Construction Of Lyapunov Functions Research Articles

Stabilization of a class of switched systems with state constraints via time-varying Lyapunov functions

This paper studies the stabilization problem of a class of continuous-time switched linear systems with state constraints under pre-specified dwell-time switchings. Such systems are defined on a closed hypercube as all state variables are constrained to the unit hypercube. The dwell time in this paper is an arbitrarily pre-specified rather than a calculated constant, which is independent of any parameters. First, a class of multiple time-varying Lyapunov functions is introduced to study the stability analysis, and sufficient conditions on stability of the studied switched systems without control input are derived in the framework of pre-specified dwell-time switchings. The distinguishing feature of the proposed Lyapunov functions is that this type of delicately constructed Lyapunov functions can efficiently eliminate the “jump” phenomena of adjacent Lyapunov functions at switching times. Second in the same framework of the dwell time, sufficient conditions for stabilization are proposed for that of the switched systems with state constraints by further designing state feedback controllers. Finally, two examples are provided to demonstrate the effectiveness of the proposed results. The results of this paper do not require to calculate the total time of each subsystem during which the state is saturated or non-saturated separately, which makes the pre-specified dwell-time switchings easy to apply.

Long time behavior of a degenerate NPZ model with spatial heterogeneity

This work proposes a nutrient–phytoplankton–zooplankton (NPZ) model which is governed by a partially degenerate reaction–diffusion equation. Due to the noncompactness of solution map, the Kuratowski measure is introduced to prove the existence of the global attractor. By an innovative construction of Lyapunov functions, sufficient conditions are given to obtain the global attractivity of steady states of the system with spatial parameters.

Vertex reinforced random walks with exponential interaction on complete graphs

We describe a model for m vertex reinforced interacting random walks on complete graphs with d≥2 vertices. The transition probability of a random walk to a given vertex depends exponentially on the proportion of visits made by all walks to that vertex. The individual proportion of visits is modulated by a strength parameter that can be set equal to any real number. This model covers a large variety of interactions including different vertex repulsion and attraction strengths between any two random walks as well as self-reinforced interactions. We show that the process of empirical vertex occupation measures defined by the interacting random walks converges (a.s.) to the limit set of the flow induced by a smooth vector field. Further, if the set of equilibria of the field is formed by isolated points, then the vertex occupation measures converge (a.s.) to an equilibrium of the field. These facts are shown by means of the construction of a strict Lyapunov function. We show that if the absolute value of the interaction strength parameters are smaller than a certain upper bound, then, for any number of random walks (m≥2) on any graph (d≥2), the vertex occupation measures converge towards a unique equilibrium. We provide two additional examples of repelling random walks for the cases m=d=2 and m∈{3,4,5} with d=2. The latter is used to study some properties of three, four and five exponentially repelling random walks on Z.

Modeling of COVID-19 spread with self-isolation at home and hospitalized classes

This work examines the impacts of self-isolation and hospitalization on the population dynamics of the Corona-Virus Disease. We developed a new nonlinear deterministic model eight classes compartment, with self-isolation and hospitalized being the most effective tools. There are (Susceptible SC(t), Exposed E(t), Asymptomatic infected IA(t), Symptomatic infected AS(t), Self-isolation TM(t), Hospitalized TH(t), Healed H(t), and Susceptible individuals previously infected HC(t)). The expression of basic reproduction number R0 comes from the next-generation matrix method. With suitably constructed Lyapunov functions, the global asymptotic stability of the non-endemic equilibria Σ0 for R0<1 and that of endemic equilibria Σ∗ for R0>1 are established. The computed value of R0=3.120277403 proves the endemic level of the epidemic. The outbreak will lessen if a policy is enforced like self-isolation and hospitalization. This is related to those policies that can reduce the number of direct contacts between infected and susceptible individuals or waning immunity individuals. Various simulations are presented to appreciate self-isolation at home and hospitalized strategies if applied sensibly. By performing a global sensitivity analysis, we can obtain parameter values that affect the model through a combination of Latin Hypercube Sampling and Partial Rating Correlation Coefficients methods to determine the parameters that affect the number of reproductions and the increase in the number of COVID cases. The results obtained show that the rate of self-isolation at home and the rate of hospitalism have a negative relationship. On the other hand, infections will decrease when the two parameters increase. From the sensitivity of the results, we formulate a control model using optimal control theory by considering two control variables. The result shows that the control strategies minimize the spread of the COVID infection in the population.

Mathematical Analysis of Efficacy of Condom as a Contraceptive on the Transmission of Chlamydia Disease

Chlamydia disease caused by the Chlamydia trachomatis is one of the major sexually transmitted infectious diseases globally. The progression of this disease has deadly effects cumulating into millions of death. Chlamydia causes numerous complications such as infertility in female, chronic pelvic pain and inflammation. Previous studies did not consider the effects of undetected infected individuals on the dynamic spread of the disease. Hence, this work investigated the effects of undetected infected individuals and efficacy of condom as a contraceptive in the dynamic spread of the disease. A six compartmental model to study the dynamic spread of chlamydia disease was developed using system of ordinary differential equation. The population was divided into susceptible, exposed, infected undetected symptomatic, infected detected symptomatic, infected asymptomatic and recovered individuals. The well-posedness of the model was investigated by the positivity of solution technique. Basic reproduction number (R0) was computed using Next Generation Matrix Method. The endemic equilibrium was investigated by the use of Lyapunov function. The results showed that the disease free equilibrium was stable whenever the basic reproduction number is less than unity (R0&lt;1). The endemic equilibrium was found to be stable as a results of constructed Lyapunov function being negative definite. Numerical analysis showed that increasing in the undetected infected individuals enhanced the spread of the disease. Findings showed that undetected infected individuals played a vital role in the spread of Chlamydia disease. Moreover, using condom as a contraceptive reduced the spread of the disease.

Construction of Lyapunov Functions for the Stability of Fifth Order Nonlinear Ordinary Differential Equations

This study employed Lyapunov function method to examine the stability of nonlinear ordinary differential equations. Using direct Lyapunov method, we constructed Lyapunov function to investigate the stability of fifth order nonlinear ordinary differential equations. V(x), a quadratic form and positive definite and U(x) which is also positive definite was chosen such that the derivative of V(x) with respect to time would be equal to the negative value of U(x). We adopted the pre-multiplication of the given equation by x..... and obtained a Lyapunov function which established local and global stability of a fifth order differential equation.

Fixed-time stabilization of IT-2 T-S fuzzy control systems with discontinuous interconnections: Indefinite derivative Lyapunov method

Using the interval type-2 Takagi–Sugeno (IT-2 T-S) fuzzy control method, this paper formulates a class of non-autonomous interconnected dynamical system (IDS) with discontinuities. Under the differential inclusion (DI) framework, the fixed-time stabilization (FXTS) problem is studied via indefinite derivative Lyapunov approach, where the time-derivative of constructed Lyapunov function doesn’t have to be negative/semi-negative. By designing novel IT-2 T-S fuzzy switching control protocol possessing time-varying control gain coefficients, several sufficient stabilization conditions are obtained to determine the system’s stability in fixed time. Furthermore, the settling time (ST) of FXTS is estimated. Due to the time-varying property of control gain coefficients and indefiniteness of system’s parameters, the advantage of the IT-2 T-S fuzzy switching control protocol designed in this paper is that its control gain coefficients are not only more flexible, but also can affect the estimation of ST. Finally, the designed control protocols and FXTS results are confirmed by numerical example.

Synchronization of fractional-order gene regulatory networks mediated by miRNA with time delays and unknown parameters

To investigate the regulatory function of miRNA on gene expression at the post-transcriptional level, the synchronization of fractional-order gene regulatory networks (FOGRNs) mediated by miRNA with time delays and unknown parameters is explored. First, a more realistic gene regulatory network model mediated by miRNA with leakage, time-varying delays and unknown parameters is established. Second, the feedback controllers and parameter adaptive laws are designed to induce synchronization and identify unknown parameters. Third, some synchronization conditions are presented and the unknown parameters are accurately identified by using the designed feedback controllers, the parameter adaptive laws and the constructed Lyapunov function. It is found that the smaller the value of the fractional order, the longer it takes to synchronize, and it takes more time to complete the identification of the unknown parameters. Finally, MATLAB simulations are given to verify the theoretical results.

Finite-time stabilization of complex-valued neural networks with proportional delays and inertial terms: A non-separation approach

This article mainly dedicates on the issue of finite-time stabilization of complex-valued neural networks with proportional delays and inertial terms via directly constructing Lyapunov functions without separating the original complex-valued neural networks into two real-valued subsystems equivalently. First of all, in order to facilitate the analysis of the second-order derivative caused by the inertial term, two intermediate variables are introduced to transfer complex-valued inertial neural networks (CVINNs) into the first-order differential equation form. Then, under the finite-time stability theory, some new criteria with less conservativeness are established to ensure the finite-time stabilizability of CVINNs by a newly designed complex-valued feedback controller. In addition, for reducing expenses of the control, an adaptive control strategy is also proposed to achieve the finite-time stabilization of CVINNs. At last, numerical examples are given to demonstrate the validity of the derived results.

Secure filtering under adaptive event-triggering protocols with memory mechanisms

The paper investigates secure filtering of nonlinear large-scale systems suffering from randomly occurring DoS attacks. By introducing an adjustable parameter, an adaptive event-triggering mechanism is proposed for the sake of decreasing the transmission burden of signals, where the memory is utilized to reflect the influence of past triggered information. The main objective is to design an event-based secure filter to ensure that the dynamics of filtering errors is input-to-state stable in the mean square. Using the constructed Lyapunov function, a sufficient condition is derived where some element matrix inequalities are utilized to handle the inherent coupling of subsystems. Furthermore, the desired filter gains are parameterized by resorting to the feasibility of matrix inequalities. Finally, a numerical simulation about a power system is provided to verify the effectiveness of the developed secure filtering algorithm.

Ergodic stationary distribution of a stochastic rumor propagation model with general incidence function

In daily lives, when emergencies occur, rumors will spread widely on the internet. However, it is quite difficult for the netizens to distinguish the truth of the information. The main reasons are the uncertainty of netizens’ behavior and attitude, which make the transmission rates of these information among social network groups be not fixed. In this paper, we propose a stochastic rumor propagation model with general incidence function. The model can be described by a stochastic differential equation. Applying the Khasminskii method via a suitable construction of Lyapunov function, we first prove the existence of a unique solution for the stochastic model with probability one. Then we show the existence of a unique ergodic stationary distribution of the rumor model, which exhibits the ergodicity. We also provide some numerical simulations to support our theoretical results. The numerical results give us some possible methods to control rumor propagation. Firstly, increasing noise intensity can effectively reduce rumor propagation when . That is, after rumors spread widely on social network platforms, government intervention and authoritative media coverage will interfere with netizens’ opinions, thus reducing the degree of rumor propagation. Secondly, speed up the rumor refutation, intensify efforts to refute rumors, and improve the scientific quality of netizen (i.e., increase the value of β and decrease the value of α and γ), which can effectively curb the rumor propagation.

Lyapunov functionals for linear neutral delay systems in the Hale's form

We construct Lyapunov functionals for linear neutral delay systems in the Hale's form. Our construction is made for no differentiable initial functions, as required for such a class of systems, but not thus considered in the existing works. The functionals depend on a special matrix-valued function which is the natural extension of the Lyapunov matrix for retarded delay systems. The constructed functionals allow deriving a converse result for exponential stability of the systems and exponential estimates for the solutions.

Convergence conditions for continuous and discrete models of population dynamics

Some classes of continuous and discrete generalized Volterra models of population dynamics are considered. It is supposed that there are relationships of the type "symbiosis", "compensationism" or "neutralism" between any two species in a biological community. The objective of the work is to obtain conditions under which the investigated models possess the convergence property. This means that the studying system admits a bounded solution that is globally asimptotically stable. To determine the required conditions, the V. I. Zubov's approach and its discrete-time counterpart are used. Constructions of Lyapunov functions are proposed, and with the aid of these functions, the convergence problem for the considered models is reduced to the problem of the existence of positive solutions for some systems of linear algebraic inequalities. In the case where parameters of models are almost periodic functions, the fulfilment of the derived conditions implies that limiting bounded solutions are almost periodic, as well. An example is presented illustrating the obtained theoretical conclusions.

Controller Synthesis of Asynchronous Periodic Event-Triggered Networked Control Systems Subject to Quantization Effects

This paper aims at analyzing the input-to-state stability (ISS) of the networked control systems (NCSs) where the sensor-to-controller and the controller-to-actuator channels are equipped with different dynamic quantizers and event-triggering laws. In the scenarios of quantization effects and external disturbances, we propose to use an asynchronous periodic event-triggering mechanism (PETM) and an improved dynamic quantization scheme. The PETM has potential to reduce the network transmissions compared with the time-based one on the one hand, and is natural for the practical applications (e.g. the digital platforms) on the other hand. The asynchronous PETM, dynamic quantization effects and external disturbances are incorporated by a hybrid dynamical framework. With the help of a technique combining the hybrid system theory and a novel constructed Lyapunov function, the co-design problem of asynchronous periodic event-triggering conditions, dynamic output-feedback control and dynamic quantizer is firstly addressed thereby ensuring the ISS of the NCSs. Moreover, the proposed controller has an intuitive and explicit form, which can be solved easily. To assess the feasibility of the proposed method, a simulation is provided.

Anti-periodic synchronization of quaternion-valued high-order Hopfield neural networks with delays

&lt;abstract&gt;&lt;p&gt;This paper proposes a class of quaternion-valued high-order Hopfield neural networks with delays. By using the non-decomposition method, non-reduced order method, analytical techniques in uniform convergence functions sequence, and constructing Lyapunov function, we obtain several sufficient conditions for the existence and global exponential synchronization of anti-periodic solutions for delayed quaternion-valued high-order Hopfield neural networks. Finally, an example and its numerical simulations are given to support the proposed approach. Our results play an important role in designing inertial neural networks.&lt;/p&gt;&lt;/abstract&gt;

Dynamic Behavior of a Stochastic Tungiasis Model for Public Health Education

In this paper, we study the dynamic behavior of a stochastic tungiasis model for public health education. First, the existence and uniqueness of global positive solution of stochastic models are proved. Secondly, by constructing Lyapunov function and using It formula, sufficient conditions for disease extinction and persistence in the stochastic model are proved. Thirdly, under the condition of disease persistence, the existence and uniqueness of an ergodic stationary distribution of the model is obtained. Finally, the importance of public health education in preventing the spread of tungiasis is illustrated through the combination of theoretical results and numerical simulation.

Stability analysis of SARS-CoV-2/HTLV-I coinfection dynamics model

&lt;abstract&gt;&lt;p&gt;Although some patients with coronavirus disease 2019 (COVID-19) develop only mild symptoms, fatal complications have been observed among those with underlying diseases. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the causative of COVID-19. Human T-cell lymphotropic virus type-I (HTLV-I) infection can weaken the immune system even in asymptomatic carriers. The objective of the present study is to formulate a new mathematical model to describe the co-dynamics of SARS-CoV-2 and HTLV-I in a host. We first investigate the properties of the model's solutions, and then we calculate all equilibria and study their global stability. The global asymptotic stability is examined by constructing Lyapunov functions. The analytical findings are supported via numerical simulation. Comparison between the solutions of the SARS-CoV-2 mono-infection model and SARS-CoV-2/HTLV-I coinfection model is given. Our proposed model suggest that the presence of HTLV-I suppresses the immune response, enhances the SARS-CoV-2 infection and, consequently, may increase the risk of COVID-19. Our developed coinfection model can contribute to understanding the SARS-CoV-2 and HTLV-I co-dynamics and help to select suitable treatment strategies for COVID-19 patients who are infected with HTLV-I.&lt;/p&gt;&lt;/abstract&gt;

Теорема об области асимптотической устойчивости и ее приложения

The article considers a theorem on the region of asymptotic stability of solutions of ordinary differential equations, which is generalized to the case of a system of equations, where the right-hand side explicitly depends on time. It is shown how it can be used to answer the question of whether the equation under consideration has periodic solutions other than the trivial one, and how, using the constructed Lyapunov functions, one can find these periodic solutions.

Analysis of an Eco-Epidemiological Model Considering Effect of Harvesting and Prey Refuge

In this work, we analyze an eco-epidemiological model with the disease in the prey, considering a constant proportion of harvesting of either species or a prey refuge. The positive invariant set, the conditions of existence, and locally asymptotically stability of the equilibria are studied using the stability theory of ordinary differential equation. The global stability of border equilibria by constructing Lyapunov functions and permanence of the system by comparison theoremare proved. The numerical simulation further proved the correctness of the theoretical analysis. The result indicates that overfishing would lead to population extinction and a reasonable fishing strategy should keep the coexistence of populations.

Dynamics of SCIR Modeling for COVID‐19 with Immigration

In this study, for COVID‐19, we divide people into four categories: susceptible S(t), closely contacted C(t), infective I(t), and removed R(t) according to the current epidemic situation and then investigate two models: the SCIR models with immigration (Model (2) and without immigration (Model 1). For the former, Model 1, we obtain the condition for global stability of its disease‐free equilibrium. For the latter, Model 2, we establish the local asymptotic stability of its endemic equilibrium by constructing Lyapunov function. Afterwards, by the bifurcation theory, we qualitatively analyze the properties of its Hopf bifurcations of the latter. Finally, numerical simulations are given to illustrate the obtained results of two models. The results imply the importance of finding closely contacted and overseas imports on epidemic control. It indicates that not only the incubation delay τ is crucial for the containment of the COVID‐19 but also the scientific and rigorous containment measures are the key factors of the success of the containment.