Global Lyapunov Function Research Articles

Novel Lagrange sense exponential stability criteria for time-delayed stochastic Cohen–Grossberg neural networks with Markovian jump parameters: A graph-theoretic approach

This paper concerns the issues of exponential stability in Lagrange sense for a class of stochastic Cohen–Grossberg neural networks (SCGNNs) with Markovian jump and mixed time delay effects. A systematic approach of constructing a global Lyapunov function for SCGNNs with mixed time delays and Markovian jumping is provided by applying the association of Lyapunov method and graph theory results. Moreover, by using some inequality techniques in Lyapunov-type and coefficient-type theorems we attain two kinds of sufficient conditions to ensure the global exponential stability (GES) through Lagrange sense for the addressed SCGNNs. Ultimately, some examples with numerical simulations are given to demonstrate the effectiveness of the acquired result.

Stability analysis of T–S fuzzy coupled oscillator systems influenced by stochastic disturbance

Coupled oscillator systems under the stochastic disturbance in practical have nonlinearity and uncertainty. To overcome the issue, the T–S fuzzy coupled oscillator system (TSFCOS) model is proposed in this paper, which provides an effective solution to coupled oscillator systems that are complex, uncertain and ill-defined. Subsequently, with the proposed nonlinear T–S fuzzy control, we give insight to the stability of the TSFCOSs with stochastic disturbance. Combined Lyapunov method and graph-theoretical technique, a systematic method to construct a global Lyapunov function for TSFCOSs is first provided, and then substantial stability criteria with the condition of the system topology property are obtained. Considering the application of our theoretical results in practical engineering, microgrids (the power networks) can be regarded as a kind of TSFCOSs with stochastic disturbance. With a numerical test of a six-generator seven-bus microgrid, the progressiveness and feasibility of our theoretical results are shown.

Graph theory-based approach for stability analysis of stochastic coupled oscillators system by energy-based synchronization control

This paper investigates the stability of stochastic coupled oscillators systems (SCOSs) by the mechanical energy of the oscillators. Assembling mechanical energy from the single oscillator system with the help of graph theory, a systematic method to construct a global Lyapunov function for SCOSs is provided. And then two types of stability criteria with the condition of the system topology property are presented. The significance of the presented results is that the stochastic global asymptotical stability of SCOSs can be achieved at equilibrium via an energy-based synchronization control law. Subsequently, the theoretical results are applied to parallel multi-inverter microgrid (a power network) with stochastic disturbance. And a numerical example about microgrid with six-generator seven-bus topology is given to verify the progressiveness and effectiveness of the proposed scheme through Matlab/Simulink simulation.

New stability criterion of fractional-order impulsive coupled non-autonomous systems on networks

This paper explores stability of new fractional-order impulsive coupled (FOIC) non-autonomous systems on networks. By utilizing graph theory and structuring individual vertex’s Lyapunov function, a systematic method is present to construct global Lyapunov functions of FOIC systems. By Lyapunov method, a new inequality as well as the graph theory, several new stable principles are derived based on the strongly connected property. Besides, several more general criteria are obtained by utilizing these principles to the FOIC systems, including uniform stability, exponential stability and Mittag-Leffler stability for a kind of FOIC systems. At last, a numerical example is provided to show the effectiveness of our findings.

Synchronization of stochastic multi-weighted complex networks with Lévy noise based on graph theory

This paper addresses the synchronization problem of stochastic multi-weighted complex networks with Lévy noise. Based on the drive-response concept and graph theory, global Lyapunov function of the error network between drive-response networks is obtained by the weighted summation of Lyapunov functions of vertex systems. According to the stochastic analysis and state feedback control technique, the rigorous synchronization analysis of drive-response networks in the pth moment and probability sense is presented. The obtained synchronization criteria are closely related with multi-weights and the intensity of Lévy noise. Finally, some numerical simulations are provided to illustrate the effectiveness of the theoretical results.

Vaccination of a multi-group model of zoonotic diseases with direct and indirect transmission

Vaccination plays an important role in preventing or reducing the spread of zoonotic diseases. In this paper, we develop a susceptible-vaccinated-exposed-infectious-pathogen multi-group epidemic model of zoonotic diseases incorporating nonlinear direct and indirect incidence rates, nonlinear pathogen shedding rates, and common environmental contamination. Under certain assumptions, we first obtained the basic reproduction number of the model. Then, we utilized the comparison principle and global Lyapunov function method to prove global stability of dynamical equilibria. Finally, we analyzed optimal vaccination strategy. All the theoretical predictions were verified by numerical simulations.

On Algebraic Proofs of Stability for Homogeneous Vector Fields

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function, which is the ratio of two polynomials (i.e., a rational function ). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares certificates and, hence, such a Lyapunov function can always be found by semidefinite programming . This generalizes the classical fact that an asymptotically stable linear system admits a quadratic Lyapunov function, which satisfies a certain linear matrix inequality. In addition to homogeneous vector fields, the result can be useful for showing local asymptotic stability of nonhomogeneous systems by proving asymptotic stability of their lowest order homogeneous component. This paper also includes some negative results: We show that in absence of homogeneity, globally asymptotically stable polynomial vector fields may fail to admit a global rational Lyapunov function, and in presence of homogeneity, the degree of the numerator of a rational Lyapunov function may need to be arbitrarily high (even for vector fields of fixed degree and dimension). On the other hand, we also give a family of homogeneous polynomial vector fields that admit a low-degree rational Lyapunov function but necessitate polynomial Lyapunov functions of arbitrarily high degree. This shows the potential benefits of working with rational Lyapunov functions, particularly as the ones whose existence we guarantee have structured denominators and are not more expensive to search for than polynomial ones.

Global attractor and Lyapunov function for one-dimensional Deneubourg chemotaxis system

We study the global-in-time existence and the asymptotic behavior of solutions to a one-dimensional chemotaxis system presented by Deneubourg (Insectes Sociaux 24 (1977)). The system models the self-organized nest construction process of social insects. In the limit as a time-scale coefficient tends to 0, the Deneubourg model reduces to a parabolic-parabolic Keller-Segel system with linear degradation. We first show the global-in-time existence of solutions. We next define the dynamical system of solutions and construct the global attractor. In addition, under the assumption of a large resting rate of worker insects, we construct a Lyapunov functional for the unique homogeneous equilibrium, which indicates that the global attractor consists only of the equilibrium.

Global asymptotic stability analysis of discrete-time stochastic coupled systems with time-varying delay

This paper is concerned with the problem of global asymptotic stability for the discrete-time stochastic coupled systems with time-varying delay (DSCSTD). Based on graph theory and Lyapunov method, a systematic method is given for constructing a global Lyapunov function for the DSCSTD. Some novel stability criteria for the DSCSTD are obtained. Furthermore, to demonstrate the significance in practice, the theoretical results are applied to stochastic coupled oscillators. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results. The above work will provide a theoretical principle for subsequent controller design of the DSCSTD in future.

Stabilization of multi-group models with multiple dispersal and stochastic perturbation via feedback control based on discrete-time state observations

This paper focuses on the multi-group models with multiple dispersal and stochastic perturbation (MGMDS). The effect of both multiple dispersal among groups and stochastic perturbation are taken into consideration. The stabilization of MGMDS is investigated via feedback control based on the discrete-time state observations. In addition, a systematic method is given to construct a global Lyapunov function for MGMDS. By means of graph theory and Lyapunov method, some sufficient conditions are obtained to ensure the stabilization in the sense of mean-square asymptotical stability. An upper bound of the duration between two consecutive state observations is estimated. Moreover, to show the applicability of our results, the main theory is employed to a stochastic coupled oscillators. Finally, a numerical example is given to illustrate the effectiveness and feasibility of the theoretical results.

Stationary distribution of stochastic multi-group models with dispersal and telegraph noise

Graph-theoretic approach as a new technique is applied to analyze the issue of stationary distribution for stochastic multi-group models (MGMs) with Markovian switching and dispersal. The feature of our model is combining the white noise, the dispersal with Markovian switching, which can model systems in practice reasonably. By constructing a global Lyapunov function for stochastic MGMs with Markovian switching and dispersal, sufficient criteria, which guarantee the existence of a stationary distribution, are presented via the combination of the graph theory, the Lyapunov method and the M-matrix method. Moreover, stochastic coupled oscillators with Markovian switching and dispersal are presented as a practical application to illustrate our results in the end.

A Nonequilibrium-Potential Approach to Competition in Neural Populations

Energy landscapes are a highly useful aid for the understanding of dynamical systems, and a particularly valuable tool for their analysis. For a broad class of rate neural- network models of relevance in neuroscience, we derive a global Lyapunov function which provides an energy landscape without any symmetry constraint. This newly obtained “nonequilibrium potential” (NEP)—the first one obtained for a model of neural circuits—predicts with high accuracy the outcomes of the dynamics in the globally stable cases studied here. Common features of the models in this class are bistability—with implications for working memory and slow neural oscillations—and population bursts, associated with signal detection in neuroscience. Instead, limit cycles are not found for the conditions in which the NEP is defined. Their nonexistence can be proven by resorting to the Bendixson–Dulac theorem, at least when the NEP remains positive and in the (also generic) singular limit of these models. This NEP constitutes a powerful tool to understand average neural network dynamics from a more formal standpoint, and will also be of help in the description of large heterogeneous neural networks.

Asymptotic analysis of endemic equilibrium to a brucellosis model.

Brucellosis is one of the worlds major infectious and contagious bacterial disease. In order to study different types of brucellosis transmission models among sheep, we propose a deterministic model to investigate the transmission dynamics of brucellosis with the flock of sheep divided into basic ewes and other sheep. The global dynamical behavior of this model is given: including the basic repro-duction number, the existence and uniqueness of positive equilibrium, the global asymptotic stability of the equilibrium. We prove the uniqueness of positive endemic equilibrium through using proof by contradiction, and the global stability of endemic equilibrium by using Lyapunov function. Especially, we give the specific coefficients of global Lyapunov function, and show the calculation method of these specific coefficients. By running numerical simulations for the cases with the basic reproduction number to demonstrate the global stability of the equilibria and the unique endemic equilibrium, re-spectively. By some sensitivity analysis of the basic reproduction number on parameters, we find that vaccination rate of sheep and seropositive detection rate of recessive infected sheep are very important factor for brucellosis.

Distributed Coordination for Nonsmooth Convex Optimization via Saddle-Point Dynamics

This paper considers continuous-time coordination algorithms for networks of agents that seek to collectively solve a general class of nonsmooth convex optimization problems with an inherent distributed structure. Our algorithm design builds on the characterization of the solutions of the nonsmooth convex program as saddle points of an augmented Lagrangian. We show that the associated saddle-point dynamics are asymptotically correct but, in general, not distributed because of the presence of a global penalty parameter. This motivates the design of a discontinuous saddle-point-like algorithm that enjoys the same convergence properties and is fully amenable to distributed implementation. Our convergence proofs rely on the identification of a novel global Lyapunov function for saddle-point dynamics. This novelty also allows us to identify mild convexity and regularity conditions on the objective function that guarantee the exponential convergence rate of the proposed algorithms for convex optimization problems subject to equality constraints. Various examples illustrate our discussion.

A Lyapunov Function Construction for a Non-convex Douglas–Rachford Iteration

While global convergence of the Douglas-Rachford iteration is often observed in applications, proving it is still limited to convex and a handful of other special cases. Lyapunov functions for difference inclusions provide not only global or local convergence certificates, but also imply robust stability, which means that the convergence is still guaranteed in the presence of persistent disturbances. In this work, a global Lyapunov function is constructed by combining known local Lyapunov functions for simpler, local sub-problems via an explicit formula that depends on the problem parameters. Specifically, we consider the scenario where one set consists of the union of two lines and the other set is a line, so that the two sets intersect in two distinct points. Locally, near each intersection point, the problem reduces to the intersection of just two lines, but globally the geometry is non-convex and the Douglas-Rachford operator multi-valued. Our approach is intended to be prototypical for addressing the convergence analysis of the Douglas-Rachford iteration in more complex geometries that can be approximated by polygonal sets through the combination of local, simple Lyapunov functions.

Global dynamics of an infinite dimensional epidemic model with nonlocal state structures

In this paper, we derive and analyze a state-structured epidemic model for infectious diseases in which the state structure is nonlocal. The state is a measure of infectivity of infected individuals or the intensity of viral replications in infected cells. The model gives rise to a system of nonlinear integro-differential equations with a nonlocal term. We establish the well-posedness and dissipativity of the associated nonlinear semigroup. We establish an equivalent principal spectral condition between the linearized operator and the next-generation operator and show that the basic reproduction number R0 is a sharp threshold: if R0<1, the disease-free equilibrium is globally asymptotically stable, and if R0>1, the disease-free equilibrium is unstable and a unique endemic equilibrium is globally asymptotically stable. The proof of global stability of the endemic equilibrium utilizes a global Lyapunov function whose construction was motivated by the graph-theoretic method for coupled systems on networks developed in [24].

Output synchronization control for a class of complex dynamical networks with non-identical dynamics

This paper is concerned with the output synchronization control problem for a class of complex dynamical networks (CDNs) with non-identical dynamics. An output feedback control protocol consisting of a feedforward control law and a feedback control law is developed to achieve the output synchronization. More specifically, the feedforward control law is designed to compensate the coupling dynamics of the CDNs, and the feedback control law is designed by solving an algebraic Riccati equation (ARE), which is established by defining a modified quadratic performance index. It is shown that the output feedback control protocol solves the output synchronization control problem, and the graph theory is used to construct a global Lyapunov function, based on which a rigorous asymptotic convergence analysis of output synchronization errors is conducted. Finally, a simulation example is given to verify the effectiveness of the theoretical results.

Adaptive Fault Estimation for T-S Fuzzy Interconnected Systems Based on Persistent Excitation Condition via Reference Signals.

This paper is concerned with the fault estimation (FE) problem for a class of interconnected nonlinear systems described by Takagi-Sugeno fuzzy models. Different from the existing FE approaches, where the reference inputs are always viewed as external disturbances, the considered reference signals are injected directly in the controller to excite the system states to ensure persistent excitation condition. Within this framework, a bank of distributed adaptive observers are then designed to effectively estimate the actuator fault parameters even in the presence of nonlinear interconnections. Moreover, in the disturbance-free case, by employing graph theory, a global Lyapunov function is constructed such that the corresponding fault parameter estimate errors are proved to be asymptotically convergent provided that the interconnected strengths are less than a given constant. Finally, two simulation examples are provided to demonstrate the validity of the presented FE scheme.

Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming

We study the global asymptotic stability in probability of the zero solution of linear stochastic differential equations with constant coefficients. We develop a sum-of-squares program that verifies whether a parameterized candidate Lyapunov function is in fact a global Lyapunov function for such a system. Our class of candidate Lyapunov functions are naturally adapted to the problem. We consider functions of the form $V(\mathbf{x}) = \|\mathbf{x}\|_Q^p: = (\mathbf{x}^\top Q\mathbf{x})^{\frac{p}{2}}$, where the parameters are the positive definite matrix $Q$ and the number $p&gt;0$. We give several examples of our proposed method and show how it improves previous results.

Global stability of an SI epidemic model with feedback controls in a patchy environment

In this paper, we investigate an SI epidemic model with feedback controls in a patchy environment where individuals in each patch can disperse among n(n ≥ 2) patches. We derive the basic reproduction number R0 and prove that the disease-free equilibrium is globally asymptotically stable if R0 ≤ 1. In the case of R0 > 1, we derive sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory. Numerical simulations are carried out to support our theoretical results.