Research On Stability Of Impulsive Control Systems With Time Delays And Applications

In many practical problems, many evolution processes are characterized by the fact thatthey experience a change of state abruptly at certain instants, namely impulsive phenomenon.It is known that the amount of money supply in a financial market, orbital transfer of satellite,biological population model, signal processing, optimal control of neural network model andso on can usually acquire better results when these mathematical models are described andanalyzed by impulsive differential control systems.Except for the impulsive effect, in many practical systems, the delay effect is inevitable.The state of system at some point or a period of time in the past may affect the current state ofsystem. For all kinds of systems, stability is always the fundamental problems. Due to allmentioned above, on stability of impulsive delay differential equations has attracted manyresearchers’ attention.This paper mainly discusses the stability criteria and related applications of most kinds ofimpulsive differential equations. The rest of this paper is organized as follows. In Chapter2,we introduce some notations and definitions first, in addition, we describe the impulsivedifferential equations. By utilizing Lyapunov’s direct method and comparison technique, weobtain several stability and exponential stability criteria of the trivial solution of impulsivedifferential equations. Finally, three examples are given to illustrate these theories. The firstone is the impulsive control of chaotic systems, the second one is the impulsive control ofan advertisement model, the last is the impulsive control of competitive model ofnormal and cancer cells in body. The corresponding results demonstrate the efficiency of thesetheories.In Chapter3, we introduce some notations and definitions of functional differentialequations first. Similar to Chapter2, in the following section, we describe the impulsivedifferential equations with delay and give some definitions related to stability. Finally, weacquire the stability criteria of impulsive differential equations with delay by utilizingLyapunov function and Razumikhin technique and Lyapunov functional and so on. For eachtheorem, we analyze the linear impulsive differential equations with delay and the resultsobtained illustrate the efficiency of theorem. In the final Chapter, based on the theoretical results obtained in Chapter3, we analyze theneural network models include Hopfield neural network and Cellular neural network. Theexponential stability criteria provide a good guidance on the practical application of models.