Conditions For Asymptotic Stability Research Articles

Dynamics of a predator–prey system with sublethal effects of pesticides on pests and natural enemies

Considering the influence of sublethal concentration of pesticides on pests and natural enemies, we propose a pest-management model with impulsive effect on chemical control and biological control strategies–periodic spraying pesticide and releasing predatory natural enemies. By using the Floquet theory and the comparison theorem of impulsive differential equations, a sufficient condition for the global asymptotic stability of the pest-eradication periodic solution is obtained. The persistence of the system is further studied, and a sufficient condition for the persistence of the system is obtained. Finally, some numerical simulations are shown to verify our theoretical works. Our works indicate that the sublethal effects of insecticides and the release of predatory natural enemies play significant roles in pest control in agricultural production.

A fractional-order model to study the dynamics of the spread of crime

Numerous crucial factors and parameters influence the dynamic process of the spread of crime. Various integer-order differential models have been proposed to capture crime spread. Most of these introduced dynamic systems have not considered the history of the criminal and the impact of crime on society. To address these shortcomings, a fractional-order crime transmission model is proposed in this manuscript considering five different classes viz law-abiding citizens, non-incarcerated criminals, incarcerated criminals, prison-released and recidivists. The primary focus of the proposed model is to study the effect of recidivism in society and decide the adequate imprisonment for repeat offenders. The existence, uniqueness, non-negativity and boundedness of the solution of the proposed model are examined. The local stability of the equilibrium points is also analysed using Routh–Hurwitz Criteria with Matignon conditions. Further, the threshold condition for the uniform asymptotic stability of the system is evaluated by the Lyapunov stability method. Moreover, the long-term impact of the imprisonment of criminals on society is also examined in the current study. The numerical simulations of the model for a range of fractional orders are obtained using power series expansion method to strengthen the theoretical results.

A Fractional-Order Eco-Epidemiological Leslie–Gower Model with Double Allee Effect and Disease in Predator

In this paper, a fractional order of a modified Leslie–Gower predator-prey model with disease and the double Allee effect in predator population is proposed. Then, we analyze the important mathematical features of the proposed model such as the existence and uniqueness as well as the non-negativity and boundedness of solutions to the fractional-order system. Moreover, the local and global asymptotic stability conditions of all possible equilibrium points are investigated using Matignon’s condition and by constructing a suitable Lyapunov function, respectively. Finally, numerical simulations are presented to verify the theoretical results. We show numerically the occurrence of two limit cycles simultaneously driven by the order of the derivative, the bistability phenomenon for both the weak and strong Allee effect cases, and more dynamic behaviors such as the forward, backward, and saddle-node bifurcations which are driven by the transmission rate. We have found that the risk of extinction for the predator with a strong Allee effect is much higher when the spread of disease is relatively high.

Novel Global Asymptotic Stability Conditions for Discrete-Time Systems With Time-Varying Delay and Generalized Overflow Arithmetic

Novel global asymptotic stability (GAS) criterion for discrete-time (DT) systems with generalized overflow arithmetic and time-varying delay is presented in this brief. The criterion is based on two unique lemmas that allow for a more precise characterization of nonlinearities by utilizing system information in greater detail. The approach leads to significantly relaxed stability conditions as compared to a recently reported criterion.

Novel Frequency-Based Approach to Analyze the Stability of Polynomial Fractional Differential Equations

This paper proposes a novel approach for analyzing the stability of polynomial fractional-order systems using the frequency-distributed fractional integrator model. There are two types of frequency and temporal stabilization methods for fractional-order systems that global and semi-global stability conditions attain using the sum-of-squares (SOS) method. Substantiation conditions of global and asymptotical stability are complicated for fractional polynomial systems. According to recent studies on nonlinear fractional-order systems, this paper concentrates on polynomial fractional-order systems with any degree of nonlinearity. Global stability conditions are obtained for polynomial fractional-order systems (PFD) via the sum-of-squares approach and the frequency technique employed. This method can be effective in nonlinear systems where the linear matrix inequality (LMI) approach is incapable of response. This paper proposes to solve non-convex SOS-designed equations and design framework key ideas to avoid conservative problems. A Lyapunov polynomial function is determined to address the problem of PFD stabilization conditions and stability established using sufficiently expressed conditions. The main goal of this article is to present an analytical method based on the optimization method for fractional order models in the form of frequency response. This method can convert it into an optimization problem, and by changing the solution of the optimization problem, the stability of the fractional-order system can be improved.

Convergence of stochastic approximation via martingale and converse Lyapunov methods

In this paper, we study the almost sure boundedness and the convergence of the stochastic approximation (SA) algorithm. At present, most available convergence proofs are based on the ODE method, and the almost sure boundedness of the iterations is an assumption and not a conclusion. In Borkar and Meyn (SIAM J Control Optim 38:447–469, 2000), it is shown that if the ODE has only one globally attractive equilibrium, then under additional assumptions, the iterations are bounded almost surely, and the SA algorithm converges to the desired solution. Our objective in the present paper is to provide an alternate proof of the above, based on martingale methods, which are simpler and less technical than those based on the ODE method. As a prelude, we prove a new sufficient condition for the global asymptotic stability of an ODE. Next we prove a “converse” Lyapunov theorem on the existence of a suitable Lyapunov function with a globally bounded Hessian, for a globally exponentially stable system. Both theorems are of independent interest to researchers in stability theory. Then, using these results, we provide sufficient conditions for the almost sure boundedness and the convergence of the SA algorithm. We show through examples that our theory covers some situations that are not covered by currently known results, specifically Borkar and Meyn (2000).

Mathematical Modelling of the Spatial Distribution of a COVID-19 Outbreak with Vaccination Using Diffusion Equation.

The formulation of mathematical models using differential equations has become crucial in predicting the evolution of viral diseases in a population in order to take preventive and curative measures. In December 2019, a novel variety of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, China, which causes a severe and potentially fatal respiratory syndrome. Since then, it has been declared a pandemic by the World Health Organization and has spread around the globe. A reaction−diffusion system is a mathematical model that describes the evolution of a phenomenon subjected to two processes: a reaction process, in which different substances are transformed, and a diffusion process, which causes their distribution in space. This article provides a mathematical study of the Susceptible, Exposed, Infected, Recovered, and Vaccinated population model of the COVID-19 pandemic using the bias of reaction−diffusion equations. Both local and global asymptotic stability conditions for the equilibria were determined using a Lyapunov function, and the nature of the stability was determined using the Routh−Hurwitz criterion. Furthermore, we consider the conditions for the existence and uniqueness of the model solution and show the spatial distribution of the model compartments when the basic reproduction rate R0<1 and R0>1. Thereafter, we conducted a sensitivity analysis to determine the most sensitive parameters in the proposed model. We demonstrate the model’s effectiveness by performing numerical simulations and investigating the impact of vaccination, together with the significance of spatial distribution parameters in the spread of COVID-19. The findings indicate that reducing contact with an infected person and increasing the proportion of susceptible people who receive high-efficacy vaccination will lessen the burden of COVID-19 in the population. Therefore, we offer to the public health policymakers a better understanding of COVID-19 management.

ASYMPTOTIC STABILITY ANALYSIS AND STABILIZATION CONTROL OF FRACTIONAL-ORDER VEHICLE SUSPENSION SYSTEM WITH TIME DELAY

Our study focuses on the analysis of asymptotic stability and the development of asymptotic stabilization control methods for fractional vehicle suspension systems (FVSS). We begin by constructing a mathematical model for FVSS using the state-space equations of Caputo fractional calculus. Initially, we utilize the fractional Routh–Hurwitz criterion to derive the necessary conditions for asymptotic stability and instability in the open-loop system of the fractional vehicle suspension. Subsequently, we propose a novel control strategy for FVSS and establish an associated asymptotic stabilization criterion by combining a new vector Lyapunov function with the [Formula: see text]-matrix method. Moreover, we extend the fractional-order vehicle suspension model to include time delay resulting from the interactions between different variables in the real system, thus creating a FVSS with time delay. Based on the vector Lyapunov function, [Formula: see text]-matrix measure, and Razumikhin interpretation, we develop a control strategy specifically tailored for FVSSs with time delay. Lastly, we compare two numerical simulations of the FVSS, one with time delay and one without, to demonstrate the accuracy, effectiveness, and applicability of the proposed method presented in our paper.

Stability of nonlinear population systems with individual scale and migration

&lt;abstract&gt;&lt;p&gt;In this paper, we study the stability of a nonlinear population system with a weighted total size of scale structure and migration in a polluted environment, where fertility and mortality depend on the density in different ways. We first prove the existence and uniqueness of the equilibrium point via a contraction mapping and give the expression for the equilibrium point. Some conditions for asymptotic stability and instability are presented by means of a characteristic equation. When the effect of density restriction on mortality is not considered, the threshold value of equilibrium stability can be obtained as $ \Lambda = 0. $ When $ \Lambda &amp;lt; 0, $ the equilibrium is asymptotically stable, and when $ \Lambda &amp;gt; 0, $ the equilibrium is unstable. In addition, the upwind difference method is used to discrete the model, and two examples are given to show the evolution of species.&lt;/p&gt;&lt;/abstract&gt;

On the stability of the zero solution with respect to a part of variables in linear approximation

The article presents the sufficient conditions for stability and asymptotic stability with respect to a part of the variables of the zero solution of a nonlinear system in the linear approximation. the case is considered when the matrix of the linear approximation may contain eigenvalues with zero real parts and the algebraic and geometric multiplicities of these eigenvalues may not coincide. The approach is based on establishing some correspondence between the solutions of the investigated system and its linear approximation. The solutions of such systems starting in a sufficiently small zero neighborhood and the systems themselves possess the same componentwise asymptotic properties in this case. Such solutions’ properties are stability and asymptotic stability with respect to some variables, and for systems componentwise local asymptotic equivalence and componentwise local asymptotic equilibrium. Considering the correspondence between the solutions of systems as an operator defined in a Banach space, there is proved that it has at least one fixed point according to the Schauder’s principle. The operator allows to construct a mapping that establishes the relationship between the initial points of the investigated system and its linear approximation. Further, a conclusion about the componentwise asymptotic properties of solutions of the nonlinear system is made on the basis of estimates of the fundamental matrix of the linear approximation rows’ entries. There is given an example of the investigation of stability and asymptotic stability with respect to a part of the variables of the zero solution of a nonlinear system is given, when the linear approximation matrix contains one negative and one zero eigenvalues, and the algebraic and geometric multiplicities of the zero eigenvalue do not coincide.

On the three dimensional competition systems arising from Arabidopsis clock

In this paper, we study mathematical models of three dimensional competitive systems originated from the represillator and their variants. There are two parts for the mathematical results. In part I, we first prove the uniqueness of the positive equilibrium $ (x^{*}, y^{*}, z^{*}) $. Then we present necessary and sufficient conditions for their local asymptotic stability and instability. In part II, we present sufficient conditions for the global asymptotic stability of $ (x^{*}, y^{*}, z^{*}) $ provided $ (x^{*}, y^{*}, z^{*}) $ is locally asymptotic stable.

Dynamic analysis of a mosquito population model with a stage structure and periodic releases of sterile males

&lt;abstract&gt;&lt;p&gt;This paper focuses on the key issues of mosquito population control, particularly exploring the impact of periodic releases of sterile males in the population model with a stage structure. We construct and analyze a model that includes only sexually active sterile mosquitoes in the dynamic interaction system. We focus on the system's dynamical behaviors under two scenarios: when the sexual lifespan $ \bar{T} $ equals the release period $ T $ of sterile mosquitoes, and when $ \bar{T} $ is less than $ T $. In the first scenario, we explore the existence and stability of equilibria, identifying a pivotal threshold $ m^* $ that determines the requisite release amount. In the second scenario, we convert the problem into an impulsive switched system and derive sufficient conditions for the local asymptotic stability of the extinction equilibrium. We also establish the existence of positive periodic solutions using the geometric method of differential equations and the fixed point theorem. Our conclusions show that the relationship between the sexual lifespan and release period of sterile mosquitoes significantly impacts the stability of the mosquito population. Additionally, our numerical simulations not only corroborate but they also complement our theoretical findings.&lt;/p&gt;&lt;/abstract&gt;

Stability and Hopf bifurcation of an epidemiological model with effect of delay the awareness programs and vaccination: analysis and simulation

The present research is concerned with studying the media coverage delay impact and vaccine impact to control outbreaks the infectious disease in the population, we suggested that our mathematical modeling divide the population into three classes: susceptible persons S(t), vaccinated person V(t) and infected persons I(t). The susceptible individuals are also divided due to media programs into awareness and unawareness of disease danger and its mode of transmission. We studied the influence of time delay in response to the media program on awareness of the epidemic. We first discussed the existence of the equilibrium points of the system and obtain sufficient conditions for the local asymptotic stability of all equilibrium points. Further, the existence of Hopf bifurcation is shown near endemic equilibrium. It is interesting to note that there exists at least one limit cycle around the unstable endemic equilibrium. In particular, sufficient conditions for a unique stable limit cycle have been presented. Finally, we verified the feasibility of the results by numerical simulation and give a brief conclusion to generalize that the delay effect of media and vaccination can cause unexpected dynamic predictions for interacting populations.

Design of robust fuzzy iterative learning control for nonlinear batch processes.

In this paper, a two-dimensional (2D) composite fuzzy iterative learning control (ILC) scheme for nonlinear batch processes is proposed. By employing the local-sector nonlinearity method, the nonlinear batch process is represented by a 2D uncertain T-S fuzzy model with non-repetitive disturbances. Then, the feedback control is integrated with the ILC scheme to be investigated under the constructed model. Sufficient conditions for robust asymptotic stability and 2D $ H_\infty $ performance requirements of the resulting closed-loop fuzzy system are established based on Lyapunov functions and some matrix transformation techniques. Furthermore, the corresponding controller gains can be derived from a set of linear matrix inequalities (LMIs). Finally, simulations on the three-tank system and the highly nonlinear continuous stirred tank reactor (CSTR) are carried out to prove the feasibility and efficiency of the proposed approach.

Dynamics of Bacterial white spot disease spreads in Litopenaeus Vannamei with time-varying delay.

In this paper, we mainly consider a eco-epidemiological predator-prey system where delay is time-varying to study the transmission dynamics of Bacterial white spot disease in Litopenaeus Vannamei, which will contribute to the sustainable development of shrimp. First, the permanence and the positiveness of solutions are given. Then, the conditions for the local asymptotic stability of the equilibriums are established. Next, the global asymptotic stability for the system around the positive equilibrium is gained by applying the functional differential equation theory and constructing a proper Lyapunov function. Last, some numerical examples verify the validity and feasibility of previous theoretical results.

Asymptotical Stability and Exponential Stability in Mean Square of Impulsive Stochastic Time-Varying Neural Network

The effect of impulse on stability of neural network is evident not only in performance, that is, impulsive control and impulsive interference. The amount of impulse has a certain impact on stability of neural network. Unlike traditional average impulsive interval, a new strategy is applied in this paper, namely, impulsive density. Based on this strategy, by constructing Lyapunov function, we establish sufficient conditions for mean square asymptotical stability of impulsive stochastic time-varying neural network without time delay. As well as, under this strategy and uniformly asymptotically stable function, by combining trajectory based approach and improved Razumikhin method, mean square exponential stability criterion of impulsive stochastic time-varying neural network with time delay is established. Finally, to demonstrate the viability of our theoretical findings, some instances are provided.

The stability of differential-difference systems with linearly increasing delay. II. Systems with additive right side

The article considers an uncontrolled system of differential-difference equations with a homogeneous additive right side and linearly increasing delay. Sufficient conditions for asymptotic stability are known for a number of special cases of such systems. Razumikhin's theorem on the asymptotic stability of homogeneous systems with proportional delay is formulated. Sufficient conditions for asymptotic stability are obtained basing on the asymptotic stability of the initial system without delay and constructing the Lyapunov function.

Data-Driven Output-Feedback Control for Unknown Switched Linear Systems

This paper proposes a data-driven control method to stabilize unknown switched linear systems under arbitrary switching. We consider the case where the system state is not measurable and design an output feedback controller only using measured input-output data. First, the system with multiple outputs is transformed into a single-output system with observability preserved. Then, a data-based state-space representation that has the same input-output relationship as the original system is constructed using the input-output data of the single-output system, based on which the data-driven controller is designed. Sufficient conditions for asymptotic stability of the closed-loop system under arbitrary switching are established in terms of linear matrix inequalities (LMIs). Compared with the existing method, the proposed method decreases the dimension of the constructed data-based state-space representation, which may reduce the computational burden of the controller design. The effectiveness of the proposed controller is illustrated by an example.

EXPONENTIAL STABILITY IN DELAY LOTKA-VOLTERRA MODELS WITH SMALL INTERACTIONS BETWEEN SPECIES

The linearisation of the Lotka-Volterra system with time delay is considered. Any type of interaction between two spices is under consideration. The sufficient conditions of asymptotic stability for this linear system are obtained

Involuntary Stabilization in Discrete-Event Physical Human–Robot Interaction

Robots are used by humans not only as tools but also to interactively assist and cooperate with humans, thereby forming physical human-robot interactions. In these interactions, there is a risk that a feedback loop causes unstable force interaction, in which force escalation exposes a human to danger. Previous studies have analyzed the stability of voluntary interaction but have neglected involuntary behavior in the interaction. In contrast to the previous studies, this study considered the involuntary behavior: a human's force reproduction bias for discrete-event human-robot force interaction. We derived an asymptotic stability condition based on a mathematical bias model and found that the bias asymptotically stabilizes a human's implicit equilibrium point far from the implicit equilibrium point and destabilizes the point near the point. The bias model, convergence of the interaction toward the implicit equilibrium point, and divergence around the point were consistently verified via behavioral experiments under three kinds of interactions using three different body parts: a hand finger, wrist, and foot. Our results imply that humans implicitly secure a stable and close relationship between themselves and robots with their involuntary behavior.