Stability in Lyapunov systems

The stability of the stationary point of a Lyapunov system [Malkin IG, Some Problems in the Theory of Non-linear Oscillations. Moscow: Gostekhizd; 1956.], which describes the perturbed motion of a dynamical system with two degrees of freedom, is investigated. It is assumed that the characteristic equation of the first approximation of the system has two pairs of pure imaginary roots and that the quadratic part of the integral is not sign-definite. Both the non-resonance case, as well as cases of lower order (second-, third- and fourth-order) resonances are considered. The necessary and sufficient conditions for stability are given in cases when the problem is solved by a combination of the first non-linear terms of normal form.     

Generalized Lyapunov inequalities involving critical Sobolev exponents

We consider Lyapunov-type inequalities generalizing the famous inequality that gives a necessary condition for the existence of solutions to a boundary value problem for a second order ordinary differential equation. For certain critical cases, when the inequalities are strict, we study the asymptotic behavior of minimizing sequences.     

Stability theory and Lyapunov regularity

We establish the stability under perturbations of the dynamics defined by a sequence of linear maps that may exhibit both nonuniform exponential contraction and expansion. This means that the constants determining the exponential behavior may increase exponentially as time approaches infinity. In particular, we establish the stability under perturbations of a nonuniform exponential contraction under appropriate conditions that are much more general than uniform asymptotic stability. The conditions are expressed in terms of the so-called regularity coefficient, which is an essential element of the theory of Lyapunov regularity developed by Lyapunov himself. We also obtain sharp lower and upper bounds for the regularity coefficient, thus allowing the application of our results to many concrete dynamics. It turns out that, using the theory of Lyapunov regularity, we can show that the nonuniform exponential behavior is ubiquitous, contrarily to what happens with the uniform exponential behavior that although robust is much less common. We also consider the case of infinite-dimensional systems.     

Some results involving Fox’s H-function and Bessel function

In this paper we have evaluated an integral involving Fox’s H-function and employed it to establish an expansion formula for the H-function involving Bessel functions.     

Exact multiplicity results for variational inequalities involving pointwise constraints on the derivatives

This paper concerns a class of nonlinear variational problems involving pointwise constraints on the second derivatives. Our aim is to describe the set of data for which these problems have solutions and to analyse the structure of the set of solutions under suitable assumptions on the asymptotic behaviour of the nonlinear term. In particular, if this term is assumed to be convex, then we can specify the number of solutions and obtain exact multiplicity results.The existence, nonexistence and multiplicity results we obtain show that the presence of constraints of this kind produces some phenomena which are typical of nonlinear elliptic equations with “jumping” nonlinearities.     

Inequalities involving partial derivatives

This paper presents a method for obtaining closed form solutions to serial and nonserial dynamic programming problems with quadratic stage returns and linear transitions. Global parametric optimum solutions can be obtained regardless of the convexity of the stage returns. The closed form solutions are developed for linear, convex, and nonconvex quadratic returns, as well as the procedure for recursively solving each stage of the problem. Dynamic programming is a mathematical optimization technique which is especially powerful for certain types of problems. This paper presents a procedure for obtaining analytical solutions to a class of dynamic programming problems. In addition, the procedure has been programmed on the computer to facilitate the solution of large problems.     

Stability of delay equations via Lyapunov functions

The importance of Lyapunov functions is well known. In the general setting of nonautonomous linear delay equations v^′=L(t)vt, we show how to characterize completely the existence of a nonuniform exponential contraction or of a nonuniform exponential dichotomy in terms of Lyapunov functions. This includes uniform exponential behavior as a very special case, and it provides an alternative (usually simpler and particularly more direct) approach to verify the existence of exponential behavior or to obtain the robustness of the dynamics under sufficiently small perturbations.     

广义大系统的Lyapunov稳定性分析

广义大系统的稳定性是一个非常重要的问题 ,由于广义大系统的复杂性 ,对其稳定性的研究也是一件相当困难的事情 .本文利用 Lyapunov方程 ,应用 Lyapunov函数法 ,研究了广义线性大系统和广义非线性大系统的稳定性和不稳定性 ,得到了系统的关联稳定参数域和不稳定域 .给出例子说明该方法的可行性 .     

Generalized sinc-Gaussian sampling involving derivatives

The generalized sampling expansion which uses samples from a bandlimited function f and its first r derivatives was first introduced by Linden and Abramson (Inform. Contr. 3, 26–31, 1960) and it was extended in different situations by some authors through the last fifty years. The use of the generalized sampling series in approximation theory is limited because of the slow convergence. In this paper, we derive a modification of a generalized sampling involving derivatives, which is studied by Shin (Commun. Korean Math. Soc. 17, 731–740, 2002), using a Gaussian multiplier. This modification is introduced for wider classes, the class of entire functions including unbounded functions on ? and the class of analytic functions in a strip. It highly improves the convergence rate of the generalized sampling which will be of exponential order. We will show that many known results included in Sampl. Theory Signal Image Process. 9, 199–221 (2007) and Numer. Funct. Anal. Optim. 36, 419–437 (2015) are special cases of our results. Numerical examples show a rightly good agreement with our theoretical analysis.     

Some results involving Bessel function and Fox’s H-function of two variables

In this paper, we have presented two integrals and employed them to establish one Fourier Bessel expansion for Fox’s H-function of two variables.     

改进的口腔微生物种群模型及其Lyapunov稳定性

考察人体口腔异味现象,利用恒化器建模方法,改进了人体口腔系统中微生物种群关系的模型,利用Lyapunov稳定性理论分析了系统的平衡点及其稳定性.进而得到结论,口腔异味作为疾病,需要专业医治才能治愈.数值模拟结果证实了理论分析的正确性.     

Stability of impulsive system by perturbing Lyapunov functions

This paper focuses on the stability of the zero solution for the impulsive differential system at fixed times by the method of perturbing Lyapunov functions. Based on the method, some sufficient conditions for the above stability to hold are given.     

Stability results for Mountain Pass and Linking type solutions of semilinear problems involving Dirichlet forms

Some stability results for Mountain Pass and Linking type solutions of semilinear problems involving a very general class of Dirichlet forms are stated. The non linear terms are supposed to have a suitable superlinear growth and the family of Dirichlet forms is required to be dominated from below and from above by a fixed diffusion type form. Some concrete examples are also given.     

A Lyapunov function for Leslie-Gower predator-prey models

A Lyapunov function for continuous time Leslie-Gower predator-prey models is introduced. Global stability of the unique coexisting equilibrium state is thereby established.