Research On The Impulsive Control Of Differential Systems With Delay And Its Applications

It is known that there’re many instantaneous mutation phenomena in the real world. Impulsive differential equations and impulsive functional differential equations are used to describe these systems. The jump mutations at a fixed time will occur in the im-pulsive differential equations. And the most prominent feature is that it can fully take into account the phenomenon of instantaneous mutation. The characters of the changes for things can be reflected and controlled more accurately. It is shown that the impul-sive differential equations have been applied in various fields such as space technology, information science, control engineering, ecological models, medicine, finance and envi-ronment etc. The earlier model equations are assumed to only related to the current state. However, as it is pointed that many variation of things are not only related with current state, but also related with past time state. The system is called time delay differential equations or functional differential equations.Now, the results on the impulsive differential systems mainly focus on constant time delay. The researches of the impulsive differential systems with the time-variant delay are very few because of the complexity. In this paper, several types of impulsive differential systems with time-variant delay will be studied. Lyapunov function method and Razumikhin techniques are often used to solve these problems. In our paper, we will construct suitable Lyapunov function to study the systems. Inequality will be applied in transforming time-variant delay differential systems to constant time delay differential systems. The thesis contains two chapters as following.In chapter one, based on the finite delay, we mainly study the followings.In part two, we study the impulsive stabilization for the following second-order delay differential equations andNew theorems on impulsive exponential stability of systems are established by using Lyapunov function. This result is obtained under the case that the system has a time-varying delay. Our result improves. In part three, it will be proved that impulses can stabilize the second-order nonlinear time-varying delay differential systems, andNew theorems on impulsive exponential stability of system (1.3.1) and (1.3.2) are established by using Lyapunov function. This result is obtained under the case that the system has a time-varying delay and the nonlinear f. Our result is the extension of. In part four, the conclusions are extended to third-order differential equations andNew theorems on impulsive exponential stability of systems are established by using Lyapunov function. This result is obtained under the case that the system has a time-varying delay. The result is the extension of [3]. In chapter two, the following control system will be studiedFirstly, new theorems on absolutely stability of linear system are established by using Lyapunov function. Then such method can be extended to nonlinear system, and the corresponding theorem be get. In the last, it is extended to a more general nonlin-ear system. The systems in this chapter have widely used range and greatly practical applications.