Stability Analysis Of Impulsive Perturbed Differential Systems

Perturbed differential systems are applied widely in the field of natural science. It is a important part of the research about differential systems. That is because in real life,when we establish a mathematical model by differential systems for the specific issue , there are always some tiny perturbed force that can not be specifically identified,and these perturbed force often have influence to the system.These perturbed force is called perturbation, and the corresponding differential systems known as the perturbed differen-tial systems. In recent years,for the theoretical significance and the value of application of the perturbed differential systems, the study of which has made some progress.At the same time,impulse as a instantaneous change phenomenon,which is very common in the nature. As the development of science research,the differential systems with im-pulses and perturbations get more and more attentions and study by scholars for his wide application in the field of life sciences and communications. Special for perturbed sys-tems with impulses at the fixed times,the stability theory become mature gradually. But,there are still a lot of room for improvement.For perturbed systems with impulses at variable times,there are few stability results.In this paper,we consider the following two perturbed systems with impulses where f(t,x)=F(t, x)+R(t,x),Ik(x)=Ik*(x)+Qk(x), R, Qk are perturbations where f(t,x)=F(t,x)+R(t,x), R,Ik are perturbations This paper mainly discuss the stability of the systems above,and establish some criterias on stability,T-stability and strict stability.There are two parts in this paperIn the first chapter,we discuss the stability of perturbed system with impulses at fixed times by the method of generalized second derivative and perturbed Lyapunov function.In the section 3,we give the concept of the generalized second derivative of vari-ational Lyapunov function.By the method of giving combined condition about discrete and continuous parts of an impulsive systems,stability criterias are established.These cri-terias do not need the Dini derivative of the variational Lyapunov function is negative definite along solution trajectory,and growth is allowed,but limit the rate of growth by the concept of bounded growth. The section 4 give a new stability criteria of perturbed systems with impulses by a row of variational Lyapunov functions'perturbations. The last section of this chapter presents the concept of T-stability of the perturbed differential systems with impulses at fixed times firstly. As far as we know, T-stability is a kind of description about the stability of solution in the case of perturbations size can be esti-mated. Based on this feature,we gave a sufficient condition by the method of Lyapunov function and gives a example to illustrate the application of it.As far as we know,for perturbed differential systems with impulses at variable times,impulsive times dependence to the solution leads to the defination of stability is different from the perturbed differential systems with impulses at fixed times,and the stability of non-trivial solution is also different from trivial solution.So far,the research of this kind of system is ont enough. In the second chapter, We first establish a new variational comparison principle by virtue of variational Lyapunov method and differen-tial inequality. And then,we can choose an appropriate function to link the solution of perturbed systems with impulses at variable times with the solution of ordinary differen-tial systems.On this basis, we give some criterias on stability of perturbed system with impulses at variable times.