The Stability Analysis Of Perturbed Differential Systems With Impulses At Variable Times

With the rapid development of science and technology,people gradually realize the importance of the impulsive differential systems,which is applied widely in each field of modern technology,such as the equipments for reducing quake and translating tech-nology about satellite orbit in aerospace technology,the study of neural net,genetics and epidemics in the medical field,controlling interest rate and industry management in the economical field and so on.So the qualitative study of the impulsive differen-tial system attracts many experts and scholars in and abroad who have made great progress.At the same time,in the real life,in the process of establishing the differential equations,there are always some tiny perturbed force of which is impossible to estimate the interference,and these perturbed forces often have the influence which is instant even continues to the path of the system.This perturbed force which caused the equation changes is called perturbation,and the corresponding differential equations known as the perturbation equation.Because of the extensive use and theoretical sig-nificance of the perturbed differential system,it caused by great attention.The method of studying perturbed differential systems are Lyapunov function method and variation of parameters formula.In recent years these two methods have been com-bined and have been a new research method of perturbed differential systems,which is called variational Lyapunov method.So far,the use of variational Lyapunov method is limited to the role of perturbed differential systems with no impulsive effect and per-turbed differential systems with impulses at fixed times,but there are few results about perturbed differential systems with impulses at variable times.In this paper,we consider the following perturbed differential systems with impulses at variable times where f(t,x)=F(t,x)+R(t,x),Ik(x)=Ik*(x)+Rk*(x),R(t,x)and Rk*(x)are perturba-tions. In this pater,mainly by using the variational Lyapunov method,we discuss the stability in terms of two measures of the system(â… ),and establish some criteria on stabil-ity and eventually stability.Here,not only the pulse depends on the state of the system solution,and we allow the solution curve of the system hitting the same surface with limited times which is called pulse phenomenon.There are three parts in this paper.In the first chapter,comparing with the ordinary differential system with no pertur-bations which is corresponding to the system(â… ),we establish a new variational comparison principle by virtue of variational Lyapunov method and differential inequality.It should be noticed that the pulse phenomenon can be allowed in this paper.It can be said that,this part is the foundation,and on this basis,in the next two sections we study the stability in terms of two measures of the perturbed differential systems with impulses.In the second chapter,we discuss the stability in terms of two measures of the per-turbed differential systems(â… ).We establish some criteria to decide that the perturbed differential systems(â… )is(h0,h)-stable under the condition that the unperturbed impul-sive differential system is (h0,h0)-stable and we also give the criteria about the strict stability in terms of two measures for the perturbed differential systems(â… ).In this part,we also give an example to explain the application of the theorem.In the third chapter,we discuss the eventually stability,the practical stability in terms of two measures for the perturbed differential systems(â… ). Similar to the second chapter,we give some criteria about the the eventually stability,the practical stability in terms of two measures for the perturbed differential systems with impulses at variable times.