| Time-delay phenomenon is often encountered in many practical control systems, and whenever the time delay is present in the systems, some erratic behavior can occur to instability and poor performances. On the other hand, one can design the delayed feedback controller to improve the control effect. Therefore, the stability analysis and control design of time-delay systems are of great important for both theoretical and practical reasons. During the past few decades, increasing attention has been devoted to the problems of stability analysis and control design for time-delay linear systems, and many nice results have been proposed for the analysis and synthesis of linear time-delay systems.However, in contrast to the existing results on the analysis and synthesis of lin-ear time-delay systems, there are fewer results for nonlinear time-delay systems. Since there are a lot of nonlinear problems in real systems, and delayed phenomenon is in-evitable, analysis and synthesis of general nonlinear delayed systems have a broader application. Due to this, nonlinear control theory has been turning into one of the hotspots in the control theory. Some nice results have been obtained on some nonlinear time-delay systems. Even so, the research outcome is still insufficient on general non-linear time-delay control systems. Particularly, the results on the finite-time problem are less. We hope that this thesis can provide some new ways to the study of nonlinear time-delay systems and further enrich the control theory of nonlinear delayed systems.In this paper, we investigate the stability, the finite-time stability, the stabilization and robust control design problems for several classes of general nonlinear time-delay systems. The main contents of this thesis are composed of the following parts:The first part investigates the stability and H∞control problems of a class of nonlin-ear constant delayed systems with Jacbian matrix being nonsingular. Firstly, an equiv-alent form is obtained for this class of systems by means of coordinate transformation and orthogonal decomposition of vector fields. Then, based on the equivalent form and free-weighting matrix method, several sufficient conditions in term of nonlinear matrix inequalities are derived for the stability analysis of the time-delay systems by construct-ing suitable Lyapunov-Krasovskii functionals. Moreover, we use the equivalent form and the obtained stability results to investigate the H∞control problem, and present a control design procedure for this class of time-delay systems. Finally, study of two illustrative examples shows that the results obtained in this part work very well in the stability analysis and control design for some nonlinear time-delay systems.The second part studies the stability and estimate of domain of attraction for a class of nonlinear time-varying delay systems. Firstly, an equivalent form is obtained for the systems by means of coordinate transformation and orthogonal decomposition of vector fields. Then, based on the equivalent form and the free-weighting matrix method, several sufficient conditions are derived for the stability analysis. Thirdly, the equivalent form and the obtained stability results are applied to the estimate of domain of attraction for this class of nonlinear time-delay systems, and a new approach is presented. Finally, study of illustrative examples shows that the results obtained in this part are effective in studying the stability and estimate of domain of attraction for the class of nonlinear time-delay systems.The third part considers the stability and H∞control for a class of nonlinear time-delay systems, and proposes a number of new results. Firstly, an equivalent form is given for this class of systems. Then, based on the equivalent form, some delay-dependent results are derived for the stability analysis of the systems by constructing a novel Lyapunov functional and using some suitable mathematical techniques. Thirdly, we use the equivalent form and the obtained stability results to investigate the H∞con-trol problem for a class of nonlinear time-delay control systems, and present a control design procedure. Finally, several illustrative examples are given to show the effective-ness of the results obtained in this part. It is shown that the main results of this part are easier to check than some existing ones, and have less conservatism.The forth part considers the stability of general nonlinear time-delay systems via the Hamiltonian functional method. Firstly, the concept of GHR of general nonlinear time-delay systems is proposed, and several new GHR methods are given. Then, based on the new GHR methods obtained, the stability of time-delay systems is investigated, and several delay-dependent sufficient conditions are derived for the stability analysis by constructing suitable Lyapunov-Krasovskii functionals. Furthermore, to reduce the computational burden, the part provides a decomposition method. By applying the method, one can avoid solving nonlinear matrix inequalities and obtain some concise conditions, which is easy to be checked. Finally, illustrative examples show that the results obtained in this part have less conservatism, and work very well in the stability analysis of some nonlinear time-delay systems. The fifth part investigates the finite-time stability (FTS) and stabilization for a class of nonlinear time-delay systems with matrix l(x), and proposes some new results. Firstly, an equivalent form is presented for the systems by a coordinate transformation, and a criterion on the time-varying FTS is obtained for general nonlinear time-delay systems via Razumikhin approach. Then, based on the criterion and equivalent form, some delay-independent and delay-dependent conditions on the FTS are derived for this class of systems by constructing a new Lyapunov function. Thirdly, we use the obtained FTS results to investigate the finite-time stabilization problem, and present a control design procedure for the time-delay systems. It has been shown that our method can avoid solving nonlinear matrix inequalities by transferring them into the form of linear matrix inequalities, which can be easily solved by the LMI Toolbox of MatLab. Finally, study of illustrative examples shows that the results obtained in this part work very well in the FTS analysis and control design for the nonlinear time-delay systems.The sixth part studies the finite-time stability of a class of nonlinear time-delay sys-tems without matrix l(x), and proposes some delay-independent and delay-dependent FTS results. Firstly, a criterion on the time-invariant FTS is proposed via Razumikhin approach, and an equivalent form is obtained for this class of systems by means of coordinate transformation and orthogonal decomposition of vector fields. Then, based on the criterion and the equivalent form, some delay-independent and delay-dependent concise conditions on the FTS are derived for the systems. Simulation shows that the finite-time stability results obtained work very well.The seventh part investigates the finite-time stability analysis and finite-time H∞control design for a class of nonlinear time-delay Hamiltonian systems, and proposes some delay-dependent results on both the FTS and finite-time control design. Firstly, a criterion on the FTS is proposed for general time-delay nonlinear systems via Razu-mikhin approach. Then, based on the criterion, some delay-independent and delay-dependent conditions on the FTS are derived for the nonlinear time-delay Hamiltonian systems by constructing a suitable Lyapunov function. Thirdly, we use the obtained FTS results to investigate the finite-time H∞control problem, and present a control de-sign procedure for a class of nonlinear time-delay port-controlled Hamiltonian systems by the energy-shaping approach. Study of two illustrative examples shows that the re-sults obtained in this part work very well in the FTS analysis and control design of some nonlinear time-delay Hamiltonian systems.Innovations of the thesis mainly include the following aspects: 1). Content Aspects:·The stabilities of general nonlinear both constant delayed systems and time-varying delayed systems are discussed, some sufficient conditions on the stability are obtained. Meantime, a estimate of domain of attraction for the systems has been given.·A new criterion on the finite-time stability for general nonlinear time-delay sys-tems is given, and some finite-time stability results are obtained for the time-delay systems. It is well worth pointing out that the criterion has more advantage than the existing one for the application.·It is first that the finite-time stabilization and finite-time H∞control design prob-lems are investigated for some nonlinear time-delay systems, and some new design procedures are given.·It is first that the finite-time stability problem is investigated for a class of non-linear time-delay Hamiltonian systems, and some delay-dependent results are obtained.·It is noted that, by using the methods proposed in the thesis, the conditions obtained are easier to check and have less conservatism.2). Method Aspects:·It is first that the stability and control design are studied for general nonlinear time-delay systems by using the orthogonal decomposition method of vector fields, and some less conservative results are given via the orthogonal condition.·It is first that the stability and control design of general nonlinear time-delay sys-tems are investigated via the Hamiltonian functional method, and the results obtained have less conservatism.·It is first that the thesis presents a constructing method on time-varying Lya-punov functional, and based on the Lyapunov functional, the thesis gives some less conservative and concise results.
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