Construction Of Lyapunov Functions Research Articles

Generalized Lyapunov functionals for the input-to-state stability of infinite-dimensional systems

This paper addresses the input-to-state stability (ISS) of infinite-dimensional systems by introducing a novel notion named generalized ISS-Lyapunov functional (GISS-LF) and the corresponding ISS Lyapunov theorem. Unlike the classical ISS-Lyapunov functional (ISS-LF) that must be positive definite, a GISS-LF can be positive semidefinite. Moreover, such a functional considers not only the relationship with elements in the state space but also takes into account the elements in the input space via a family of certain functionals. Consequently, this notion provides more options in constructing Lyapunov functionals for the ISS assessment of infinite-dimensional systems. In particular, we provide a positive answer to the open question raised by A. Mironchenko and C. Prieur, “Input-to-state stability of infinite-dimensional systems: recent results and open questions”, (Mironchenko and Prieur, 2020), regarding the existence of a coercive ISS-LF for the heat equation with Dirichlet boundary disturbances. To demonstrate the application of the proposed method, which we refer to as the generalized Lyapunov method, we present two examples, showing how to construct GISS-LFs by using positive semidefinite and non-coercive functionals for nonlinear parabolic equations defined over higher dimensional domains with Dirichlet boundary disturbances, and to derive small-gain conditions for guaranteeing the ISS with respect to distributed in-domain disturbances for coupled nonlinear degenerate parabolic equations, which contain ordinary differential equations as special cases.

Predefined-time adaptive robust simultaneous stabilisation control for two nonlinear singular time-delay systems

This paper mainly studies the simultaneous stabilisation control problem of a class of nonlinear singular systems with predefined time. We first develop two singular system in differential algebraic forms via an appropriate state feedback. And then the two systems are extended to one system for further research. We have solved the difficulty of constructing Lyapunov functions by constructing appropriate Hamiltonian functions for our research. In addition, when considering nonlinear singular systems with unknown disturbances, parameter uncertainties and time delays, a control strategy is developed to achieve simultaneous stabilisation within a predefined time. Finally, the effectiveness of the proposed controller was verified through simulation.

Adaptive Command Filtering Control for High-order Strict Feedback Systems

In this paper, a command-filter based neural network adaptive control method is proposed for high-order nonlinear systems with strict feedback. The command filter is used to avoid the computational complexity explosion problem of the traditional backstepping method, and the filtering error compensation mechanism is designed to reduce the impact of filtering error on the performance of the closed-loop system. In addition, the radial basis function neural network is used to deal with the uncertainty of the system, so as to improve the robustness and tracking performance of the controller. By constructing Lyapunov function, the asymptotic stability of the closed-loop system is proved theoretically. Simulation results show that the designed controller can achieve accurate and fast tracking of the reference signal and significantly improve the system performance.

Design of fault-tolerant non-fragile control for parabolic PDE systems with semi-Markov jump: the finite-time case

This study looks into the problem of non-fragile control design for PDE systems in the presence of Neumann-type boundary conditions. In particular, the considered PDE is a parabolic type having semi-Markov jump and is subject to parametric uncertainties and external disturbances. The primary objective of this study is to provide a fault-tolerant non-fragile control protocol that ensures the targeted stabilisation of the system despite gain variations and actuator faults. Moreover, the impact of external disturbances is minimised by employing mixed H ∞ and passivity performance. After that, adequate criteria are established in the framework of linear matrix inequalities through the construction of a proper Lyapunov function together with the employment of integral-based Wirtinger's inequality such that it ensures the accomplishment of finite-time boundedness with a satisfactory performance level for the closed-loop system. Consequently, the explicit configuration of the developed controller gain matrices is computed, which is obtained by solving the developed criteria. Finally, to show the effectiveness of the devised control approach, simulation results with numerical examples are made available.

Fractional-order input-to-state stability and its converse Lyapunov theorem

This paper establishes input-to-state stability (ISS) for fractional-order nonlinear systems, providing a stability constraint framework for fractional-order systems affected by external inputs. The general fractional comparison principle aids in constructing Lyapunov functions, offering criteria for input-to-state stability of fractional-order nonlinear systems. The concept of weak robustness and the converse Lyapunov theorem for fractional-order nonlinear systems further enhance the completeness of input-to-state stability in these systems. By designing systems to maintain fractional-order input-to-state stability (FOISS), the system can remain robust against the effects of bounded external inputs. Serving as a foundational ”toolbox,” this work provides insights into the stability analysis of fractional-order nonlinear systems, benefiting researchers and practitioners in the field of control theory.

Modeling Covid-19 Virus after Lifting Preventive Measures: A Case Study of Kisii County

Purpose: This research is about a new COVID-19 SIR model containing three classes; susceptible S(t), infected I(t), and recovered R(t) with the convex incident rate. Methodology: The NCOVID-19 model was formulated in the following system, the whole population N(t) was divided into three classes S(t), I(t), and R(t), which represented Susceptible, Infected, and Recovered compartments in the form of differential equations. Lyapunov functions were used to validate the stability of the equilibrium of the ordinary differential equations, linearization of the system was also done using Jacobian matrices by finding the derivatives of f(x) for x. Findings: Covid-19 is an infectious disease caused by the novel coronavirus identified as Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2). The people infected by COVID-19 experience mild respiratory problems such as; Fever, dry cough, throat infection, and fatigue. People may also have symptoms such as nasal infection, aches, and sore throat. The pandemic has led to a dramatic loss of human life in Kenya, Africa, and the whole world as it presents an unprecedented challenge to public health, food systems, and the world of work. This case study seeks to model covid-19 virus after lifting preventive measures with a major focus on Kisii County, the subject model was presented in the form of differential equations and the disease-free and endemic equilibrium was calculated for the model. Also, the basic reproduction number R0 = 0.7831 was calculated and the disease-free equilibrium was found to be asymptotically stable meaning that the virus could be eliminated from the population, this showed that the county government of Kisii was in good control of the COVID-19 situation., in addition, The global stability of the model was calculated using the Lyapunov function construction while the Local stability was calculated using the Jacobian matrices. The numerical solutions were calculated using the non-standard finite difference scheme (NFDS) and MATLAB software. Unique Contribution to Theory, Practice and Policy: This study has laid a foundation for future research in the area. In the future, a study that can include the rate of COVID-19 virus mutation and its impacts is recommended.

Stability, uniqueness and existence of solutions to McKean–Vlasov SDEs: a multidimensional Yamada–Watanabe approach

We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean–Vlasov equations with random non-Lipschitz continuous coefficients. In the deterministic case, we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada–Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise exponential stability. Furthermore, Lyapunov exponents are computed explicitly.

Results on a nonlinear wave equation with acoustic and fractional boundary conditions coupling by logarithmic source and delay terms: Global existence and asymptotic behavior of solutions

The nonlinear wave equation with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, is notable for its capacity to model complex systems, contribute to the advancement of mathematical theory, and exhibit wide-ranging applicability to real-world problems. This paper investigates the global existence and general decay of solutions to a wave equation characterized by the inclusion of logarithmic source and delay terms, governed by both fractional and acoustic boundary conditions. The global existence of solutions is analyzed under various hypotheses, and the general decay behavior is established through the construction and application of a suitable Lyapunov function.

Analyzing Sumability in Mean Square for Stochastic Difference Equations via Lyapunov Methods

This paper investigates solutions to the second-kind stochastic volterra difference equation through two case studies. It focuses on, concentrates on the sumability of these solutions in the context of mean square and mean fourth criteria, utilizing the technique of constructing Lyapunov functionals.

Results from a Nonlinear Wave Equation with Acoustic and Fractional Boundary Conditions Coupling by Logarithmic Source and Delay Terms: Global Existence and Asymptotic Behavior of Solutions

The nonlinear wave equation with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, is significant for its ability to model complex systems, its contribution to the advancement of mathematical theory, and its wide-ranging applicability to real-world problems. This paper examines the global existence and general decay of solutions to a wave equation characterized by coupling with logarithmic source and delay terms, and governed by both fractional and acoustic boundary conditions. The global existence of solutions is analyzed under a range of hypotheses, and the general decay behavior is established through the construction and application of an appropriate Lyapunov function.

Observer-based adaptive fuzzy control of stabilized platform in rotary steerable system with input saturation and output constraint

This paper explores the control strategy for toolface tracking in the stabilized platform of a rotary steerable system, considering both input saturation and output constraint. To mitigate the impact of unknown friction torque and modeling error on the stabilized platform, we propose an adaptive fuzzy backstepping control approach. A fuzzy state observer is specifically devised to handle the uncertain state arising from unknown friction torque and modeling errors. Introducing an auxiliary system effectively compensates for input saturation, and the resolution of the output error constraint is achieved through the construction of a barrier Lyapunov function. Furthermore, employing the Lyapunov method establishes that all signals in the entire closed-loop control system are semi-globally uniformly ultimately bounded. Simulation results confirm the efficacy of the proposed control methodology.

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.

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.

Unified analysis on multistablity of fraction-order multidimensional-valued memristive neural networks

This article provides a unified analysis of the multistability of fraction-order multidimensional-valued memristive neural networks (FOMVMNNs) with unbounded time-varying delays. Firstly, based on the knowledge of fractional differentiation and memristors, a unified model is established. This model is a unified form of real-valued, complex-valued, and quaternion-valued systems. Then, based on a unified method, the number of equilibrium points for FOMVMNNs is discussed. The sufficient conditions for determining the number of equilibrium points have been obtained. By using 1-norm to construct Lyapunov functions, the unified criteria for multistability of FOMVMNNs are obtained, these criteria are less conservative and easier to verify. Moreover, the attraction basins of the stable equilibrium points are estimated. Finally, two numerical simulation examples are provided to verify the correctness of the results.

On stability in probability for a cooperative system of two equations with Nicholson’s growth and distributed delay under stochastic perturbations

A method of stability investigation for stochastic nonlinear dynamical systems is demonstrated on a system of two connected Nicholson’s blowflies type equations with distributed delay and exponential nonlinearity. It is supposed that this system is affected by stochastic perturbations of the white noise type that are directly proportional to the deviation of the system state from its equilibrium. Stability conditions for the zero and positive equilibria of the system under consideration are obtained by virtue of the general method of Lyapunov functionals construction and are formulated in terms of Linear Matrix Inequalities (LMIs), that can be checked by MATLAB. Numerical examples and figures illustrate the obtained theoretical results. The described method can be applied in various applications to many stochastic systems of higher dimensions with different types of high-order nonlinearities and different forms of delays.

Modeling the impact of non-human host predation on the transmission of Chagas disease

In addition to the traditional transmission route via the biting-and-defecating process, non-human host predation of triatomines is recognized as another significant avenue for Chagas disease transmission. In this paper, we develop an eco-epidemiological model to investigate the impact of predation on the disease’s spread. Two critical thresholds, Rvp (the basic reproduction number of triatomines) and R0p (the basic reproduction number of the Chagas parasite), are derived to delineate the model’s dynamics. Through the construction of appropriate Lyapunov functions and the application of the Bendixson–Dulac theorem, the global asymptotic stabilities of the equilibria are fully established. The vector-free equilibrium E0 is globally stable when Rvp<1. E1, the disease-free equilibrium, is globally stable when Rvp>1 and R0p<1, while the endemic equilibrium E∗ is globally stable when both Rvp>1 and R0p>1. Numerical simulations highlight that the degree of host predation on triatomines, influenced by non-human hosts activities, can variably increase or decrease the Chagas disease transmission risk. Specifically, low or high levels of host predation can reduce R0p to below unity, while intermediate levels may increase the infected host populations, albeit with a reduction in R0p. These findings highlight the role played by non-human hosts and offer crucial insights for the prevention and control of Chagas disease.

Analysis of a fractional-order model for dengue transmission dynamics with quarantine and vaccination measures

A comprehensive mathematical model is proposed to study two strains of dengue virus with saturated incidence rates and quarantine measures. Imperfect dengue vaccination is also assumed in the model. Existence, uniqueness and stability of the proposed model are proved using the results from fixed point and degree theory. Additionally, well constructed Lyapunov function candidates are also applied to prove the global stability of infection-free equilibria. It is also demonstrated that the model is generalized Ulam-Hyers stable under some appropriate conditions. The model is fitted to the real data of dengue epidemic taken from the city of Espirito Santo in Brazil. For the approximate solution of the model, a non-standard finite difference(NSFD) approach is applied. Sensitivity analysis is also carried out to show the influence of different parameters involved in the model. The behaviour of the NSFD is also assessed under different denominator functions and it is observed that the choice of the denominator function could influence the solution trajectories. Different scenario analysis are also assessed when the reproduction number is below or above one. Furthermore, simulations are also presented to assess the epidemiological impact of dengue vaccination and quarantine measures for infected individuals.

A new co-infection model for HBV and HIV with vaccination and asymptomatic transmission using actual data from Taiwan

The co-infection of Human Immunodeficiency Virus (HIV) and Hepatitis B virus (HBV) poses a major threat to public health due to their combined negative impacts on health and increased risk of complications. A novel fractional mathematical model of the dynamics of co-infection between HBV and HIV for Taiwan is presented in this paper. Detailed analyses are conducted on the possible impact of HBV vaccination on the dynamics of HBV and HIV co-infection. The next-generation matrix technique is used to calculate the fundamental reproduction number R 0 = max{R 1, R 2}, where R 1 and R 2 are the reproduction numbers for HBV and HIV, respectively. The disease-free and endemic equilibria of the co-infection model are calculated. An extensive investigation is carried out to determine the local and global stability of the disease-free equilibrium point through Rough Hurtwiz criteria and the construction of Lyapunov function, respectively. We demonstrate that when R 1 < 1 < R 2, HBV infection is eradicated, but HIV remains prevalent. If R 2 < 1 < R 1, the opposite outcome occurs. The real data from 2000-2023 for Taiwan is used to fit the model. The fitting results show how effectively our model handles the data. In addition, numerical simulations are run for different scenarios to observe how the vaccine and fractional parameters changed the model state variables, as well as how the solutions behaved and how quickly they reached the model’s equilibrium points. According to the model’s numerical analysis, greater vaccination efforts against HBV have a positive effect on the propagation of co-infection.

Dynamics for a Nonlinear Stochastic Cholera Epidemic Model under Lévy Noise

In this study, we develop a comprehensive mathematical model to analyze the dynamics of epidemic cholera, characterized by acute diarrhea due to pathogen overabundance in the human body. The model is first developed from a deterministic point of view, and then it is modified to include the randomness by stochastic differential equations. The study selected Lévy noise above other well-known types of noise, emphasizing its importance in epidemic modeling. Besides presenting a biological justification for the stochastic system, we demonstrate that the equivalent deterministic model exhibits possible equilibria. The introduction is followed by theoretical analysis of the model. Through rigorous analysis, we establish that the stochastic model ensures a unique global solution. Lyapunov function theory is applied to construct necessary conditions, which on average, guarantee the model’s stability for R0s&gt;1. Our findings suggest the likelihood of eradicating the disease when Rs is below one, a significant insight supported by graphical simulations of the model. Graphical illustrations were generated from simulating the model in order to increase the analytical results’ robustness. This work provides a strong theoretical framework for a thorough comprehension of a range of such diseases. This research not only provides a deeper understanding of cholera dynamics but also offers a robust theoretical framework applicable to a range of similar diseases, alongside a novel approach for constructing Lyapunov functions for nonlinear models with random disturbances.

Asymptotic Analysis for the Generalized Langevin Equation with Singular Potentials

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori–Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small-mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard–Jones and Coulomb functions.