LaSalle's Invariance Principle Research Articles

An SVIQR model with vaccination-age, general nonlinear incidence rate and relapse: Dynamics and simulations

This work is concerned with the global dynamics of a Susceptible–Vaccinated–Infected–Isolated–Recovered model, denoted by SVIQR, with vaccination-age. Moreover, the considered model contains a relapse term and a general form of the incidence function. This model is formulated to show the effect of sanitary isolation and vaccination on the disease prevalence when the relapse phenomena occurs. First, we show that the model is mathematically well posed by using the integrated semigroup theory. Second, we prove the existence of equilibrium points based on the basic reproduction number [Formula: see text]. Moreover, by developing appropriate Lyapunov functionals and applying the LaSalles invariance principle, we established the global dynamics of the model, especially, we show that if [Formula: see text] the the disease-free equilibrium is globally asymptotically stable and, if [Formula: see text] the endemic equilibrium is globally asymptotically stable. Finally, using the finite difference method, numerical simulations are made to reinforce the theoretical findings and to prove the contribution of the vaccination and the isolation on the disease disappearing.

Stable cargo transportation for underactuated partial space elevator with model uncertainties and disturbances

An adaptive neural network energy-based control approach is proposed in this paper to address the issue of stabilizing a partial space elevator during cargo transportation. The method handles the challenges presented by external disturbances and model uncertainties. The dynamic model of the partial space elevator, with a typical multi-input and multi-output underactuated system, is presented. A control scheme using energy-based techniques is proposed to guarantee the stable configuration of the system, and auxiliary kinematic vector variables are designed to incorporate the uncontrollable variables to constitute controllable state subsets. Under these frameworks, adaptive control schemes are formulated using multi-layer neural networks to mitigate model uncertainty. Robust compensators are designed to effectively counteract the unknown boundaries of external disturbances and approximation errors. The stability of the closed control system is rigorously demonstrated through the application of the Lyapunov function and LaSalle's invariance principle. Numerical simulations demonstrate that the proposed control method is highly effective in suppressing oscillations during the transfer period of a partial space elevator.

Earth-Venus Mission Analysis via Weak Capture and Nonlinear Orbit Control

Exploration of Venus is recently driven by the interest of the scientific community in understanding the evolution of Earth-size planets, and is leading the implementation of missions that can benefit from new design techniques and technology. In this work, we investigate the possibility to implement a microsatellite exploration mission to Venus, taking advantage of (i) weak capture, and (ii) nonlinear orbit control. This research considers the case of a microsatellite, equipped with a high-thrust and a low-thrust propulsion system, and placed in a highly elliptical Earth orbit, not specifically designed for the Earth-Venus mission of interest. In particular, to minimize the propellant mass, phase (i) of the mission was designed to inject the microsatellite into a low-energy capture around Venus, at the end of the interplanetary arc. The low-energy capture is designed in the dynamical framework of the circular restricted 3-body problem associated with the Sun-Venus system. Modeling the problem with the use of the Hamiltonian formalism, capture trajectories can be characterized based on their state while transiting in the equilibrium region about the collinear libration point L1. Low-energy capture orbits are identified that require the minimum velocity change to be established. These results are obtained using the General Mission Analysis Tool, which implements planetary ephemeris. After completing the ballistic capture, phase (ii) of the mission starts, and it is aimed at driving the microsatellite toward the operational orbit about Venus. The transfer maneuver is based on the use of low-thrust propulsion and nonlinear orbit control. Convergence toward the desired operational orbit is investigated and is proven analytically using the Lyapunov stability theory, in conjunction with the LaSalle invariance principle, under certain conditions related to the orbit perturbing accelerations and the low-thrust magnitude. The numerical results prove that the mission profile at hand, combining low-energy capture and low-thrust nonlinear orbit control, represents a viable and effective strategy for microsatellite missions to Venus.

Passivity and underactuated modeling-based load energy coupling control for three-dimensional overhead crane systems

The conventional control strategies and methods with the three-dimensional model of overhead crane system are excessively dependent on model parameters and lead to the controllers design complex on position and anti-swing of overhead crane systems. With analysis on the coupling relationship among various state variables of overhead crane, the three-dimensional dynamic model of overhead crane based on mechanical virtual displacement and Lagrangian equation is constructed. A novel control strategy based on load energy coupling is designed to realize system positioning and eliminate the problem of residual load swing. The passivity of the three-dimensional overhead crane system is judged, and the energy composition of the system is deduced. The energy coupling function based on load displacement and swing angle is built. The control strategy of three-dimensional overhead crane system based on load energy coupling is designed by the Lyapunov stability theory and function construction method. The stability of the system is verified by the Lasalle invariance principle. With the experimental and programming, the proposed method is compared with different control methods. The results show that this method can realize the accurate positioning of the trolley more quickly and restrain the load swing more effectively.

Global dynamics of a tuberculosis model with age-dependent latency and time delays in treatment.

Since there exists heterogeneity in incubation periods of tuberculosis and a time lag between treatment and recovery. In this study, we develop a tuberculosis model that takes into account age-dependent latency and time delays in treatment to describe the transmission of tuberculosis. We first show that the solution semi-flow of the model is well-posed and has a global attractor [Formula: see text] within an infinite dimensional space [Formula: see text]. Then we define the basic reproduction number [Formula: see text] and prove that it determines the global dynamics of the model. If [Formula: see text], the global attractor [Formula: see text] reduces to the disease-free equilibrium state, indicating that the disease-free equilibrium state is globally asymptotically stable. When [Formula: see text], the semi-flow generated by the model is uniformly persistent, and there exists an interior global attractor [Formula: see text] for this uniformly persistent model. By constructing a suitable Lyapunov function and applying LaSalle's Invariance Principle, we show that the global attractor [Formula: see text] is reduced to the endemic equilibrium state, which means that the endemic equilibrium state is globally asymptotically stable. Based on the tuberculosis data in China from 2007 to 2020, we simulate the parameters and initial values of the proposed model. Furthermore, we calculate the sensitivity of [Formula: see text] to the parameters and find the most sensitive parameters to [Formula: see text]. Finally, we present an improved strategy to achieve the WHO's goal of reducing the incidence of tuberculosis by 90% by 2035 compared to 2015.

Decentralized formation of multi‐agent conformable fractional nonlinear robot systems

AbstractA novel approach to address the asymptotic stability problem of the decentralized formation control of multi‐agent robot systems considering a cyclic pursuit formation graph is derived by using the classical Lyapunov's stability method and LaSalle's invariance principle. Theoretical development of a local potential function‐based control law and the formulation of the position and formation error dynamics, from the viewpoint of the multi‐variable conformable fractional‐order calculus, is given. In addition, a comparison between the integer‐order case and conformable fractional‐order case is provided. Finally, a numerical example illustrates the performance of the developed results.

The Volterra-Lyapunov matrix theory and nonstandard finite difference scheme to study a dynamical system

A compartmental model is considered to study the transmission dynamics of COVID-19. The proposed model is investigated for different results by using Volterra-Lyapunov (V-L) matrix theory. In this regard, first we presented a modified form of SEIR model by incorporating three new compartments C (protected), D (death due to corona) and Q (quarantined). Both equilibrium points are computed together with basic reproductive number. In addition, local stability of both equilibrium points for our proposed model is examined by assuming that wearing of mask, testing of the unaware infected individuals and medical care of the individuals that got infected should constantly be maintained. Hence, subsequently by combining the V-L stable matrix theory with the traditional methodology of constructing the Lyapunov functions, a procedure for the global stability analysis of COVID-19 is presented. Furthermore, based on LaSalle and Lipschitz invariance principle, the global stability of disease free equilibrium point is also examined. The technique we introduced in this paper will provide the more profound comprehension to understand the basic structure of COVID-19. Moreover, for numerical interpretation of our proposed model non-standard finite difference (NSFD) scheme is utilized for simulations. Different graphical illustrations are provided to understand the transmission dynamics.

On the global stabilization of 6‐DOF quadrotors: A direct Lyapunov approach

AbstractNumerous linear and nonlinear controller structures are available to stabilize 6‐DOF quadrotors, however, those structures are capable of stabilizing the quadrotors in a local region only containing the hovering configuration. This article focuses on the design of a nonlinear controller to achieve the stabilization in large for these unmanned aerial vehicles in hovering configuration at the desired attitude. In this manner, a specific set of state variables is introduced to avoid the singularity problems. Afterwards, a candidate Lyapunov function is considered and the stability is proved based on that function by solving a set of ordinary and partial differential equations, the so‐called matching conditions. Then, the asymptotic stability of the desired configuration in 6‐DOF is proved using LaSalle's invariant set principle with the region of attraction containing almost all state space. Numerical simulation results are included to validate the feasibility of the proposed almost globally stabilizing controller.

PDE-based prescribed performance adaptive attitude and vibration control of flexible spacecraft

This article presents a prescribed performance adaptive control scheme for the attitude tracking and vibration reduction of flexible spacecraft modeled by partial differential equations (PDEs). First, the original constrained problem is converted to an equivalent unconstrained one by adopting a prescribed error transformation. Then, a boundary control torque law and a boundary control force law are developed to track the desired attitude and reduce the vibration, respectively. Moreover, the parametric adaptation technique is integrated with the boundary control laws to estimate the disturbances. The closed-loop system is evaluated to be asymptotically stable through the Lyapunov stability theorem and LaSalle's invariance principle. The developed controller has two distinctive features as follows. One is the controller can guarantee the attitude tracking error and tip deformation always satisfy the prescribed performance constraints. Another is the controller can acquire the strong robustness owing to the feedforward disturbance compensation. Lastly, the derived results are demonstrated by simulated studies.

Hopf Bifurcation Analysis of a Housefly Model with Time Delay

The oscillatory dynamics of a delayed housefly model is analyzed in this paper. The local and global stabilities at the non-negative equilibria are obtained via analyzing the distribution of eigenvalues and Lyapunov–LaSalle invariance principle, and the model undergoes the supercritical Hopf bifurcation and the transient oscillation. Based on Wu’s global Hopf bifurcation theory, the existence of the global bifurcation is established under certain conditions.

Convex Optimization-Based Adaptive Fuzzy Control for Uncertain Nonlinear Systems With Input Saturation Using Command Filtered Backstepping

This paper presents a modified command filter backstepping tracking control strategy for a class of uncertain nonlinear systems with input saturation based on the convex optimization method and the adaptive fuzzy logic system control technique. Firstly, the effect of complex uncertainties is eliminated by introducing n command filters and a single fuzzy logic system (FLS). And then, the update laws of FLS weights are designed based on the convex optimization technique. Next, a new piece-wise continuous function is employed to deal with the input saturation problem. The closed-loop system performance is also analyzed using the Lyapunov stability theorem and the Lasalle invariant principle. Finally, simulation and experimental results are presented to show the effectiveness of our controller.

Distributed Algorithm for Nonsmooth Resource Allocation Problems With Nonlinear Constraints and Its Application to Smart Grids

In this article, we study nondifferentiable resource allocation problems (RAPs). In our problem, the decisions of agents are subject to coupling inequality constraints, local inequality constraints, and local convex set constraints. In contrast to existing RAPs, the cost functions are nonsmooth and the inequality constraints are nonsmooth and nonlinear in our problem. Based on differential inclusions and projection methods, we exploit a fully distributed subgradient-based resource allocation algorithm to optimally allocate the network resources. With the help of the set-valued LaSalle invariance principle, we prove the global convergence of the algorithm to the optimal resource allocation of our problem. Finally, our method is applied to the economic dispatch problems of smart grids. With our method, the generations of generating units converge to the optimal power generation.

An Epidemiological Model for Tuberculosis Considering Environmental Transmission and Reinfection

As tuberculosis (TB) patients do not have lifetime immunity, environmental transmission is one of the key reasons why TB has not been entirely eradicated. In this study, an SVEIRB model of recurrent TB considering environmental transmission was developed to explore the transmission kinetics of recurrent TB in the setting of environmental transmission, exogenous infection, and prophylaxis. A more thorough explanation of the effect of environmental transmission on recurrent TB can be found in the model’s underlying regeneration numbers. The global stability of disease-free and local equilibrium points can be discussed by looking at the relevant characteristic equations. The Lyapunov functions and the LaSalle invariance principle are used to show that the local equilibrium point is globally stable, and TB will persist if the basic reproduction number is larger. Conversely, the disease will disappear if the basic reproduction number is less than one. The impact of environmental transmission on the spread of tuberculosis was further demonstrated by numerical simulations, which also demonstrated that vaccination and reducing the presence of the virus in the environment are both efficient approaches to control the disease’s spread.

Dynamical behaviors of a delayed SIR information propagation model with forced silence function and control measures in complex networks.

Due to the advanced network technology, there is almost no barrier to information dissemination, which has led to the breeding of rumors. Intended to clarify the dynamic mechanism of rumor propagation, we formulate a SIR model with time delay, forced silence function and forgetting mechanism in both homogeneous and heterogeneous networks. In the homogeneous network model, we first prove the nonnegativity of the solutions. Based on the next-generation matrix, we calculate the basic reproduction number . Besides, we discuss the existence of equilibrium points. Next, by linearizing the system and constructing a Lyapunov function, the local and global asymptotically stability of the equilibrium points are found. In the heterogeneous network model, we derive the basic reproduction number through the analysis of a rumor-prevailing equilibrium point . Moreover, we conduct the local and global asymptotic stability analysis for the equilibrium points according to the LaSalle's Invariance Principle and stability theorem. As long as the maximum spread rate is large enough, the rumor-prevailing point is locally asymptotically stable when . Additionally, it hits that the system exists bifurcation behavior at due to the newly added forced silence function. Later, after adding two controllers to the system, we research the problem of optimal control. Finally, aimed at authenticating the above theoretical results, a serious of numerical simulation experiments are carried out.

A Lyapunov functional for vaccination model on the dynamics of cholera epidemic with non-linear incidence of infection

A new deterministic susceptible-exposed-infectious-vaccinated-removed-pathogen (SEIVRB) cholera epidemic model with combined mass action incidence and saturated incidence rates is proposed and analyzed. A threshold level of vaccine coverage necessary for controlling or eradicating cholera has been determined and analyzed using the next generation matrix approach. It is shown that the higher values of vaccine coverage that are lower than the threshold value significantly reduces the number of infected individuals whenever basic reproduction number is less than unity, and the cholera would persist in the populations whenever the model basic reproduction number exceeds unity. The global stability for cholera free equilibrium state and cholera endemic equilibrium state of the model system is investigated using Lyapunov functional approach and Lasalle invariance principle, which are found to be globally asymptotically stable at both equilibrium states. Numerical simulations and graphical illustrations is presented to support the analytical results found in the study.

Stability analysis and optimal control of epidemic-like transmission model with nonlinear inhibition mechanism and time delay in both homogeneous and heterogeneous networks

In this paper, considering the bilingual environment, we establish the 2SIR rumor models with a nonlinear inhibition mechanism and time delay in both homogeneous networks and heterogeneous networks. In the homogeneous network model, we prove the non-negativity of the solutions for the system and obtain the basic reproduction number R01 through the next-generation matrix. Then we discuss the local and global stability of the equilibrium point by means of analyzing the characteristic equation of linearized system and Lyapunov functional. In the heterogeneous model, based on the analysis of the existence of the rumor-spreading equilibrium points, we derive the basic reproduction numbers R02. Besides, by utilizing the linearized system and LaSalle's invariance principle, the stability of the rumor-free equilibrium point was proved. What's more, the system exhibits bifurcation behavior at R02=1 by choosing a system parameter as the bifurcation parameter. Additionally, optimal control problems in both homogeneous and heterogeneous networks with three control measures are discussed. Moreover, numerical simulations are carried out to verify the correctness of theoretical results.

Traveling wave solutions of a class of multi-species non-cooperative reaction–diffusion systems *

In this paper we establish a sharp existence result on weak traveling wave solutions for a general class of multi-species reaction–diffusion systems. Moreover, the minimal speed of the traveling waves is explicitly determined. Such a weak traveling wave solution connects the predator-free equilibrium point E 0 at but needs not to connect the coexistence equilibrium at . We apply this result to three important non-cooperative systems: the classical diffusive SIS system for the spread of infectious disease, a predator–prey system with age structure and a generalised Lotka–Volterra predator–prey system of one predator species feeding on n prey species, and prove with the aid of Lyapunov functions and the LaSalle invariance principle that their weak traveling wave solutions are actually traveling wave solutions that connect at . For the SIS system and the generalised Lotka–Volterra predator–prey system, we develop additional techniques to establish the boundedness of their weak traveling wave solutions before applying the LaSalle’s invariance principle.

Adaptive Exact Penalty Design for Optimal Resource Allocation.

In this article, a distributed adaptive continuous-time optimization algorithm based on the Laplacian-gradient method and adaptive control is designed for resource allocation problem with the resource constraint and the local convex set constraints. In order to deal with local convex sets, a distance-based exact penalty function method is adopted to reformulate the resource allocation problem instead of the widely used projection operator method. By using the nonsmooth analysis and set-valued LaSalle invariance principle, it is proven that the proposed algorithm is capable of solving the nonsmooth resource allocation problem. Finally, two simulation examples are presented to substantiate the theoretical results.

On stability analysis study and strategies for optimal control of a mathematical model of hepatitis HCV with the latent state

In this work, we analyze a viral hepatitis C model. This epidemic remains a major problem for global public health, in all communities, despite the efforts made. The model is analyzed using the stability theory of systems of nonlinear differential equations. Based on the results of the analysis, the proposed model has two equilibrium points: a disease-free equilibrium point E0 and an endemic equilibrium point E∗. We investigate the existence of equilibrium point of the model. Furthermore, based on the indirect Lyapunov method, we study the local stability of each equilibrium point of the model. Moreover, by constructing the appropriate Lyapunov function and by using LaSalle invariance principle, we get some information on the global stability of equilibrium points under certain conditions. The basic reproduction number R0 is calculated using the Next Generation method. The positivity of the solutions and their bornitude have been proven, the existence of the solutions has also been proven. Optimal control of the system was studied by proposing three types of intervention: awareness program, early detection, isolation and treatment. The maximum principle of Pontryagin was used to characterize the optimal controls found. Numerical simulations were carried out with a finite numerical difference diagram and using MATLAB to confirm acquired results.

Stability analysis of a nonlinear mathematical model for COVID-19 transmission dynamics

The whole world had been plagued by the COVID-19 pandemic. It was first detected in the Wuhan city of China in December 2019, and has then spread worldwide. It has affected each one of us in the worst possible way. In the current study, a differential equation-based mathematical model is proposed. The present model highlights the infection dynamics of the COVID-19 spread taking hospitalization into account. The basic reproduction number is calculated. This is a crucial indicator of the outcome of the COVID-19 dynamics. Local stability of the equilibrium points has been studied. Global stability of the model is proven using the Lyapunov second method and the LaSalle invariance principle. Sensitivity analysis of the model is performed to distinguish the factor responsible for the faster spread of the infection. Finally, the theoretical aspects have been corroborated via numerical simulations performed for various initial conditions and different values of the parameters.