Stability Analysis Of Impulisive Functional Differential Systems With Infinite Delays

In this paper we mainly study impulisive functional differential systems with infinite delavs(1)and consider system(2)where F(t,x_t)=f(t,x)+R(t,x_t),and x_t(θ)=x(t+θ).t≥t₀≥0≥a≥-∞,a can be-∞,θ∈|a,0|.In many fields of modern science and technology,such as control theory,physics.chemistry,population dynamics,biotechnology,industrial robotics,economies and etc,some mathematical models of practical questions can be summed up as impulsive functionaldifferential systems.So it is of great significance to study it,and it has become a hot in recent years.At present,it mainly studies the impulsive functional differential systemswith finite delays(papers[9]-[l6]).but they have few researches with infinite delays.In[9,12.13],the major research methods arc Lyapunov method,several Lyapunov functionsincluding partial components and Razumikhin technique. They are effective tools,butit is difficult to construct appropriate Lyapunov functions.In [17],they combine parametervariational method and Lyapunov functions methods to make a new method,thatis Variational Lyapunov method. At the same time.they give a new idea that choosea suitable cone other than R_n⁺to work in a given situation in [29],where every conevalued Lyapunov functions stabilities weaker conditions and is easier to construct.Basedon these ideas,we employ variational Lyapunov functions method and cone-valued Lyapunov functions method to study the properties of impulsive functional differential systemwith infinite delays.This paper is divided into two chapters.In chapter one,we construct variational Lyapunov functions to combine two systems(1)and (2),by means of direct method and Razumikhin technique,using middle measure h^*,then we get the corresponding ((?),h)- stability of system (1)by the (h₀,h^*)- stabilityproperties of system (2).Our results are not only effective but suitable for many applications.Finally we give an example to illustrate the advantages of our results.In chapter two,firstly we give the defination of the cone and the order relation onthe cone.And we introduce the conception of cone-valued Lyapunov functions and itsderivative along the solution of system (1).Then we study the uniform boundedness and uniformly ultimate boundedness properties in terms of two measures of system(3)where x_t(s)=x(t+s),t≥t₀≥0≥a≥-∞,a can be-∞,s∈[a,0],By using conevaluedLyapunov functions coupled with Razumikhin techinque.it is not necessary torequire our results to satisfy quasimonotone nondecreasing property of the comparationsystem after choosing a suitable cone other than R_n⁺.So we make the method more usful.