Stability Analysis Of Impulsive Diferential Control Systems

In this paper,we consid er the fxed-time impulsive diferential control systemsand the state-dependent impulsive control sys temwhere impulse times, and klâ†'im∞tk=∞, u is any admissible vector in given admissible controlsetIt is by impulsive diferential control systems that various dynamic models are ex-plained from a mathematical perspective. It makes people know the internal law of thesystem, so we can control the system much better. For example, in the feld of the bio-logical control system, the interesting and meaningful biological control of the group hasa very wide range of applications, arousing a large number of researchers’ attention.The stability of impulsive diferential control system aroused the interest of manyresearchers[814,33]. Control of vector control system under study is mostly defned inthe control collection of Ω={u∈Rm: U (t, u)≤r(t), t≥t0}.In Chapter I, we frst introduce the basic idea of the cone-valued Lyapunov functionmethod and then establish a new comparison principle. Based on the comparison theo-rem, we study (h0, h)-stability, asymptotic stability, uniform stability, practical stability,ultimate stability, bounded, uniformly bounded, ultimately bounded and other propertiesof system(1). The results in this chapter are more efective in determining the scope of abroader and have the conclusions of promotion. At the end of this chapter we illustratean applicability of the theorem.In Chapter II section3of this paper, we establish a new comparison principle todiscuss the stability of the state-dependent impulsive control system (2) in the condi-tion of allowing the solution curve of system (2) to collide the same pulse-face collision with limited times. Next we use the cone-valued Lyapunov function method[3]to studythe stability properties of system (2). In Chapter II section4of this paper, we use thecone-valued variational Lyapunov function method[11]to study the stability properties ofsystem (2). In the comparison results of the study, we allow the solution curve of system(2) to collide the same pulse-face collision with limited times.With regard to the direct results of the stability of the state-dependent impulsivecontrol system (2), the control vector of impulsive control system which is studied isusually defned in the control set Ω={u∈Rm: U (t, u)≤r(t), t≥t0}, and thesolution of system (2) x(t)=x(t, t0, x0, u), Ï„k1(x0)<t0<Ï„k(x0), is assumed colli-sion of each pulse surface Si, i≥k only once. In Chapter II, we defne the control setE={u∈Rm: U (t, u)≤λ0(t), t≥t0}, where λ0(t) is a given function and allow thesolution curve of system (2) to collide the same pulse-face collision with limited times.By employing the cone-valued variational Lyapunov function method and Razumikhintechnique which is used in the study of impulsive functional diferential systems, a Razu-mikhin type of the cone-valued variational Lyapunov function method is obtained. Atthe end of this chapter we illustrate an applicability of the theorem.