A hybrid system is in essence a combination of continuous dynamical system and discrete event systems that exhibit simultaneously several kinds of dynamic behavior. Although hybrid systems have become popular, their use in such applications as physical, chemical, and control systems is vulnerable to time delays which often lead to instability. Consequently, this thesis is to investigate the stability problems of hybrid systems with time delay. It mainly covers switched delay systems, impulsive delay systems and impulsive neutral systems. In this thesis, Lyapunov function method and inequality techniques are applied to switched delay systems. It is shown that slowly switched stable systems is stable. Then, the results are extended to systems that include their stable and unstable subsystems. Some relations of the dwell times among the stable subsystems and unstable subsystems, based on the inequalities, are derived to guarantee stability. Impulsive delay systems and large scale impulsive delay systems are then investigated, resulting in some sufficient conditions for stability. It is demonstrated that when the system matrix is unstable, time delay and impulses can stabilize the system. By developing some new inequalities, the relations among the delay, impulses, and system matrices are given. Furthermore, absolute stability, based on Lyapunov functional, are studied for some impulsive delay systems. Conditions for the system matrix are developed so that impulsive delay systems are absolutely stable. Neutral impulsive systems are also studied. The emphasis is on the sufficient conditions which guarantee that the large scale neutral impulsive system is stable. After establishing some inequalities, the results on their stability are developed for neutral impulsive systems. Finally, applications to Internet congestion control and impulsive control system are discussed.
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