Lyapunov type inequalities for second-order half-linear differential equations

In this paper, we establish some new Lyapunov type inequalities for second-order half-linear differential equations, which almost generalize and extend all related existing results in the literature.     

Construction of Lyapunov functionals for delay differential equations in virology and epidemiology

In the present paper, we present a method for constructing a Lyapunov functional for some delay differential equations in virology and epidemiology. Here some delays are incorporated to the original ordinary differential equations, for which a Lyapunov function is already obtained. We present simple and clear explanation of our method using some models whose Lyapunov functionals are already obtained. Moreover, we present several new results for constructing Lyapunov functionals using our method.     

Stabilization of Linear Nonautonomous Systems with Norm-Bounded Controls

In this paper, we study the stabilization problem for a class of linear nonautonomous systems with norm-bounded controls. Using the Lyapunov function technique, we establish simple verifiable stabilizability conditions without solving any Riccati differential equation. Numerical examples are given to illustrate the results.Communicated by F. E. UdwadiaThis work was supported by the National Basic Program in Natural Sciences, Vietnam and by a Thailand Research Fund Grant. The authors thank the anonymous referees for valuable comments and remarks which have improved the paper.     

On the stability of impulsive functional differential equations with infinite delays

In this paper, the stability problem of impulsive functional differential equations with infinite delays is considered. By using Lyapunov functions and the Razumikhin technique, some new theorems on the uniform stability and uniform asymptotic stability are obtained. The obtained results are milder and more general than several recent works. Two examples are given to demonstrate the advantages of the results. Copyright © 2014 John Wiley & Sons, Ltd.     

平面不可压缩的Navier-Stokes方程的一个新七模截断(英文)

A new seven-modes truncation of Fourier series of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained.And its stationary solutions,the existence of attractor and the global stability of the equations are firmly proved.At the same time,several issues such as some basic dynamical behaviors and routs to chaos are shown numerically by changing Reynolds number.The system exhibits a stochastic behavior approached through an involved sequence of bifurcations.     

New Lyapunov‐type inequalities for a class of even‐order linear differential equations

In this paper, we obtain some new Lyapunov‐type inequalities for a class of even‐order linear differential equations, the results are new and generalize and improve some early results in this field.     

LAGRANGE STABILITY IN MEAN SQUARE OF STOCHASTIC REACTION DIFFUSION EQUATIONS

This work is devoted to the discussion of stochastic reaction diffusion equations and some new theorems on Lagrange stability in mean square of the solution are established via Lyapunov method which is nothing to be done in the past.     

Asymptotic stability and instability with respect to part of variables for solutions to impulsive systems

We obtain sufficient conditions for asymptotic stability with respect to part of variables for the zero solution to an impulsive system with the fixed moments of impulse effects.     

Lyapunov-type integral inequalities for certain higher order differential equations

In this paper, first we will give a short survey of the most basic results on Lyapunov inequality, and next we obtain this-type integral inequalities for certain higher order differential equations. Our results are sharper than some results of Yang (2003) [20].     

Lyapunov inequalities for partial differential equations

This paper is devoted to the study of Lp Lyapunov-type inequalities (1?p?+∞) for linear partial differential equations. More precisely, we treat the case of Neumann boundary conditions on bounded and regular domains in RN. It is proved that the relation between the quantities p and N/2 plays a crucial role. This fact shows a deep difference with respect to the ordinary case. The linear study is combined with Schauder fixed point theorem to provide new conditions about the existence and uniqueness of solutions for resonant nonlinear problems.     

Delay independent stability of linear switching systems with time delay

We consider a switching system with time delay composed of a finite number of linear delay differential equations (DDEs). Each DDE consists of a sum of a linear ODE part and a linear DDE part. We study two particular cases: (a) all the ODE parts are stable and (b) all the ODE parts are unstable and determine conditions for delay independent stability. For case (a), we extend a standard result of linear DDEs via the multiple Lyapunov function and functional methods. For case (b) the standard DDE result is not directly applicable, however, we are able to obtain uniform asymptotic stability using the single Lyapunov function and functional methods.     

Asymptotic Stability of Solutions to Perturbed Linear Difference Equations with Periodic Coefficients

We consider perturbed linear systems of difference equations with periodic coefficients. The zero solution of a nonperturbed system is assumed asymptotically stable, i.e., all eigenvalues of the monodromy matrix belong to the unit disk {||<1}. We obtain conditions on the perturbation of this system under which the zero solution of the system is asymptotically stable and also establish continuous dependence of one class of numeric characteristics of asymptotic stability of solutions on the coefficients of the system.     

非Lipschitz条件下非线性时变微分方程的稳定性

利用非Lipschitz李雅普诺夫函数给出了一类非线性时变微分方程指数稳定性的新的充分条件,改进了文献中的某些结果.     

Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations

Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.     

On the stability and boundedness of solutions of a kind of third order delay differential equations

In this paper, we study Eq. (1.1) for asymptotic stability of the zero solution when and uniformly bounded and uniformly ultimate bounded of all solutions when      

Estimates and exact expressions for lyapunov exponents of stochastic linear differential equations

We consider the asymptotic behavior of the solutions of a stochastic linear differential equation driven by a finite states Markov process. We consider the sample path Lyapunov exponent λ and the p-moment Lyapunov exponents g(p) for positive p. We derive relations between X and g{p\ which are extensions to our situation of results of Arnold [1] in a different context. Using a Lyapunov function approach, an exact expression forg(2) and estimates for g(p) are obtained, thus leading to upper and lower bounds for λ     

On Stability of Solutions of a Certain Fourth—Order Functional Differential Equations

Using a Razumikhin-type theorem,we obtain sufficient conditions for the global asymptoticstability of the zero solution of a certain fourth order functional differential equations.The resultgeneralizes the well known results.     

矩阵微分方程的等度有界性

用矩阵李雅普诺夫函数研究了矩阵微分方程的等度有界性,给出了非自治矩阵微分方程等度有界性的几个判定定理,实例说明了主要定理的实用性