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Chapter 1 Introduction 1.1 A Brief History Hybrid systems are dynamical systems that exhibit typical characteristics of both continuous time systems and discrete time systems. Some of the earlier references that studied the hybrid systems can be found in [1�� 2]. There are many subclasses of hybrid systems in the literature�� see the books [3-11] for some interesting results. Note that in many real world systems and natural processes�� a special hybrid behavior can be often observed. For instance�� some biological systems such as biological neural networks and epidemic models in pathology�� as well as optimal control models in economics�� pulse-frequency modulated systems�� and spacecraft relative motions�� and the like�� are characterized by sudden changes of system states at discrete moments. Such phenomena can be described by so-called impulsive systems. Generally�� an impulsive system consists of three main components [8]: an ordinary differential equation�� which determines the continuous evolution of the system between impulses; a difference equation�� which determines the way in which the system state is suddenly changed at impulse times; and an impulsive law�� which governs when the impulses occur. In the 1980s��the mathematical theory of impulsive systems (systems of impulsive differential equations)�� including the existence and continuity theorems�� asymptotic properties of solutions�� and Lyapunov stability theory�� has been formulated in [3��12]. During the past decade�� impulsive systems have been receiving increasing attention in science and engineering because they provide a natural framework for the mathematical modeling of a variety of practical systems [5��8��13-15]. Moreover�� impulsive systems not only are the basis of impulsive control theory�� but also have found important applications in many fields of modem control theory�� such as networked control systems [16-18]�� sampled-data control [19��20]�� state estimation [21�� 22]�� secure communication [23-26]�� etc. Generally�� the study on dynamics of impulsive systems can be divided into two classes: impulsive disturbance problem (IDP) and impulsive control problem (ICP). In the case where a given system without impulse possesses certain performances�� such as periodic solution�� attractor�� stability�� and boundedness�� while the correspon-ding performance can be preserved when the system is subject to sudden disturbances. Such phenomenon is regarded as IDP. Roughly speaking�� IDP can be considered as a class of robustness analysis of the systems subject to discontinuous disturbances. Many interesting works on IDP of impulsive systems have been reported in the literature�� see [27-39]. While in the case where a given system without impulse does not possess a certain performance�� but it may possess it via an admissible impulsive control�� it is then regarded as ICP. In fact�� ICP is a much attractive problem because it is discontinuous and usually has simple structure�� only the discrete control is needed to achieve the desired performance. It has been widely used in many fields��such as electrical engineering [5]��nuclear spin generator [40��41]�� aerospace engineering [42�� 43]�� population management [14��44��45]��and secure communication [23-26]. We also refer to monographs [5��46] on ICP. To now various impulsive control strategies have been proposed [5��23��46-68] and the references in these works�� such as [47] dealt with the dual-stage impulsive control�� [48��55] with the impulsive distributed control�� [49��50��52] with the pinning-impulsive control�� [61��63��64] with the event-triggered impulsive control (ETIC)�� [65�� 66] with the observer-based impulsive control�� [54��59] with the impulsive time window�� and [56��57��60��67��68] with the delayed impulsive control. From the impulsive magnitude point of view�� the impulses in IDP actually are a class of destabilizing impulses which will potentially destroy the dynamics and the impulses in ICP are a class of stabilizing impulses from which the dynamics will benefit. As we know�� time delays are ubiquitous in physical systems and engineering applications�� such as biology�� chemistry�� economy�� medical science and so on�� see [69-75]. They are often source of the degradation of performance or the instability of the system. The study on stability and control problem of delay systems is one of the hot issues in control theory and related fields. It has been shown that time delay has PN effects (i.e.�� positive effects and negative effects) on stability of a system�� namely�� it may not only cause the degradation of performance�� or instability of a system but also�� inversely�� make an unstable system stable or possess some good performances [76-86]. In particular�� impulsive delayed systems�� a natural generalization of (delay-free) impulsive systems and of time-delay systems�� has been investigated intensively during the past decades. The earlier works for stability problems of impulsive delayed systems were done by [87�� 88]. Now there are