Several Problems Of Stabilization On Degenerate Differential Systems With Delays

Stability is one of the important topics in system control theory research all the time. It's not only systems' basic structure property but also the premise condition to ensure the system operated normally. The phenomenon of delays can't avoided but existed extensively all the nature world. In many dynamical systems, the existences of delays lead to system instability, and also make the system analysis more difficult and complicated. In the delay systems currently, The result of stability could divide into two types:one type is stability with delays but another is not.Stability relate to delays means that the condition which systems'parameter satisfied could ensure it stable only when the delay smaller than the upper bound. Recently, with the development of the social sciences and technology, specific model of delay differential equations existed in many science and area of engineering. Such as modern physics, aerospace controls, ecology, management etc. Above these lead to the analysis of stability and the controller synthesize about system with delays become to one important direction in the control systems. So, introduction of controller make the degenerate delay systems stability has important meaning.This article mainly discusses some stabilization problems of degenerate delay systems, and gave some necessary and sufficient conditions of it. In addition, we proved the feasibility of these conditions in detail. The result which obtained in the article was used to enrich the theory and promoting development of degenerate delay systems.In this article, the research methods mainly make use of the feedback controller designer, the descriptor form, and combine Lyapunov, Lyapunov-Krasovskii functions, Schur complement lemma and solving linear matrix inequality to discuss stabilization about these degenerate systems with delays. Designer of controller is confirmed effectively via proving the negative of LMI, and solve the feedback gain matrix by the toolbox of MATLAB.There are four parts in this article.The first part described the background, significance, current develop situation in domestic and foreign of the problem which involved in this paper.Then,introduce main work in this article, finally lemmas and symbols involved.The second part divided into two parts, The first one mainly discusses stabilization of linear neutral degenerate time-delayed systems. By constructing dynamical feedback controller, the method of descriptor form, Lyapunov-Krasovskii function and LMI form, the necessary and sufficient conditions of stabilization obtained about the system. The second one consider stabilization for a type of degenerate systems with discrete delays, and get related conclusion via the same method, At last numerical examples illustrate the efficiency and feasibility of the result.The third part consider different assumptions which the system satisfied, we discuss robust stability and robust feedback stabilization for grey degenerate neutral systems with delays under different conditions via designer of state feedback controller, decomposed technique of grey matrix norm inequality, and functions of Lyapunov and Lyapunov-Krasovskii. By the end of the chapter, numerical examples verify the result's feasibility.The four part mainly discuss sliding mode control for linear delay degenerate systems with nonlinear disturbances. The uncertainties which contained not only matched disturbances and also mismatched ones. By using restricted equivalent decomposition, efficient sliding surface and designer of controller make the system attach the surface in limited time and maintains motion on it. At last, sufficient conditions of stabilization obtained via Lyapunov functions. At last, numerical examples verify the result's efficiency.