Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations

This paper studies the practical stability of the solutions of nonlinear impulsive functional differential equations. The obtained results are based on the method of vector Lyapunov functions and on differential inequalities for piecewise continuous functions. Examples are given to illustrate our results.     

Stability and boundedness criteria of nonlinear impulsive systems employing perturbing Lyapunov functions

This paper develops a new comparison principle for nonlinear impulsive differential systems, then the stability, practical stability and boundedness of impulsive differential systems are proved by using the method of perturbing Lyapunov functions. The notion of perturbing Lyapunov functions enables us to discuss stability properties of impulsive systems under much weaker assumptions. The reported novel results complement the existing results. It may provide a greater prospect for solving problems which exhibit impulsive effects.     

Lyapunov vector functions and partial boundedness of solutions with partially controlled initial conditions

On the basis of the method of Lyapunov vector functions, we obtain a sufficient test for the uniform partial boundedness of solutions with partially controlled initial conditions. We introduce the notions of partial equiboundedness, partial equiboundedness in the limit, and partial uniform boundedness in the limit of solutions with partially controlled initial conditions. By the method of Lyapunov vector functions, we obtain sufficient tests for the partial equiboundedness of solutions and for the partial uniform boundedness in the limit and partial equiboundedness in the limit of solutions with partially controlled initial conditions.     

Higher-order derivatives of Lyapunov functions and partial boundedness of solutions with partially controllable initial conditions

Mathematical Notes - Certain sufficient criteria for the types of partial boundedness of solutions with partially controllable initial conditions are obtained in terms of higher-order derivatives...     

Non-Autonomous stochastic differential equations in hubert spaces: Lyapunov Function,stability and ultimate boundedness

In this paper, we establish the Lyapunov function characterization which is a sufficient and necessary condition for mild solutions of semilinear stochastic evolution equations to be exponentially stable in mean square. We also study the Lyapunov function characterization of ultimate exponential boundedness, a concept which is closely related to the existence of invariant measures of non-stationary stochastic evolution equations     

Liapunov functions and boundedness

In this paper we give conditions under which one can conclude that all solutions of a differential equation are bounded when there is a Liapunov function which is not radially unbounded. The problem of Lurie and the Liénard equation are given as examples.     

Perturbing Lyapunov functions and stability criteria for initial time difference

Stability criteria for differential equations where the initial time for each solution is different is developed using the method of perturbing Lyapunov functions.     

Stability and boundedness of nonlinear impulsive systems in terms of two measures via perturbing Lyapunov functions

This paper develops the concepts of stability, practical stability and boundedness in terms of two measures for nonlinear impulsive differential systems using the method of perturbing Lyapunov functions. The notion of perturbing Lyapunov functions enables us to discuss stability properties of solutions of nonlinear impulsive differential systems in terms of two measures under much weaker assumptions. The novel results offer a way to unify a variety of stability results found in the relative literature.     

Cone-valued Lyapunov functions and stability for impulsive functional differential equations

The stability criteria in terms of two measures for impulsive functional differential equations are established via cone-valued Lyapunov functions and Razumikhin technique. The stability can be deduced from the (Q₀,Q)-stability of comparison impulsive differential equations. An example is given to illustrate the advantages of the results obtained.     

Nonlinear stability in reaction-diffusion systems via optimal Lyapunov functions

We define optimal Lyapunov functions to study nonlinear stability of constant solutions to reaction-diffusion systems. A computable and finite radius of attraction for the initial data is obtained. Applications are given to the well-known Brusselator model and a three-species model for the spatial spread of rabies among foxes.     

Lyapunov functions for investigating the global stability of non-linear systems

The sufficient conditions of the sign determinacy of a sum of polylinear forms are obtained. From these systems the Lyapunov functions which are used to derive the sufficient conditions of the global asymptotic stability of the non-perturbed motion of non-linear systems are set up. At the same time the perturbed motion is described by a set of ordinary differential equations with the right-hand side in the form of the sum of homogeneous polynomials. An application to the analysis of the stability of winged aircraft is considered.     

Lyapunov modular functions

We prove a Lyapunov type theorem for modular functions on complemented lattices.     

Modular functions: Uniform boundedness and compactness

We prove the Nikodym Boundedness, Brooks-Jewett and Vitali-Hahn-Saks theorems for modular functions on orthomodular lattices with SIP and on particular complemented or sectionally complemented lattices, and the equivalence, for any complemented or sectionally complemented lattice, between the Brooks-Jewett and Vitali-Hahn-Saks theorems for group-valued modular functions. As consequence, we obtain characterizations of relative, sequential and weak compactness in spaces of modular functions.     

Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models

Lyapunov functions for classical SIR, SIRS, and SIS epidemiological models are introduced. Global stability of the endemic equilibrium states of the models is thereby established.