Construction Of Lyapunov Functions Research Articles

SOS Construction of Compatible Control Lyapunov and Barrier Functions

We propose a novel approach to certify closed-loop stability and safety of a constrained polynomial system based on the combination of Control Lyapunov Functions (CLFs) and Control Barrier Functions (CBFs). For polynomial systems that are affine in the control input, both classes of functions can be constructed via Sum Of Squares (SOS) programming. Using two versions of the Positivstellensatz we derive an SOS formulation seeking a rational controller that — if feasible — results in compatible CLF and multiple CBFs.

Asymptotic Stability Investigation of the Zero Solution for a Class of Nonlinear Nonstationary Systems by the Averaging Method

A system of nonlinear differential equations is considered that describes the interaction of two coupled subsystems, one of these subsystems is linear, and the other is nonlinear and homogeneous with an order of homogeneity greater than one. It is assumed that this system is affected by nonstationary perturbations with zero mean values. Using the averaging method, sufficient conditions are determined under which perturbations do not disturb the asymptotic stability of the zero solution. The derivation of these conditions is based on the use of a special construction of the nonstationary Lyapunov function which takes into account the structure of the acting perturbations. In addition, we consider the case where there is a constant delay in the right-hand sides of the system. An original approach to the construction of the Lyapunov-Krasovskii functional for such a system is proposed. Using this functional, conditions are found that guarantee the preservation of the asymptotic stability for any positive delay.

Modeling mathematical and analysis of an alcohol drinking with n complications

In this article, we propose a new continuous mathematical approach to model and analyze the dynamics of the population of heavy drinkers and their health complications. The model has several compartments, including a new one that represents the number of heavy drinkers with different complications associated with excessive alcohol consumption. We study the stability of the model using mathematical theories such as the Routh-Hurwitz criteria for local stability, in order to study the equilibrium without and with excessive alcohol consumption, and the construction of Lyapunov functions makes it possible to study global stability. A sensitivity analysis is performed to determine which parameters have the most significant impact on the number of reproductions R0. The results were validated using numerical simulations carried out under MATLAB. This model may be useful in guiding public health policies aimed at reducing the number of drinkers and the complications associated with alcohol consumption.

A survey on constructing Lyapunov functions for reaction-diffusion systems with delay and their application in biology

Motivated by some biological and ecological problems given by reaction-diffusion systems with delays and boundary conditions of Neumann type and knowing their associated Lyapunov functions for delay ordinary differential equations, we consider a method for determining their Lyapunov functions to establish the local/global stability. The method is essentially based on adding integral terms to the corresponding Lyapunov function for ordinary differential equations. The new approach is not general but it is applicable in a wide variety of delays reaction-diffusion models with one discrete delay or more, distributed delay, and a combination of both of them. To illustrate our results, we present the method application to a reaction-diffusion epidemiological model with time delay (latency period) and indirect transmission effect.

Modified refinement algorithm to construct Lyapunov functions using meshless collocation

<p style='text-indent:20px;'>Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sublevel sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. One of the numerical construction methods for Lyapunov functions uses meshless collocation with radial basis functions.</p><p style='text-indent:20px;'>Recently, this method was combined with a grid refinement algorithm (GRA) to reduce the number of collocation points needed to construct Lyapunov functions. However, depending on the choice of the initial set of collocation point, the algorithm can terminate, failing to compute a Lyapunov function. In this paper, we propose a modified grid refinement algorithm (MGRA), which overcomes these shortcomings by adding appropriate collocation points using a clustering algorithm. The modified algorithm is applied to two- and three-dimensional examples.</p>

Global Properties of a Diffusive SARS-CoV-2 Infection Model with Antibody and Cytotoxic T-Lymphocyte Immune Responses

A severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection can lead to morbidity and mortality. SARS-CoV-2 infects the epithelial cells of the respiratory tract and causes coronavirus disease 2019 (COVID-19). The immune system’s response plays a significant role in viral progression. This article develops and analyzes a system of partial differential equations (PDEs), which describe the in-host dynamics of SARS-CoV-2 under the effect of cytotoxic T-lymphocyte (CTL) and antibody immune responses. The model characterizes the interplay between six compartments, healthy epithelial cells (ECs), latent infected ECs, active infected ECs, free SARS-CoV-2 particles, CTLs, and antibodies. We consider the logistic growth of healthy ECs. We first investigate the properties of the model’s solutions, then, we calculate all steady states and determine the conditions of their existence and global stability. The global asymptotic stability is examined by constructing Lyapunov functions. The analytical findings are supported via numerical simulations.

Global Stability of Delayed SARS-CoV-2 and HTLV-I Coinfection Models within a Host

The aim of the present paper is to formulate two new mathematical models to describe the co-dynamics of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and human T-cell lymphotropic virus type-I (HTLV-I) in a host. The models characterizes the interplaying between seven compartments, uninfected ECs, latently SARS-CoV-2-infected ECs, actively SARS-CoV-2-infected ECs, free SARS-CoV-2 particles, uninfected CD4+T cells, latently HTLV-I-infected CD4+T cells and actively HTLV-I-infected CD4+T cells. The models incorporate five intracellular time delays: (i) two delays in the formation of latently SARS-CoV-2-infected ECs and latently HTLV-I-infected CD4+T cells, (ii) two delays in the reactivation of latently SARS-CoV-2-infected ECs and latently HTLV-I-infected CD4+T cells, and (iii) maturation delay of new SARS-CoV-2 virions. We consider discrete-time delays and distributed-time delays in the first and second models, respectively. We first investigate the properties of the model’s solutions, then we calculate all equilibria and study their global stability. The global asymptotic stability is examined by constructing Lyapunov functionals. The analytical findings are supported via numerical simulation. The impact of time delays on the coinfection progression is discussed. We found that, increasing time delays values can have an antiviral treatment-like impact. Our developed coinfection model can contribute to understand the SARS-CoV-2 and HTLV-I co-dynamics and help to select suitable treatment strategies for COVID-19 patients with HTLV-I.

Global stability and Hopf bifurcation of a delayed cooperative species model with density-dependent diffusion

In this article, we have proposed a newly delayed cooperative species model with density-dependent diffusion. Firstly, we prove the existence and uniqueness of positive equilibrium of this model through mathematical analysis method. Then, for this model, we investigate the persistence properties in the case of self-diffusion and global stability of positive equilibrium by constructing Lyapunov function. Further, we discuss the existence problem of Hopf bifurcation deduced by delay. Finally, the theoretical results in this article are verified by carrying out some numerical simulations. The research results show that density dependent diffusion does not affect the stability of the positive equilibrium of the model, but delay does.

Stability of equilibria of exponential type system of three differential equations under stochastic perturbations

A system of three differential equations with exponential nonlinearity is considered. It is shown that the considered system has both the zero and nonzero (positive or negative) equilibria. It is supposed that the system is exposed to stochastic perturbations that are directly proportional to the deviation of a system state from an equilibrium. Via the general method of Lyapunov functionals construction and the method of linear matrix inequalities sufficient conditions of stability in probability for an each equilibrium are obtained. Numerical simulations and figures are presented to demonstrate the obtained results. The proposed investigation method can be applied for many other types of nonlinear systems.

Dynamics Analysis of a Class of Stochastic SEIR Models with Saturation Incidence Rate

In this article, a class of stochastic SEIR models with saturation incidence is studied. The model is a symmetric and compatible distribution family. This paper studies various properties of the system by constructing Lyapunov functions. First, the gradual properties of the systematic solution near the disease-free equilibrium of the deterministic model is studied, followed by the final behavior of the model, including stochastic persistence and final extinction. Finally, the existence conditions of the stationary distribution of the model are given, and then it is proved that it is traversed, and the corresponding conclusions are verified through numerical simulation.

About Stability of Nonlinear Stochastic Differential Equations with State-Dependent Delay

A nonlinear stage-structured population model with a state-dependent delay under stochastic perturbations is investigated. Delay-independent and delay-dependent conditions of stability in probability for two equilibria of the considered system are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The model under consideration is not the aim of the work and was chosen only to demonstrate the proposed research method, which can be used for the study of other types of nonlinear systems with a state-dependent delay.

Time-/Event-Triggered Adaptive Neural Asymptotic Tracking Control for Nonlinear Systems With Full-State Constraints and Application to a Single-Link Robot.

This study proposes the time-/event-triggered adaptive neural control strategies for the asymptotic tracking problem of a class of uncertain nonlinear systems with full-state constraints. First, we design a time-triggered strategy. The effect caused by the residuals of the estimation via radial basis function (RBF) neural networks (NNs), and the reasonable upper bounds on the first derivative of the reference signal and the derivative of each virtual control, can be eliminated by designing appropriate adaptive laws and utilizing the basic properties of RBF NNs. Moreover, the construction of the barrier Lyapunov functions (BLFs) in this work ensures the compliance of the full-state constraints and also holds the asymptotic output tracking performance. Then, based on the time-triggered strategy, we further design a relative threshold event-triggered strategy. The proposed event-triggered adaptive neural controller can solve the main control objective of this work, that is: 1) the full-state constraint requirements of the system are not violated and 2) the output signal asymptotically tracks the reference signal. Compared with the traditional method, the event-triggered strategy can improve the utilization of communication channels and resources and has greater practical significance. Finally, an example of single-link robot under the proposed two strategies illustrates the validity of the constructed controllers.

A Semiglobal Approach for Stabilizing Nonlinear Systems With State Delays by Memoryless Linear Feedback

This article investigates the problem of how to control nonlinear systems with delays in the state by memoryless linear feedback. The notions of semiglobal asymptotic stabilization and sublevel sets are introduced in the context of time-delay systems. With the aid of Razumikhin theorem, we develop a semiglobal design method for the construction of Lyapunov functions, associated sublevel sets, and delay-free linear state feedback laws, step-by-step, achieving semiglobal asymptotic stabilization for time-delay nonlinear systems in a lower triangular form. In contrast to the global stabilization of nonlinear systems with delays in the state, which is usually achieved by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dynamic state</i> feedback (Lin and Zhang, 2020), the significance of this work is to point out that a tradeoff of the control objectives, e.g., semiglobal versus global stabilization, makes it possible to control a class of time-delay nonlinear systems by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">static linear</i> feedback. Extensions to nonlinear systems with globally asymptotically locally expoentially stable (GALES)-like inverse dynamics are also included in this article.

Adaptive fault-tolerant flutter control based on dynamic output feedback

In this paper, the problem of two-dimensional wing flutter control with sensor and actuator failures is studied. Firstly, the aeroelastic equations of motion for two-dimensional wing with plunge and pitch are established; then the wing flutter model with failure modes is established considering the failure modes of sensor and actuator. Secondly, by designing a novel adaptive fault-tolerant controller, the wing flutter can be suppressed effectively and the external disturbance can be dealt with successfully. The stability of the system is proved by the constructed Lyapunov function. Finally, numerical simulation results show that the adaptive flutter fault-tolerant controller can effectively suppress wing flutter when sensors and actuators fail, and the controller has good robustness to the changes of external disturbances.

Path planning in formation and collision avoidance for multi-agent systems

This paper investigates, in a centralized manner, the motion planning problem for a team of unicycle-like mobile robots in a known environment. In particular, a multi-agent collision-free patrolling and formation control algorithm is presented, which combines outcomes of: (i) stability analysis of hybrid systems, (ii) algebraic geometry, and (iii) classical potential functions. The objective is achieved by designing a Lyapunov-based hybrid strategy that autonomously selects the navigation parameters. Tools borrowed from algebraic geometry are adopted to construct Lyapunov functions that guarantee the convergence to the desired formation and path, while classical potential functions are exploited to avoid collisions among agents and the fixed obstacles within the environment. The proposed navigation algorithm is tested in simulation and then validated by using the robots of a remote accessible robotic testbed.

Numerical Construction of Lyapunov Functions Using Homotopy Continuation Method

Lyapunov functions are frequently used for investigating the stability of linear and nonlinear dynamical systems. Though there is no general method of constructing these functions, many authors use polynomials in $ p-forms $ as candidates in constructing Lyapunov functions while others restrict the construction to quadratic forms. By focussing on the positive and negative definiteness of the Lyapunov candidate and the Hessian of its derivative, and using the sum of square decomposition, we developed a method for constructing polynomial Lyapunov functions that are not necessarily in a form. The idea of Newton polytope was used to transform the problem into a system of algebraic equations that were solved using the polynomial homotopy continuation method. Our method can produce several possibilities of Lyapunov functions for a given candidate. The sample test conducted demonstrates that the method developed is promising

An improved two-layer model for rumor propagation considering time delay and event-triggered impulsive control strategy

The rapid development of the Internet has broadened the existing channels of rumor propagation, it has added many new features during the dynamic process of the rumor propagation that are different from those in the past. In this paper, we construct an improved XYZ−ISR two-layer model considering time delay to describe the dynamic process of rumor propagation in multiple channels. Firstly, we obtain the basic regeneration number R0 by the next generation matrix method. Secondly, we derive the stability condition of the equilibrium point by constructing Lyapunov functions. In order to reduce the negative impact caused by rumor propagation at minimum cost, we design an event-triggered impulsive control strategy and optimize the control parameters by using a particle swarm optimization algorithm. Then, by comparing the rumor propagation process in the XYZ−ISR two-layer model and the classical rumor DK model, we find that although rumors propagate faster in the XYZ−ISR model in the initial stage, the rumor spreads on a smaller scale than in the DK model in terms of the whole life cycle of the rumor due to the official supervision and timely release of corresponding rumor-debunking information. Besides, individuals need a certain amount of careful consideration time between receiving the information and deciding whether to forward it or not, and this time delay slows the behavior spreading and reduces its scale. Finally, we verify the reasonableness of the above theoretical results through numerical simulations. Moreover, by simulating a rumor case on the microblog with real data, we find that the model proposed in this paper can fit the real rumor propagation process well, and the event-triggered impulsive control strategy can control the rumor propagation more effectively in multiple channels.

A new regulation control method for underactuated 3D overhead cranes

Interconnection and damping assignment passivity-based control (IDA-PBC) has been successfully applied to many mechanical, electrical, and electromechanical systems. For the control of such systems, existing IDA-PBC methods have concentrated on systems with underactuation degree one, such as the ball and beam system, the vertical takeoff and landing aircraft, and the pendulum on a cart. In this paper, we focus on the application of the IDA-PBC methodology to the regulation control of a three-dimensional (3D) overhead crane system with underactuation degree two. The goal of this study is to design and analyze a regulation controller that can drive the cart to the desired position precisely and eliminate the payload swing effectively. To achieve this challenging objective, we first find an ingenious way to solve partial differential equations for constructing a Lyapunov function candidate. Then, we devise a nonlinear controller on the basis of the constructed Lyapunov function and analyze the stability of the closed-loop system by using Lyapunov techniques and LaSalle’s invariance principle. Finally, both simulation and experimental results demonstrate the proposed strategy can achieve excellent positioning accuracy and significant swing elimination, and a comparison study between the proposed method and the existing PBC method is included as well.

Bifurcations in Asymptotically Autonomous Hamiltonian Systems Subject to Multiplicative Noise

The effect of multiplicative stochastic perturbations on planar Hamiltonian systems is investigated. It is assumed that perturbations fade with time and preserve a stable equilibrium of the limiting system. The paper investigates bifurcations associated with changes in the stability of the equilibrium and with the appearance of new stochastically stable states in the perturbed system. It is shown that depending on the structure and the parameters of the decaying perturbations, the equilibrium can remain stable or become unstable. In some intermediate cases, a practical stability of the equilibrium with estimates for the length of the stability interval is justified. The performed stability analysis is based on a combination of the averaging method and the construction of stochastic Lyapunov functions.

Averaging method for the stability analysis of strongly nonlinear mechanical systems

A mechanical system under strongly nonlinear potential and dissipative forces, with nonlinear nonstationary perturbations having zero mean values, is studied. Proposing a special construction of Lyapunov function, the conditions are found, under which the perturbations do not influence the asymptotic stability of the trivial equilibrium position of the system. These conditions include the requirements on asymptotic stability of the disturbance-free system and the relations of the nonlinearity orders between potential and dissipative forces. The developed approach is extended to the problem of monoaxial stabilization of a rigid body.