THE ALL-DELAY STABILITY OF DEGENERATE DIFFERENTIAL SYSTEMS WITH DELAY

In this paper,the all-delay stability of degenerate differential systems with delay is discussed.We come up with some new criteria for evaluating the all-delay stability of degenerate differential systems with delay and degenerate neutral differential systems with delay.Also,we give an example to illustrate the main results.     

OSCILLATION THEOREM TO SYSTEMS OF IMPULSIVE NEUTRAL DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper,we study the oscillation of solutions to the systems of impul- sive neutral delay parabolic partial differential equations.Under two different boundary conditions,we obtain some sufficient conditions for oscillation of all solutions to the systems.     

ON LIAPUNOV FUNCTIONAL IN THEORY OF STABILITY OF SYSTEMS WITH TIME-LAG

It is well known,that in the theory of stability in differential equations,Liapunov's second method may be the most important The center problem of Liapunov's second method is construction of Liapunov function for concrete problems.Beyond any doubt,construction of Liapunov functions is an art.In the case of functional differential equations,there were also many attempts to establish various kinds of Liapunov type theorems.Recently Burton[2]presented an excellent theorem using the Liapunov functional to solve the asymptotic stability of functional differential equation with bounded delay. However,the construction of such a Liapunov functional is still very hard for concrete problems. In this paper, by utilizing this theorem due to Burton,we construct concrete Liapunov functional for certain and nonlinear delay differential equations and derive new sufficient conditions for asymptotic stability.Those criteria improve the result of literature[1]and they are with simple forms,easily checked and applicable.     

OSCILLATION OF SYSTEMS OF IMPULSIVE DELAY PARABOLIC EQUATIONS ABOUT BOUNDARY VALUE PROBLEMS

In this paper,oscillation of solutions to a class of impulsive delay para- bolic partial differential equations system with higher order Laplace operator is studied.Under two different boundary value conditions,we establish some sufficient criteria with respect to the oscillations of such systems,employing first-order impulsive delay differential inequalities.The results fully reflect the influence action of impulsive and delay in oscillation.     

STABILITY OF TIME VARYING SINGULAR DIFFERENTIAL SYSTEMS WITH DELAY

In this paper, stability of time varying singular differential systems with delay is considered. Based on variation formula and Gronwall-Bellman integral inequality, we obtain the exponential estimation of the solution and the sufficient conditions under which the considered system is stable and exponentially asymptotically stable. These results will be very useful to further research on Roust stability and control design of uncertain singular control systems with delay.     

ON THE EQUIVALENCE OF AUTONOMOUS AND NONAUTONOMOUS DIFFERENTIAL SYSTEMS

Using reflecting function of Mironenko we construct some nonautonomous differential systems which are equivalent to the given autonomous differential systems. The results are applied to discuss the existence and stability of periodic solution of these nonautonomous nonlinear differential systems.     

时不变和时变的Caputo分数阶时滞微分系统的常数变易公式

This paper studies the variation of constant formulae for linear Caputo fractional delay differential systems. We discuss the exponential estimates of the solutions for linear time invariant fractional delay differential systems by using the Gronwall＇s integral inequality. The variation of constant formula for linear time invariant fractional delay differential systems is obtained by using the Laplace transform method. In terms of the superposition principle of linear systems and fundamental solution matrix, we also establish the variation of constant formula for linear time varying fractional delay differential systems. The obtained results generalize the corresponding ones of integer-order delayed differential equations.     

A RESULT OF SYSTEMS OF NONLINEAR COMPLEX ALGEBRAIC DIFFERENTIAL EQUATIONS

In this article, we mainly investigate the behavior of systems of complex differential equations when we add some condition to the quality of the solutions, and obtain an interesting result, which extends Gaekstatter and Laine＇s result concerning complex differential equations to the systems of algebraic differential equations.     

EXISTENCE OF MILD SOLUTIONS TO NEUTRAL FRACTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper, we consider the existence of mild solution to a class of neutral fractional differential equations with infinite delay. By means of fixed points methods, we obtain some sufficient conditions for the existence and uniqueness of mild solutions, which extend some known results.     

On liapunov functional in theory of stability of systems with time-lag

It is well known, that in the theory of stability in differential equations, Liapunov's second method may be the most important. The center problem of Liapunov's second method is construction of Liapunov function for concrete problems. Beyond any doubt, construction of Liapunov functions is an art. In the case of functional differential equations, there were also many attempts to establish various kinds of Liapunov type theorems. Recently Burton [2] presented an excellent theorem using the Liapunov functional to solve the asymptotic stability of functional differential equation with bounded delay. However, the construction of such a Liapunov functional is still very hard for concrete problems. In this paper, by utilizing this theorem due to Burton, we construct concrete Liapunov functional for certain and nonlinear delay differential equations and derive new sufficient conditions for asymptotic stability. Those criteria improve the result of literature [1] and they are with simple forms, easily checked and applicable.This project is supported by the National Natural Science Foundation of China.     

Two families of liapunov functions for functional differential systems

Two families of Liapunov functions are employed to study the global stability and boundedness of functional differential systems. New stability and boundedness theorems are obtained. Applications of these theorems to some nonlinear differential systems with infinite delay are discussed. Project supported by the National Science Foundation of China Under Grants 69871005     

时滞微分方程依照两种测度的稳定性

运用变异Liapunov方法，讨论了时滞微分方程依照两种测度的稳定性。借助于中间测度h^*(t,x），在未扰动系统为常微分方程的情形下，得到了关于时滞微分方程非一致和一致稳定性的判定定理。     

变时滞的退化滞后型微分系统的稳定性

主要研究了具有变时滞的退化时滞微分系统的稳定性.利用退化时滞微分系统的变易公式和Gronwall-Bellman积分不等式给出了该系统的指数估计以及稳定和指数渐近稳定的充分条件.     

非线性无穷时滞微分系统的稳定性

本文主要内容是：（１）在局部条件下，建立了一类具有无穷时滞的非线性的正、反向积分微分不等式；（２）结合常数变易法和Ｌｉａｐｕｎｏｖ函数法（不必寻找Ｐ－函数），利用所建立的不等式研究了非线性无穷时滞微分系统的稳定性；（３）举例说明了本文定理的应用和优点。     

Opial's inequality and oscillation of 2nd order equations

For a second-order differential equation, we obtain from Opial's inequality lower bounds for the spacing between two zeros of a solution or between a zero of a solution and a zero of its derivative. These bounds are expressed in terms of antiderivatives of the potential, and in particular we derive some new Liapunov type inequalities from them.

     

无穷时滞脉冲方程的一致渐近稳定性

本文利用Liapunov泛函与Liapunov函数方法建立了无穷时滞脉冲泛函微分方程基于两种测度的一致稳定和一致渐近稳定的一个新的定理,并通过实例说明了所获结论的应用.     

Some Solvable Stochastic Control Problems With Delay

We consider optimal harvesting of systems described by stochastic differential equations with delay. We focus on those situations where the value function of the harvesting problem depends on the initial path of the process in a simple way, namely through its value at 0 and through some weighted averages

A verification theorem of variational inequality type is proved. This is applied to solve explicitly some classes of optimal harvesting delay problems     

概周期系统概周期解的存在唯一性

应用构造Ляпунов函数的方法，讨论了非线性微分方程系概周期解的存在唯一性.同时给出了Liénard方程存在唯一概周期解的一组充分条件.     

具有无穷时滞中立型泛函微分方程概周期解的存在性

对具有无穷时滞中立型泛函微分方程，本文利用Ｌｉａｐｕｎｏｖ泛函，建立概周期解的存在性定理．