Qualitative Analysis Of Impulsive Functional Diferential Systems With Infinite Delays

In this paper, we mainly study impulsive functional diferential systems with infinitedelayswhere xt(θ)=x(t+θ), t≥t0≥0≥a≥∞, θ∈[a,0], a can be∞.At present, mainly impulsive functional diferential systems with finite delays arestudied, but there are few researches of impulsive functional diferential systems withinfinite delays because of its complexity. The method of Lyapunov functions and theRazumikhin technique have been very efective in the study of impulsive functional dif-ferential equations. But it is difcult to construct appropriate Lyapunov functions. Inaddition, documentation [22] developed a new technique in studying stability of FDE))Lyapunov function of partial components, in which the components of x are divid-ed into several groups. Correspondingly, several Lyapunov functions are adopted, thentheorems of stability are established, where every Lyapunov function satisfies weaker con-ditions and is easier to be constructed. However, when some specific impulsive functionaldiferential systems are studied, more convenient and useful results may be obtained byemploying appropriate inequality technique according to the structural features of thesystems. Based on the ideas above, the three methods will be used to study the system(I) in this paper. This paper is divided into three chapter.In chapter one, mainly asymptotic stability of zero solution of system (I) is studied.Firstly, new criteria on global asymptotic stability of system (I) are established by usingLyapunov function and Razumikhin technique. The proof idea of new theorem is diferentfrom the usual idea in which uniform asymptotic stability is usually studied(see[10,11]).The given Razumikhin condition needs weaker request, and can be verified more easily. The sign of Dini derivative of V can change. Then, new comparison principles are estab-lished for system (I) by using Razumikhin technique and several Lyapunov functions ofpartial components of the state variable x. By employing the comparison principles, wecan obtain the corresponding stability of system (I) according to the stability of impulsivediferential system.In chapter two, mainly exponential stability of zero solution of system (I) is studied.Firstly, new Razumikhin-type theorems on global exponential stability are obtained byusing Razumikhin technique and Lyapunov function. It allows V to increase significantlyat impulse times, as long as the decrease of V between impulse times balances themproperly. Then, by using Razumikhin technique and several Lyapunov functions of partialcomponents of the state variable, global exponential stability of system (I) is studed, andnew Razumikhin-type theorems on global exponential stability are obtained. Due to theefect of impulse, in the theorems the Dini derivative of V about the system (I) needn’t benegative definite or negative and can be allowed to be positive, highlighting the influenceto the property of the solutions of system (I) that the impulse acts.In chapter three, we mainly study stability of the impulsive Hopfield Neural Net-works modelFirstly, new extended Halanay inequality is established. By using Lyapunov functionand the Hanalay inequality, we study global asymptotic stability of the equilibrium pointof model (II), and obtain some new criteria. Then, by employing the results obtainedin chapter two, some criteria of global exponential stability of the equilibrium point aregiven, which permit that the distance between the solutions and equilibrium point isincreased at impulse times, which improve the results in [41] in some degree.