Stability Analysis Of Nonlinear Impulsive Hybrid Differential Systems

.In this paper,we mainly study the following two kinds of nonlinear impulsive hybrid differential systems:the impulsive switched systems with time delay: where xt(θ)=x(t+θ),θ∈[-h,0], fκ-1(t, xt):R+×PC([-h,0])→Rn,and the impulsive hybrid differential systems: where f∈C[R+×Rn×Rm,Rn],Iκ∈C[Rn,Rn],λκ∈[Rn,Rm],κ= 0,1,2,we obtain the strict stability and exponential stability of the trivial solution properties for sys-tem(1.2.1),and the stability of set for system(2.2.1).It is known that the phenomena of impulsive widely exist in the practical problems of many fields in the modern technology,many of them can be described by impulsive differential systems.But it is far from enough,in view of this,we introduce the impulsive hybrid differential system,such as workshop distribution, current differential equa-tion is different at different period,and even depends on the previous period.Impulsive hybrid differential system is a important impulsive differential system, and its features is that different period differential system can differently, and a period of differential system relies on the previous period.Switch system is an important kind of hybrid systems,which is composition with a group of limited (limitless) subsystems,and in accordance with some switching rules in subsystems switching power system, switch system exist widely in transportation, aerospace scheduling, engineering, and other fields, these systems typical examples in-cluding computer hard disk, drive system, restricted robot control system, auto steering system, the plane's steering control system, the television channel switching system,etc. In practice, due to various reasons, many production and business development pro-cess will inevitably occur delay phenomenon, For example, element aging or information interference.In many cases, pulse, switch,delay are together proceed.how to control these factors make the production or the development of things according to the established goal, is a worthy of discussion. This kind of system research has caused the domestic and foreign scientist's interest,and it has gradually became popular research fields, Many scientists for its in-depth research.However,impulsive and switched system with finite delay on strictly stability, ex-ponential stability, the stability of sets for the pulse hybrid differential system was not enough.So in this field we have many problems to do.In chapter 1, we mainly studies the stability of the impulsive and switched system with time delay.First of all, according to the proof of literature, we provide the comparison theorem of this chapter, and then used the method of Lyapunov-Razumikhin functional to study strict stability of the trivial solution properties for system(1.2.1), obtain some new results, and it generalized related conclusions of the functional differential equation and of impulsive differential equations.Secondly, Lyapunov function and Razumikhin skills and the Lyapunov functional method were used to study the exponential stability and global exponential stability of nonzero solution, and obtained some exponential stability results of system (1.2.1). At last, an example verified the effectiveness of the results.In chapter 2, we mainly studies the stability of sets for the nonlinear impulsive hybrid differential system.In practical problems, the zero solution of the system may be unstable, but we can find a set that has some kind of stability of the system, which is called a set of stability, this chapter used Lyapunov function direct method and comparison principle to study the set of stability which based in paper [20] for system (2.2.1), and we got the set on uniform stability, consistent asymptotically stable. And different from ever, we weaken the derivative limitations because the pulses in the influence of the theorem of Lyapunov function,and we allowing the derivative positive, highlighted pulse influence on system solution properties.