some examples.

Impulsive Stabilization of Nonautonomous Timescale-Type Neural Networks With Constant and Unbounded Time-Varying Delays

This article focuses on the impulsive stabilization of nonautonomous timescale-type neural networks (NTNNs) with constant and unbounded time-varying delays. By choosing two discontinuous piecewise linear functions and employing the convex combination method, several algebraic criteria are demonstrated to achieve globally asymptotic stabilization of NTNNs with constant and unbounded time-varying delays. First, the asymptotic stability theorem for timescale-type systems is constructed by utilizing the timescale theory. Based on this asymptotic stability theorem, an impulsive control scheme is designed to guarantee global asymptotic stabilization of NTNNs with constant delay. Second, we propose several impulsive control schemes to achieve globally asymptotic stabilization of NTNNs with unbounded delay by virtue of the comparison strategy. Globally asymptotic stabilization criteria for NTNNs include the theoretical results of continuous-time neural networks (NNs), their discrete-time forms and NNs on continuous-discrete hybrid time scales. Especially, impulsive control for globally asymptotic stabilization of NTNNs with proportional delay is demonstrated without any variable transformation. Finally, the effectiveness of our proposed theoretical results is verified by four numerical examples.

Reduction theorems for stability of compact sets in time-varying systems

Reduction theorems provide a framework for stability analysis that consists in breaking down a complex problem into a hierarchical list of subproblems that are simpler to address. This paper investigates the following reduction problem for time-varying ordinary differential equations on Rn. Let Γ1 be a compact set and Γ2 be a closed set, both positively invariant and such that Γ1⊂Γ2⊂Rn. Suppose that Γ1 is uniformly asymptotically stable relative to Γ2. Find conditions under which Γ1 is uniformly asymptotically stable. We present a reduction theorem for uniform asymptotic stability that completely addresses the local and global version of this problem, as well as two reduction theorems for uniform stability and either local or global uniform attractivity. These theorems generalize well-known equilibrium stability results for cascade-connected systems as well as previous reduction theorems for time-invariant systems. We also present Lyapunov characterizations of the stability properties required in the reduction theorems that to date have not been investigated in the stability theory literature.

Global stability of a general HTLV-I infection model with Cytotoxic T-Lymphocyte immune response and mitotic transmission

Mathematical models have been considered as a robust tool to support biological and medical studies of human viral infections. The global stability of viral infection models remains an important and largely open research problem. Such results are necessary to evaluate treatment strategies for infections and to establish thresholds for treatment rates. Human T-lymphotropic virus class I (HTLV-I) is a retrovirus which infects the CD4+T cells and causes chronic and deadly diseases. In this article, we developed a general nonlinear system of ODEs which describes the within-host dynamics of HTLV-I under the effect Cytotoxic T-Lymphocytes (CTLs) immunity. The mitotic division of actively infected cells are modeled. We consider general nonlinear functions for the generation, proliferation and clearancerates for all types of cells. The incidence rate of infection is also modeled by a general nonlinear function. These general functions are assumed to satisfy a set of suitable conditions and include several forms presented in the literature. We determine a bounded domain for the system’s solutions. We prove the existence of the system’s equilibrium points and determine two threshold numbers, the basic reproductive number R0 and the CTL immunity stimulation number R1. We establish the global stability of all equilibrium points by constructing Lyapunov function and applying Lyapunov-LaSalle asymptotic stability theorem. Under certain conditions it is shown that ifR0≤1, then the infection-free equilibrium point is globally asymptotically stable (GAS) and the HTLV-I infection is cleared. If R1≤1<R0, then the infected equilibrium point without CTL immunity is GAS and the HTLV-I infection becomes chronic with no sustained CTL immune response. If R1>1, then the infected equilibrium point with CTL immunity is GAS and the infection becomes chronic with persistent CTL immune response. We present numerical simulations for the system by choosing special shapes of the general functions. The effect of Crowley-Martin functional response and mitotic division of actively infected cells on the HTLV-I progression are studied. Our results cover and improve several ones presented in the literature.

Asymptotic Synchronization of Fractional-Order Complex Dynamical Networks with Different Structures and Parameter Uncertainties

This paper deals with the asymptotic synchronization of fractional-order complex dynamical networks with different structures and parameter uncertainties (FCDNDP). Firstly, the FCDNDP model is proposed by the Riemann–Liouville (R-L) fractional derivative. According to the property of fractional calculus and the Lyapunov direct method, an original controller is proposed to achieve the asymptotic synchronization of FCDNDP. Our controller is more adaptable and effective than those in other literature. Secondly, a sufficient condition is given for the asymptotic synchronization of FCDNDP based on the asymptotic stability theorem and the matrix inequality technique. Finally, the numerical simulations verify the effectiveness of the proposed method.

The proof of Lyapunov asymptotic stability theorems for Caputo fractional order systems

A recent comment paper (Wu, 2021) pointed out that the proof for the existing asymptotic stability theorems of Caputo fractional order systems is questionable. Since these theorems are fundamental and widely adopted in many scenarios, such a comment indeed causes panic in fractional community and doubt from other academic communities. Under this background, the mentioned theorems are revisited and confirmed.

Stability of a secondary dengue viral infection model with multi-target cells

Dengue is one of the vector-borne diseases spread in most parts of the world. The number of infected individuals is increased each year. This paper proposes a mathematical model describing the secondary dengue viral infection in micro-environment. This model takes into consideration that the dengue virus can infect multiple classes of target cells. Due to the secondary infection, the model incorporates two types of antibodies, heterologous and homologous. We establish the well-posedness of the model. We compute three threshold parameters which characterize the existence and stability conditions for the four steady states of the model. Global stability analysis for all steady states is carried out by formulating Lyapunov function and using Lyapunov-LaSalle asymptotic stability theorem. We demonstrate the analytical findings via numerical simulations.

Stability of Nonlinear Neutral Mixed Type Liven-Nohel Integro-Differential Equations

In this paper, we use the contraction mapping theorem to obtain asymptotic stability results about the zero solution for a nonlinear neutral mixed type Levin-Nohel integro-differential equation. An asymptotic stability theorem with a necessary and sufficient condition is proved. An example is also given to illustrate our main results.

Lyapunov-Razumikhin techniques for state-dependent delay differential equations

We present Lyapunov stability and asymptotic stability theorems for steady state solutions of general state-dependent delay differential equations (DDEs) using Lyapunov-Razumikhin methods. Our results apply to DDEs with multiple discrete state-dependent delays, which may be nonautonomous for the Lyapunov stability result, but autonomous (or periodically forced) for the asymptotic stability result. Our main technique is to replace the DDE by a nonautonomous ordinary differential equation (ODE) where the delayed terms become source terms in the ODE. The asymptotic stability result is based on a contradiction argument together with the Arzelà-Ascoli theorem. This approach alleviates the need to construct auxiliary functions to ensure the asymptotic contraction, which is a feature of other Lyapunov-Razumikhin asymptotic stability results of which we are aware. We apply our results to a state-dependent model equation which includes Hayes equation as a special case, to directly establish asymptotic stability in parts of the stability domain together with lower bounds on the size of the basin of attraction.

Novel asymptotic stability criterion for fractional‐order gene regulation system with time delay

AbstractAt present, Routh–Herwitz criterion is the mostly applied characteristic equation of the time‐lag fractional‐order gene regulatory network systems (FGRNs) for stability analysis. However, it is limited by linearization, which makes the result have a certain error. In addition, due to the processes of obtaining the characteristic roots are too complicated, there are too many constraints on stability criterion. To solve these problems, the vector Lyapunov function method combined with the M‐matrix is used in the present study. Based on this method, the corresponding asymptotic stability theorem is derived. Our results show that this novel method not only had fewer limitations compared with conventional methods, but also is suitable to all fractional‐order operators between 0 and 1 with a large time delay.

Modeling and analysis of a within-host HIV/HTLV-I co-infection.

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that attack the {text {CD4}}^{+}T cells and impair their functions. Both HIV and HTLV-I can be transmitted between individuals through direct contact with certain body fluids from infected individuals. Therefore, a person can be co-infected with both viruses. HIV causes acquired immunodeficiency syndrome (AIDS), while HTLV-I is the causative agent for adult T-cell leukemia (ATL) and HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP). Several mathematical models have been developed in the literature to describe the within-host dynamics of HIV and HTLV-I mono-infections. However, modeling a within-host dynamics of HIV/HTLV-I co-infection has not been involved. The present paper is concerned with the formulation and investigation of a new HIV/HTLV-I co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible {text {CD4}}^{+}T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell transmission. On the other side, HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact through the virological synapse, and (ii) vertical transmission through the mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We define a set of threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and Lyapunov–LaSalle asymptotic stability theorem. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

Random iterations of maps on : asymptotic stability, synchronisation and functional central limit theorem

We study independent and identically distributed random iterations of continuous maps defined on a connected closed subset S of the Euclidean space . We assume the maps are monotone (with respect to a suitable partial order) and a ‘topological’ condition on the maps. Then, we prove the existence of a pullback random attractor whose distribution is the unique stationary measure of the random iteration, and we obtain the synchronisation of random orbits. As a consequence of the synchronisation phenomenon, a functional central limit theorem is established.

Time-Varying Halanay Inequalities With Application to Stability and Control of Delayed Stochastic Systems

In this article, the Halanay inequality is generalized into the time-varying version under simple and natural conditions, without constant to bound the time-varying coefficients. The solution of the underlying inequality is estimated by virtue of the feasible function pair associated with the coefficients and delays. Corollaries are given for typical special cases. Some approaches are illustrated for the searching of the initial feasible pairs, and an iterative procedure is proposed for the optimization of the feasible pairs. The usage and superiority of the obtained result are exhibited by numerical examples with simulations. As a base, the comparison principle for the functional differential inequalities and equations with arbitrary time-varying delays is studied first. As an application of the main result, the stability and control of the stochastic systems with time-varying coefficients and delays are investigated, with the main concern to the time variance. An asymptotic stability theorem and a criterion are established, and an interesting control strategy with unbounded time delay, namely the pantograph delay sampled data based feedback, is proposed. The stability criterion and the control strategy are illustrated and verified with numerical experiments.

Comments on “Stability analysis of Caputo fractional-order nonlinear systems revisited”

This note points out that the proof for a widely and mostly used (uniform) asymptotic stability theorem for Caputo fractional-order systems, presented by the article “Stability analysis of Caputo fractional-order nonlinear systems revisited” published in Nonlinear Dynamics, is incorrect.

Stability of HTLV/HIV dual infection model with mitosis and latency.

In this paper, we formulate and analyze an HTLV/HIV dual infection model taking into consideration the response of Cytotoxic T lymphocytes (CTLs). The model includes eight compartments, uninfected CD4+T cells, latent HIV-infected cells, active HIV-infected cells, free HIV particles, HIV-specific CTLs, latent HTLV-infected cells, active HTLV-infected cells and HTLV-specific CTLs. The HIV can enter and infect an uninfected CD4+T cell by two ways, free-to-cell and infected-to-cell. Infected-to-cell spread of HIV occurs when uninfected CD4+T cells are touched with active or latent HIV-infected cells. In contrast, there are two modes for HTLV-I transmission, (ⅰ) horizontal, via direct infected-to-cell touch, and (ⅱ) vertical, by mitotic division of active HTLV-infected cells. We analyze the model by proving the nonnegativity and boundedness of the solutions, calculating all possible steady states, deriving a set of key threshold parameters, and proving the global stability of all steady states. The global asymptotic stability of all steady states is proven by using Lyapunov-LaSalle asymptotic stability theorem. We performed numerical simulations to support and illustrate the theoretical results. In addition, we compared between the dynamics of single and dual infections.