Time delays appear commonly in many practical systems which may induce performance degradation or even instability. Hence considerable attention has been devoted to analysis and control of delayed systems with a large volume of results in the existing literature. It is natural that the theories for stability analysis and control synthesis are mainly developed for continuous-time systems since most practical systems are described in continuous time. However, the controller is usually implemented digitally in practice. In this case, the resulting control system is normally described by a discrete-time model. Even though relatively less attention has been paid to discrete-time systems with time delays in comparison with continuous-time case, several methods have been developed for analysis and synthesis of discrete-time systems with time delays. One widely adopted method to deal with discrete-time systems with time delays is the so-called augmentation approach which can convert a time-delayed discrete-time system to a higher-dimensional delay-free system. Nevertheless, this augmentation approach will result in large-dimensional or infinite-dimensional discrete-time systems which could incur high computational cost or even difficulty in analysis and synthesis of the augmented systems. Another effective method which handles the original discrete-time systems with time delays directly is the so-called Lyapunov approach which can be typically classified into two types of stability theorems, the Lyapunov-Krasovskii theorems and Lyapunov-Razumikhin theorems.

Most existing works are concerned with discrete-time systems with bounded time delays. However, infinite delays, also named unbounded delays, exist widely in practical systems, such as, teleoperation systems, biology, mechanics, social science, electrodynamics and so on. More recently, stability and control problems of continuous-time systems with infinite delays have been studied. However, it is noted that the results on discrete-time systems with infinite delays are rather limited. There exist some results on functional difference equations with infinite delays reported in the 1990s and 2000s, such as asymptotic behavior, periodic solutions, convergence theory and so on. However, stability analysis and controller synthesis problems of discrete-time systems with infinite delays remain to be investigated.

This thesis focuses on stability analysis and control synthesis problems of discrete-time systems with infinite delays. The main results of this thesis can be summarized as follows.

1. The stability analysis problem of discrete-time systems with infinite delays is considered. Several new theorems for stability analysis of the concerned delayed systems are developed. Specifically, the results on uniform stability, uniform asymptotic stability, global uniform asymptotic stability, exponential stability and global exponential stability are obtained. Furthermore, our results include some existing works on bounded delays as special cases.

2. The stabilization problem of discrete-time linear systems with infinite distributed input delays is investigated. A novel framework is adopted to address the concerned problem. Under this framework, two truncated predictor feedback controllers are developed for two classes of discrete-time linear systems with infinite distributed input delays via the low gain method respectively. It is shown that under the designed controllers, those two classes systems are globally exponentially stabilized. To the best of our knowledge, this is the first time that the stabilization problem of discrete-time linear systems with infinite distributed input delays is considered.

3. The semi-global stabilization problem of discrete-time systems with infinite distributed input delays and actuator saturations is studied. Two low gain controllers are developed for two classes of discrete-time systems with both infinite distributed input delays and actuator saturations respectively. To the best of our knowledge, the concerned stabilization problem is investigated for the case of discrete-time systems for the first time in this thesis. Compared with existing works on infinite delays, a more general framework is adopted and more accurate scaling is developed in this work. Our results include the stabilization results for discrete-time systems with only saturations and discrete-time systems with both bounded delays and actuator saturations as special cases. Furthermore, to handle the nonlinearity induced by saturations, a novel converse Lyapunov theorem for discrete-time linear systems with infinite delays and a novel stability theorem for perturbed discrete-time linear systems with infinite delays are developed.

4. The stabilization problem of discrete-time time-varying systems with infinite distributed input delays is considered. A low gain feedback controller is developed for discrete-time Α-periodic time-varying systems with infinite distributed input delays. Under some mild assumptions, the resulting closed-loop system is shown to be globally exponentially stable. To the best of our knowledge, this is the first time that the stabilization problem of the concerned discrete-time system are addressed. Furthermore, the proposed result can include some existing works on discrete-time time-varying systems with a single constant delay and bounded distributed delays as special cases.

5. The stabilization of discrete-time time-varying systems with infinite distributed input delay where time-varying system matrix and input matrices converge to constant matrices as time goes to infinity is investigated. Two low gain controllers are developed for the concerned systems which achieve global exponential stabilization. Furthermore, since the conditions of Lyapunov theorems developed in previous chapter are rather difficult to be satisfied for discrete-time varying systems with infinite delays, two stability theorems are proposed for exponential stability and global exponential stability of discrete-time systems with infinite delays. Compared with previous corresponding stability theorems, the conditions in these new stability theorems are easier to be satisfied.