Asymptotic Stability Properties Research Articles

Global dynamics and numerical simulation of a modified epidemiological model for viral marketing on social networks

The aim of this work is to conduct a rigorous mathematical analysis for global dynamics and numerical simulation of a recognized viral marketing (VM) model, which is described by a system of ordinary differential equations (ODEs). We first establish positivity and boundedness of solutions and then investigate local and global asymptotic stability properties of possible equilibrium points. As an important consequence, complex dynamics of the VM model is determined fully.Secondly, we develop the Mickens’ methodology to design a nonstandard finite difference (NSFD) scheme, which is useful in numerical simulation of the VM model. The main advantage of the constructed NSFD scheme is that it has the ability to preserve important mathematical features of the continuous-time model for all finite values of the step size. These features include the positivity and boundedness of solutions, positively invariant sets, equilibrium points and their asymptotic stability properties. Consequently, the NSFD scheme is not only effective to simulate dynamics of the VM model, but also easy to be implemented.Thirdly, to emphasize implications of the constructed mathematical analysis, an extended version combining the integer-order ODE model under consideration with the Caputo fractional derivative is considered and analyzed. From the mathematical analysis performed for the integer-order VM model, global dynamics of the fractional-order VM model is also investigated rigorously.Finally, the theoretical insights are supported by a set of illustrative numerical experiments.The findings of this research not only improve some existing results in the literature, but may also provide several useful real-life applications.

Dynamics of Oscillator Populations Globally Coupled with Distributed Phase Shifts.

We consider a population of globally coupled oscillators in which phase shifts in the coupling are random. We show that in the maximally disordered case, where the pairwise shifts are independent identically distributed random variables, the dynamics of a large population reduces to one without randomness in the shifts but with an effective coupling function, which is a convolution of the original coupling function with the distribution of the phase shifts. This result is valid for noisy oscillators and/or in the presence of a distribution of natural frequencies. We argue also, using the property of global asymptotic stability, that this reduction is valid in a partially disordered case, where random phase shifts are attributed to the forced units only. However, the reduction to an effective coupling in the partially disordered noise-free situation may fail if the coupling function is complex enough to ensure the multistability of locked states.

A novel one-bit dynamic quantizer for event-triggered control systems

We investigate the problem of feedback stabilization of networked control linear systems with unknown initial state. In particular, the sampling and the quantization effects of network are considered and we propose a quantized event-triggered control technique to handle this case. To cope with unknown initial state, a novel exponential dynamic quantizer is synthesized, which enables to capture the state in a finite time. Moreover, the dynamic quantization policy only depends on the sign of the quantization error and allows for one-bit transmission rather than packet-based transmission, which is desirable in practice. Then, the quantized state measurements are transmitted to the controller using an event-triggering mechanism to reduce the amount of transmissions over the network. The approach ensures global asymptotic stability property for the closed-loop system and prevents the occurrence of Zeno behavior. The entire closed-loop system is represented as a hybrid dynamical system and the closed-loop stability is assessed using suitable Lyapunov functions. The effectiveness of the proposed approach is demonstrated through numerical simulations.

Stability analysis of Caputo fractional time-dependent systems with delay using vector lyapunov functions

<p>In this study, we investigate the stability and asymptotic stability properties of Caputo fractional time-dependent systems with delay by employing vector Lyapunov functions. Utilizing the Caputo fractional Dini derivative on Lyapunov-like functions, along with a new comparison theorem and differential inequalities, we derive and prove sufficient conditions for the stability and asymptotic stability of these complex systems. An example is included to showcase the method's practicality and to specifically illustrate its advantages over scalar Lyapunov functions. Our results improves, extends, and generalizes several existing findings in the literature.</p>

A dynamically consistent nonstandard finite difference scheme for a generalized SEIR epidemic model

This work is devoted to the proposal and analysis of a new mathematical study of the transmission dynamics of infectious diseases. First, a generalized SEIR epidemic model is presented that uses general nonlinear incidence rates to describe the ‘psychological’ effect. Then, a rigorous mathematical analysis is performed for the proposed SEIR model. We establish positivity and boundedness, calculate the basic reproduction number, determine possible equilibrium points (disease-free and endemic), and investigate their asymptotic stability properties of the SEIR model. The obtained results improve and extend a SEIR model constructed in a recent work; moreover, the proposed model is useful for studying the COVID-19 epidemic in particular and other infectious diseases in general. For the purpose of numerical simulation, the Mickens method is applied to construct a dynamically consistent non-standard finite difference (NSFD) model for the proposed SEIR epidemic model. The constructed NSFD scheme is able to provide reliable approximations that not only preserve the dynamic properties of the SEIR model for all values of the step size, but also are easy to implement. Finally, a series of illustrative numerical experiments are performed to support the theoretical findings and confirm the advantages of the NSFD scheme over some well-known standard methods.

Effect of Impaired B-Cell and CTL Functions on HIV-1 Dynamics

This paper formulates and analyzes two mathematical models that describe the within-host dynamics of human immunodeficiency virus type 1 (HIV-1) with impairment of both cytotoxic T lymphocytes (CTLs) and B cells. Both viral transmission (VT) and cellular infection (CT) mechanisms are considered. The second model is a generalization of the first model that includes distributed time delays. For the two models, we establish the non-negativity and boundedness of the solutions, find the basic reproductive numbers, determine all possible steady states and establish the global asymptotic stability properties of all steady states by means of the Lyapunov method. We confirm the theoretical results by conducting numerical simulations. We conduct a sensitivity analysis to show the effect of the values of the parameters on the basic reproductive number. We discuss the results, showing that impaired B cells and CTLs, time delay and latent CT have significant effects on the HIV-1 dynamics.

Modelling and Feedback Control for Reversing a Nonholonomic Mobile Robot Platoon

This paper addresses the control problem of a platoon of nonholonomic mobile robots in the case of backward motion of the leading robot. By using a virtual non-static with respect to the lead robot reference frame as a target, a kinematic model of the robot platoon in error coordinates is developed. A nonlinear feedback controller based on the entire platoon model is designed by means of high-gain control. Asymptotic stability property of the closed-loop system is established using Lyapunov theory. For backward motion of the leading robot along a circular path, at steady-state, the two-robot platoon should travel concentric arcs of same radii with prescribed inter-robot spacing. The performance of the proposed tracking controller is illustrated through numerical simulations.

Output-Based Dynamic Periodic Event-Triggered Control with Application to the Tunnel Diode System

The integration of communication channels with the feedback loop in a networked control system (NCS) is attractive for many applications. A major challenge in the NCS is to reduce transmissions over the network between the sensors, the controller, and the actuators to avoid network congestion. An efficient approach to achieving this goal is the event-triggered implementation where the control actions are only updated when necessary from stability/performance perspectives. In particular, periodic event-triggered control (PETC) has garnered recent attention because of its practical implementation advantages. This paper focuses on the design of stabilizing PETC for linear time-invariant systems. It is assumed that the plant state is partially known; the feedback signal is sent to the controller at discrete-time instants via a digital channel; and an event-triggered controller is synthesized, solely based on the available plant measurement. The constructed event-triggering law is novel and only verified at periodic time instants; it is more adapted to practical implementations. The proposed approach ensures a global asymptotic stability property for the closed-loop system under mild conditions. The overall model is developed as a hybrid dynamical system to truly describe the mixed continuous-time and discrete-time dynamics. The stability is studied using appropriate Lyapunov functions. The efficiency of the technique is illustrated in the dynamic model of the tunnel diode system.

Stability analysis of optimal control problems with time-dependent costs

We present stability conditions for deterministic time-varying nonlinear discrete-time systems whose inputs aim to minimize an infinite-horizon time-dependent cost. Global asymptotic and exponential stability properties for general attractors are established. This work covers and generalizes the related results on discounted optimal control problems to more general systems and cost functions.

Lunar descent and landing via two-phase explicit guidance and pulse-modulated reduced-attitude control

This work considers the three-dimensional descent path of a space vehicle, from periselenium of its operational orbit to the lunar surface. The trajectory is split in two arcs: (1) descent path, up to an altitude of 50 m, and (2) terminal approach and soft touchdown. For phase 1, a new, three-dimensional locally-flat near-optimal guidance is introduced that is based on the local projection of the position and velocity variables. A minimum-time problem is defined using the locally flat coordinates of position and velocity. This leads to identifying closed-form functions of time for the two thrust angles, which identify the commanded thrust direction. During terminal approach (phase 2), correct vertical alignment, modest velocity, and negligible angular rate at touchdown are pursued. With this intent, a predictive bang-off-bang guidance algorithm is proposed that is capable of guaranteeing the desired final conditions. In both phases, the attitude control system has the objective of aligning the actual thrust direction with the commanded one, provided by the guidance algorithm. The resulting reduced-attitude control problem is addressed through the use of a new quaternion-based nonlinear control algorithm, which is proven to enjoy asymptotic stability properties. The attitude actuation system is composed of 12 monopropellant thrusters, ignited using pulse width modulation. Monte Carlo simulations are run, assuming significant displacements from the nominal initial conditions and including several harmonics of the selenopotential. The numerical results unequivocally prove that the guidance and control architecture proposed in this study is effective to achieve lunar descent and safe touchdown in nonnominal flight conditions.

Exact numerical solutions and high order nonstandard difference schemes for a second order delay differential equation

In this work, we obtain an expression, given in terms of some special functions, for the exact numerical solution of the second order delay differential equation with general initial condition for . Based on this exact solution, we propose a family of nonstandard finite difference methods that combine high order accuracy and efficient computational properties. We also show that the numerical solutions obtained with the new schemes are consistent with asymptotic stability properties of the exact solutions.

On Asymptotic Stability of the Sine-Gordon Kink in the Energy Space

We consider the sine-Gordon (SG) equation in 1+1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space H^1times L^2. It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space. In this paper we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in Hoffman and Wayne (Differ Int Equ 26(3–4):303–320, 2013), Muñoz and Palacios (Ann. IHP C Analyse Nonlinéaire 36(4):977–1034, 2019) and Virial estimates around odd perturbations of the vacuum solution, in the spirit of Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result and Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and phi ^4 at the level of linear Bäcklund transformations.

A nonlinear optimal control approach for voltage source inverter-fed three-phase PMSMs

A nonlinear optimal control approach for voltage source inverter-fed three-phase PMSMs

A multiconsistent computational methodology to resolve a diffusive epidemiological system with effects of migration, vaccination and quarantine

BackgroundWe provide a compartmental model for the transmission of some contagious illnesses in a population. The model is based on partial differential equations, and takes into account seven sub-populations which are, concretely, susceptible, exposed, infected (asymptomatic or symptomatic), quarantined, recovered and vaccinated individuals along with migration. The goal is to propose and analyze an efficient computer method which resembles the dynamical properties of the epidemiological model. Materials and methodsA non-local approach is utilized for finding approximate solutions for the mathematical model. To that end, a non-standard finite-difference technique is introduced. The finite-difference scheme is a linearly implicit model which may be rewritten using a suitable matrix. Under suitable circumstances, the matrices representing the methodology are M-matrices. ResultsAnalytically, the local asymptotic stability of the constant solutions is investigated and the next generation matrix technique is employed to calculate the reproduction number. Computationally, the dynamical consistency of the method and the numerical efficiency are investigated rigorously. The method is thoroughly examined for its convergence, stability, and consistency. ConclusionsThe theoretical analysis of the method shows that it is able to maintain the positivity of its solutions and identify equilibria. The method’s local asymptotic stability properties are similar to those of the continuous system. The analysis concludes that the numerical model is convergent, stable and consistent, with linear order of convergence in the temporal domain and quadratic order of convergence in the spatial variables. A computer implementation is used to confirm the mathematical properties, and it confirms the ability in our scheme to preserve positivity, and identify equilibrium solutions and their local asymptotic stability.

Distributed Adaptive Nash Equilibrium Solution for Differential Graphical Games.

This article investigates differential graphical games for linear multiagent systems with a leader on fixed communication graphs. The objective is to make each agent synchronize to the leader and, meanwhile, optimize a performance index, which depends on the control policies of its own and its neighbors. To this end, a distributed adaptive Nash equilibrium solution is proposed for the differential graphical games. This solution, in contrast to the existing ones, is not only Nash but also fully distributed in the sense that each agent only uses local information of its own and its immediate neighbors without using any global information of the communication graph. Moreover, the asymptotic stability and global Nash equilibrium properties are analyzed for the proposed distributed adaptive Nash equilibrium solution. As an illustrative example, the differential graphical game solution is applied to the microgrid secondary control problem to achieve fully distributed voltage synchronization with optimized performance.

Classification of radial solutions to −Δgu = eu on Riemannian models

We provide a complete classification with respect to asymptotic behaviour, stability and intersections properties of radial smooth solutions to the equation −Δgu=eu on Riemannian model manifolds (M,g) in dimension N≥2. Our assumptions include Riemannian manifolds with sectional curvatures bounded or unbounded from below. Intersection and stability properties of radial solutions are influenced by the dimension N in the sense that two different kinds of behaviour occur when 2≤N≤9 or N≥10, respectively. The crucial role of these dimensions in classifying solutions is well-known in Euclidean space; here the analysis highlights new properties of solutions that cannot be observed in the flat case.

A Global Tracking Sensorless Adaptive PI-PBC Design for Output Voltage Regulation in a Boost Converter Feeding a DC Microgrid

The problem of the output voltage regulation in a DC-DC boost converter feeding a DC microgrid is addressed in this research via the passivity-based control theory with a proportional–integral action (PI-PBC). Two external input estimators were implemented in conjunction with the proposed controller to make it sensorless and adaptive. The first estimator corresponds to the immersion & invariance (I&I) approach applied to calculate the expected value of the DC load, which is modeled as an unknown DC current. The second estimator is based on the disturbance–observer (DO) approach, which reaches the value of the voltage input. The main advantage of both estimators is that these ensure exponential convergence under steady-state operating conditions, and their parametrization only requires the definition of an integral gain. A comparative analysis with simulations demonstrates that the proposed PI-PBC approach is effective in regulating/controlling the voltage profile in unknown DC loads as compared to the adaptive sliding mode controller. Experimental validations have demonstrated that the proposed PI-PBC approach, in conjunction with the I&I and the DO estimators, allowed regulation of the voltage output profile in the terminals of the DC load with asymptotic stability properties and fast convergence times (1.87 ms) and acceptably overshoots (6.1%) when the voltage input varies its magnitude (from 10 to 12 V and from 10 to 8 V) considering that the DC load changed with a square waveform between 1 and 2 A with 100 Hz.

Balanced implicit methods with strong order 1.5 for solving stochastic differential equations

The aim of this article is to propose a family of balanced implicit methods and split-step balanced implicit methods for solving Itô stochastic differential equations. These numerical methods represent a class of highly efficient linear-implicit schemes which covered the implicitness in the stochastic terms. The convergence in mean-square with strong order 1.5 and the asymptotic mean-square stability properties of these methods are analysed. What is more, the linear mean-square stability regions of some proposed methods are given. Several numerical examples are carried out to illustrate the theoretic findings.

Chaos synchronization using super-twisting sliding mode control applied on Chua’s circuit

Chua?s circuit is the classic chaotic system and the most widely used in serval areas due to its potential for secure communication. However, developing an accurate chaos control strategy is one of the most challenging works for Chua?s circuit. This study proposes a new application of super twisting algorithm (STC) based on sliding mode control (SMC) to eliminate or synchronize the chaos behavior in the circuit. Therefore, the proposed control strategy is robust against uncertainty and effectively regulates the system with a good regulation tracking task. Using the Lyapunov stability, the property of asymptotical stability is verified. The whole of the system including the (control strategy, and Chua?s circuit) is implemented under a suitable test setup based on dSpace1104 to validate the effectiveness of our proposed control scheme. The experimental results show that the proposed control method can effectively eliminate or synchronize the chaos in the Chua's circuit.

Nonautonomous impulsive differential equations of alternately advanced and retarded type

A variation of parameters formula and Gronwall type integral inequality are proved for an impulsive differential equation involving piecewise alternately advanced and retarded argument. At the same time, we study its oscillation and asymptotic stability properties. Our results are new, extend and improve earlier ones. Several numerical examples and simulations are also given to show the feasibility of our results.