Global Asymptotic Stability Research Articles

The Stabilization of a Nonlinear Permanent-Magnet- Synchronous-Generator-Based Wind Energy Conversion System via Coupling-Memory-Sampled Data Control with a Membership-Function-Dependent H∞ Approach

This study presents the coupling-memory-sampled data control (CMSDC) design for the Takagi–Sugeno (T-S) fuzzy system that solves the stabilization issue of a surface-mounted permanent-magnet synchronous generator (PMSG)-based wind energy conversion system (WECS). A fuzzy CMSDC scheme that includes the sampled data control (SDC) and memory-sampled data control (MSDC) is designed by employing a Bernoulli distribution order. Meanwhile, the membership-function-dependent (MFD) H∞ performance index is presented, mitigating the continuous-time fuzzy system’s disturbances. Then, by using the Lyapunov–Krasovskii functional with the MFD H∞ performance index, the data of the sampling pattern, and a constant signal transmission delay, sufficient conditions are derived. These sufficient conditions are linear matrix inequalities (LMIs), ensuring the global asymptotic stability of a PMSG-based WECS under the designed control technique. The proposed method is demonstrated by a numerical simulation implemented on the PMSG-based WECS. Finally, Rossler’s system demonstrates the effectiveness and superiority of the proposed method.

Global stability and optimal control of an age-structured SVEIR epidemic model with waning immunity and relapses.

The efficacy of vaccination, incomplete treatment and disease relapse are critical challenges that must be faced to prevent and control the spread of infectious diseases. Age heterogeneity is also a crucial factor for this study. In this paper, we investigate a new age-structured SVEIR epidemic model with the nonlinear incidence rate, waning immunity, incomplete treatment and relapse. Next, the asymptotic smoothness, the uniform persistence and the existence of interior global attractor of the solution semi-flow generated by the system are given. We define the basic reproduction number and prove the existence of the equilibria of the model. And we study the global asymptotic stability of the equilibria. Then the parameters of the model are estimated using tuberculosis data in China. The sensitivity analysis of is derived by the Partial Rank Correlation Coefficient method. These main theoretical results are applied to analyze and predict the trend of tuberculosis prevalence in China. Finally, the optimal control problem of the model is discussed. We choose to take strengthening treatment and controlling relapse as the control parameters. The necessary condition for optimal control is established.

Neimark–Sacker bifurcation of two second-order rational difference equations

We investigate the Neimark–Sacker bifurcation of the equilibrium of two special cases of the difference Equation x n + 1 = β x n x n − 1 + γ x n − 1 2 + δ x n B x n x n − 1 + C x n − 1 2 + D x n where the parameters β, γ, δ, B, C, D are non-negative numbers which satisfy B + C + D>0 and the initial conditions x − 1 and x 0 are arbitrary nonnegative numbers such that B x n x n − 1 + C x n − 1 2 + D x n > 0 for all n ≥ 0 . More precisely, we consider special cases where either γ = D = 0 or β = D = 0 . As we will show both equations exhibit Neimark–Sacker bifurcation, where one of equations ( γ = D = 0 ) probably exhibits Chenciner bifurcation, with two invariant curves, while another Equation ( β = D = 0 ) exhibits simple Neimark–Sacker bifurcation with one invariant curve. We will also obtain some global asymptotic stability result for each equation in the subset of the parametric region of local stability.

Constrained model predictive control for networked jump systems under denial‐of‐service attacks and time delays

AbstractThis article addresses the problem of model predictive control of networked jump systems in the presence of DoS attacks and time delays. In the structural framework of the network predictive control system, we mathematically model the networked jump system by using Markov chains to describe the time delays and a polytope model to describe the jump system phenomenon, considering the properties of DoS attacks and time delays. Based on this, we propose a strategy to lessen the effect of network constraints on the control performance of the system. This strategy involves the corresponding control inputs from the control sequence for real‐time active compensation. It includes adjusting the control sequence application length variation based on the duration of the DoS attacks and time delays at each moment. In addition, we demonstrate the recursive feasibility of the control strategy and the global asymptotic stability of the control system from a theoretical perspective through the Lyapunov stability theory. Finally, the effectiveness of the proposed strategy is verified by simulation arithmetic.

Identifying numerical bifurcation structures of codimensions 1 and 2 in interacting species system

The present study has focused on the examination of a Holling type III nonlinear mathematical model that incorporates the influence of fear and Michaelis–Menten‐type predator harvesting. The incorporation of fear with respect to the prey species has been observed to result in a reduction in the survival probability of the prey population and a concurrent reduction in the reproduction rate of the prey species. The existence and stability of ecologically significant equilibria have been ascertained through mathematical analysis. Emphasis within the proposed model primarily centers on numerical bifurcations of codimensions 1 and 2. Numerical validation has been performed on all simulated outcomes within the feasible range of parametric values. Dynamical characteristics of the model have subsequently undergone investigation through a series of numerical simulations, successfully revealing various forms of local and global bifurcations. In addition to the identification of saddle‐node, Hopf, Bogdanov–Takens, transcritical, cusp, homoclinic, and limit point cycle (LPC) bifurcations, the model has also demonstrated bistability and global asymptotic stability. These bifurcation phenomena serve as illustrative examples of the intricate dynamical behavior inherent to the model. Numerical validation through graphical representations has been utilized to elucidate the effects of factors such as fear, nonlinear predator harvesting, and predation rate on the dynamics of the interacting species under different parametric conditions.

Dynamic analysis of a rumor propagation model based on a familiarity mechanism and refuters

The spread of rumors not only causes massive economic losses, but also seriously threatens social harmony and stability of human livelihoods. As a result, preventing rumors from spreading is critical. Individuals' familiarity with rumors and refuters has a significant impact on their spread. We develop a rumor propagation model that introduces familiar mechanisms and refuters, as well as demonstrating the existence of an equilibrium point and its local and global asymptotic stability in the model. We also calculate the model's basic regeneration number. Subsequently, we select two appropriate control variables: the contact rate between the ignorant, who are familiar with the rumor, and the disseminators; and the probability that the disseminators will become refuters. We discover that the control effect is optimal when the first variable is minimum and the second variable is maximum. Numerical simulation verifies the accuracy of the obtained results, and sensitivity analysis of the parameters is investigated. According to the research, an increase in the proportion of disseminators who become refuters will slow the spread of rumors, while an increase in the population of ignorant people who are familiar with rumors contacting the disseminators will accelerate the spread of rumors.

Robust adaptive control of delayed systems with multiple uncertainties and polynomial nonlinearities

This work discusses the issue of robust adaptive control for delayed systems with multiple uncertain factors. The system uncertainties cover uncertain dynamics, unknown system parameters, and external disturbances. Besides, the growing conditions have both low-order and high-order polynomial nonlinearities, which bring in numerous challenges for control design. The suitable transformations are first introduced to obtain a new system with a dynamic gain. Then, the control design of an auxiliary system is considered. By introducing the Lyapunov–Krasovskii (L–K) functional method, and providing an important result for an auxiliary system, the adaptive homogeneous control strategy is proposed to achieve global asymptotic stability (GAS) of the entire system. Besides, by considering the external disturbances, the method is also extended to design robust adaptive tracking controller. Numerical and practical examples are provided to verify the theory.

Asymptotic stability results of generalized discrete time reaction diffusion system applied to Lengyel-Epstein and Dagn Harrison models

In this research paper, we delve into the analysis of a generalized discrete reaction-diffusion system. Our study begins with the discretization of a generalized reaction-diffusion model, achieved through second-order and L1-difference approximations. We explore the local stability of its unique solution, both in the absence and presence of the diffusion term. To determine the conditions for global asymptotic stability of the steady-state solution, we employ suitable techniques including the direct Lyapunov method. To illustrate the practical application of this theoretical framework, we provide several numerical simulations that examine both the Lengyel-Epstein reaction-diffusion model and the discrete Degn-Harrison reaction-diffusion model. These simulations serve to validate the predictions of asymptotic stability.

Neural Network Energy Management-Based Nonlinear Control of a DC Micro-Grid with Integrating Renewable Energies

The broad acceptance of sustainable and renewable energy sources as a means of integrating them into electrical power networks is essential to promote sustainable development. Microgrids using direct currents (DCs) are becoming more and more popular because of their great energy efficiency and straightforward design. In this work, we discuss the control of a PV-based renewable energy system and a battery- and supercapacitor-based energy storage system in a DC microgrid. We describe a hierarchical control approach based on sliding-mode controllers and the Lyapunov stability theory. To balance the load and generation, a fuzzy logic-based energy management system has been created. Using a neural network, maximum power defects for the PV system were determined. The global asymptotic stability of the framework has been verified using Lyapunov stability analysis. In order to simulate the proposed DC microgrid and controllers, MATLAB/SimulinkR (2019a) was utilized. The outcomes show that the system operates effectively with changing production and consumption.

Lyapunov functionals for a virus dynamic model with general monotonic incidence, two time delays, CTL and antibody immune responses

In this paper, we study global asymptotic stability of all equilibria of a virus dynamic model with general monotonic incidence, two time delays, CTL and antibody immune responses by constructing Lyapunov functionals and applying LaSalle’s invariance principle.

Dynamics of reaction–diffusion–advection system and its impact on river ecology in the presence of spatial heterogeneity

This study considers a two-species reaction–diffusion–advection (RDA) model in a heterogeneous advective environment with zero Neumann boundary conditions. We suppose that two species compete for the same food resource but have different diffusion and advection rates. The primary objective of this study is to investigate the global asymptotic stability and coexistence of steady state based on different and unequal diffusion and advection rates through theoretical and numerical analysis. We establish the local stability of two semi-trivial steady states of the competing species. Also, the non-existence of coexistence steady state is proved with the help of some non-trivial assumptions. Finally, we show the global stability with the help of monotone dynamical systems and combine the local stability and non-existence of coexistence steady state. If one species adopts a smaller diffusion and advection rate, the competition’s result depends on the advection and diffusion rate ratio. If the ratio is smaller, then the species will prevail. Also, the species with higher diffusion and smaller advection will win. Lastly, the effectiveness of the model in one and two-dimensional situations is shown by a series of numerical calculations, which is particularly important for environmental consideration.

Design Principles for Perfect Adaptation in Biological Networks with Nonlinear Dynamics.

Establishing a mapping between the emergent biological properties and the repository of network structures has been of great relevance in systems and synthetic biology. Adaptation is one such biological property of paramount importance that promotes regulation in the presence of environmental disturbances. This paper presents a nonlinear systems theory-driven framework to identify the design principles for perfect adaptation with respect to external disturbances of arbitrary magnitude. Based on the prior information about the network, we frame precise mathematical conditions for adaptation using nonlinear systems theory. We first deduce the mathematical conditions for perfect adaptation for constant input disturbances. Subsequently, we translate these conditions to specific necessary structural requirements for adaptation in networks of small size and then extend to argue that there exist only two classes of architectures for a network of any size that can provide local adaptation in the entire state space, namely, incoherent feed-forward (IFF) structure and negative feedback loop with buffer node (NFB). The additional positiveness constraints further narrow the admissible set of network structures. This also aids in establishing the global asymptotic stability for the steady state given a constant input disturbance. The proposed method does not assume any explicit knowledge of the underlying rate kinetics, barring some minimal assumptions. Finally, we also discuss the infeasibility of certain IFF networks in providing adaptation in the presence of downstream connections. Moreover, we propose a generic and novel algorithm based on non-linear systems theory to unravel the design principles for global adaptation. Detailed and extensive simulation studies corroborate the theoretical findings.

Stability analysis of a fractional-order [formula omitted] epidemic model for the COVID-19 pandemic

To explore the impact of the COVID-19 vaccination rate and immune loss rate on the pandemic, this paper proposes a fractional-order SIV1V2S epidemic model with vaccination ineffectiveness and infection differences. And we compare and analyze the dynamic differences between integer-order and Caputo fractional-order operators. First, we show the non-negativity and boundedness of solutions for the Caputo fractional-order model. Based on the basic reproduction number, we use the fractional-order matrix eigenvalue method and the Routh–Hurwitz criterion to determine the local asymptotic stability of disease-free equilibrium points and endemic equilibrium points, construct Lyapunov functions, and use the fractional-order Barbalat’s Lemma to prove the global asymptotic stability of the two equilibrium points. Second, real data from India were used to fit the model parameters and get the optimal fractional-order parameter, and then sensitivity analysis was performed on the basic reproduction number. Further, through numerical simulation of different fractional-order parameter epidemic models, we verify the global asymptotic stability of the equilibrium point of the fractional-order SIV1V2S COVID-19 model. The comparison shows that the fractional-order model is more closely related to actual virus transmission than the integer-order model. In the case of the SIV1V2S model, which only considers vaccination measures, simultaneously increasing the vaccination rate of susceptible individuals and reducing the rate of immunization loss can effectively control epidemic transmission.

A simple approach for studying stability properties of an SEIRS epidemic model

Abstract In this work, we study stability properties of a well-known integer-order SEIRS model with nonlinear incidence and vertical transmission. Firstly, we introduce a simple approach to the analysis of global asymptotic stability (GAS) of the integer-order model. This approach is based on general quadratic Lyapunov functions and characteristic of quadratic forms associated with real matrices. The result is that the GAS of disease-free and disease-endemic equilibrium points is completely established. This provides an important improvement for results constructed in two previous works. Secondly, we generalize the integer-order SEIRS model by considering it in the context of the Caputo fractional-order derivative. After that, the present approach is utilized to investigate the GAS of the proposed fractional-order model. As an important consequence, not only the GAS but also the uniform stability of the fractional-order model are determined fully. Therefore, the applicability of the approach is shown. Finally, a series of numerical experiments is conducted to illustrate and support the theoretical findings.

Dynamic modeling of the glucose–insulin system with inhibitors impulsive control

Pulse injection of insulin analogues is an important strategy to control glucose concentrations and can be combined with ‐glucosidase inhibitors to reduce adverse effects to improve glucose control. To elucidate this combination therapy strategy, we assumed dietary intake in the form of pulse glucose injection and proposed a novel mathematical model incorporating pulse injection insulin and ‐glucosidase inhibitors. In type 1 diabetes, the existence and uniqueness of the positive periodic solution is confirmed utilizing the Lambert W function. The global asymptotic stability of the positive periodic solution is achieved through the application of Floquet multiplier theory and the comparison principle. Furthermore, in type 2 diabetes, the permanence of the system is also confirmed through the comparison theorem. Numerical analysis validated the theoretical calculations, highlighting the significance of insulin injection dosage and frequency, as well as ‐glucosidase inhibitor therapy. Additionally, we systematically assessed a rational approach for diabetes treatment combined with ‐glucosidase inhibitors, providing more practical clinical strategies. This facilitates the extension of our model to encompass other drugs that may be utilized in future clinical interventions.

A Mathematical Model for Differentiated Thyroid Cancer with Combination Therapy

Objectives: Differentiated thyroid cancer (DTC) is usually treated with surgery and radioactive iodine therapy. However, in advanced stages, molecular changes can lead to a progressive loss of sensitivity to iodine, making the cancer resistant to radioactive iodine treatment (known as radioactive iodine-refractory, or RAIR). For cases of RAIR-DTC, alternative treatments such as tyrosine kinase inhibitors and immunotherapy are explored. To better understand the dynamics of these alternative treatments, a new mathematical model has been developed. In this model, Sorafenib represents the tyrosine kinase inhibitor, while Pembrolizumab is the immunotherapy agent under investigation. Methods: Two mathematical models were developed based on systems of ordinary differential equations to study differentiated thyroid cancer treated with (1) Sorafenib alone and (2) a combination of a Sorafenib and Pembrolizumab. The first model included three variables: drug concentration of Sorafenib, number of cancer cells, and number of immune cells with fourteen parameters. The second model additionally incorporated the concentration of the Pembrolizumab. Parameter values were estimated through simulations utilizing the fmincon optimization technique in MATLAB. Local asymptotic stability analysis was performed to investigate the models' behavior around equilibrium points. Moreover, the Lyapunov method was employed to establish global asymptotic stability of the models. Findings: The findings revealed insights into the dynamics of treating RAIR-DTC patients with tyrosine kinase inhibitors and immunotherapy. The stability analysis contributed to the understanding of the system's behavior under different conditions, offering valuable information for clinical application. Numerical simulations were then performed using Simbiology in MATLAB to simulate the treatment dynamics over time. Novelty: This research contributes to the field by presenting a novel mathematical model that integrates tyrosine kinase inhibitors and immunotherapy for the treatment of RAIR-DTC. The comprehensive analysis of equation stability enhances the reliability of the model, while the numerical simulations provide a practical and visual tool for assessing the potential outcomes of the proposed therapeutic strategy. This study offers a valuable framework for further exploration and refinement of treatment approaches for patients with challenging DTC cases. Keywords: Mathematical modeling, Routh Hurwitz Criteria, Jacobian, Differentiated thyroid Cancer, Stability Analysis

Dynamic behavior of a stochastic HIV model with latent infection and Ornstein–Uhlenbeck process

HIV spreads in the body through two infection patterns: virus-to-cell (VTC) infection and cell-to-cell (CTC) transmission. This study introduces an HIV dynamic model incorporating both transmission patterns, where CTC transmission occurs when healthy CD4+T cells come into contact with latently and actively infected cells. The research first analyzes the local asymptotic stability of the disease-free equilibrium and the global asymptotic stability of the endemic equilibrium in the deterministic model. Subsequently, a corresponding stochastic HIV model is developed by introducing log-normal Ornstein–Uhlenbeck (OU) process to perturb the infection rates. The study establishes the conditions for the extinction of HIV infection in the stochastic system and examines the existence of an ergodic stationary distribution by constructing a series of stochastic Lyapunov functions. Specifically, the paper proposes the explicit expression of the probability density function for the linearized stochastic system near the quasi-equilibrium when all infected cells transform into latently infected cells. Finally, the theoretical conclusions are validated through numerical simulations.

Dynamics of simplicial SEIRS epidemic model: global asymptotic stability and neural Lyapunov functions.

The transmission of infectious diseases on a particular network is ubiquitous in the physical world. Here, we investigate the transmission mechanism of infectious diseases with an incubation period using a networked compartment model that contains simplicial interactions, a typical high-order structure. We establish a simplicial SEIRS model and find that the proportion of infected individuals in equilibrium increases due to the many-body connections, regardless of the type of connections used. We analyze the dynamics of the established model, including existence and local asymptotic stability, and highlight differences from existing models. Significantly, we demonstrate global asymptotic stability using the neural Lyapunov function, a machine learning technique, with both numerical simulations and rigorous analytical arguments. We believe that our model owns the potential to provide valuable insights into transmission mechanisms of infectious diseases on high-order network structures, and that our approach and theory of using neural Lyapunov functions to validate model asymptotic stability can significantly advance investigations on complex dynamics of infectious disease.

Forward completeness does not imply bounded reachability sets and global asymptotic stability is not necessarily uniform for time-delay systems

An example of a time-invariant time-delay system that is uniformly globally attractive and exponentially stable, hence forward complete, but whose reachability sets from bounded initial conditions are not bounded over compact time intervals is provided. This gives a negative answer to two current conjectures by showing that (i) forward completeness is not equivalent to robust forward completeness (i.e. boundedness of reachability sets) and (ii) global asymptotic stability is not equivalent to uniform global asymptotic stability. In addition, a novel characterization of robust forward completeness for systems having a finite number of discrete delays is provided. This characterization relates robust forward completeness of the time-delay system with the forward completeness of an associated nondelayed finite-dimensional system.

Geometric tracking on 𝒮3 based on sliding mode control

AbstractAttitude tracking on the unit sphere of dimension 3 based on sliding mode is considered in this article. The tangent bundle of Lagrangian dynamics that describes the rotational motion of a rigid body is first equipped with the structure of a Lie group, then a sliding subgroup emerged on it is defined. Next, a sliding‐mode controller is designed for attitude tracking that relies on an intrinsic error defined on the Lie group. Almost global asymptotic stability of the closed loop is demonstrated using the Lyapunov analysis. Comparisons of the proposed geometric sliding mode controller designed on the nonlinear manifold with one designed in the Euclidean space showed that under the same working conditions the former is more energy efficient, more robust to measurement noises, and has a shorter and smoother transient response.