Several Problems Of Stability On Degenerate Differential Systems With Delays

In recent years, with the development of society and the progress of science and technology, the specific models of delay differential equations extensively exist in modern physics, space control study, ecology, management, economics and many other scientific and engineering fields. As ordinary differential equations usually be an ideal state of the classical model, sometimes the small delay will have any significant impact on the system. Therefore, delay differential equations depict the movements of things more accurately. Thus, people are interested in the research of delay in differential systems, which bring a global tide. Delay is the widespread physical phenomenon of nature. The existence of delay makes the stability analysis more difficult. The stability analysis of dynamical system theory is an important issue, because it is a basic requirement in dynamic system. The stability is the necessary precondition in order to make all control systems operate normally.We also provided some suggestions about the stability, asymptotic stability of delay differential systems, and also provided suggestions about the numerical stability of the generalized delay differential systems. For these stability issues, I gave some checkable conditions, and confirmed the feasibility of these conditions in details. I also gave the criteria for determining the types of stability problems. I think the results might play a certain role in promoting the research and development in the field of stability of the solutions on degenerate differential delay systems. The main methods are Lyapunov functional (V-functional) and linear multi-step methods. I gave V-functional criteria for the stability of degenerate delay differential systems. Then for a special degenerate delay differential system—the mixed system of difference differential systems with difference systems, I gave a concrete V-functional and the corresponding theorem. For linear multi-step methods, this paper proved that linear multistep methods is asymptotic stability if and only if it is A-stable.This paper has five chapters which mainly discusses the asymptotic stability of a neutral delay differential system, the stability of singular uncertain differential system with multiple time-varying delays, numerical stability of linear multiple methods for delay differential equations, and asymptotic stability and numerical analysis for a kind of singular differential delay equations.Chapter 1 Described the background and significance of the problem and the work what the paper done.Chapter 2 Discussed the asymptotic stability of a class of linear neutral system. By using linear matrix inequalities and constructing Lyapunov functional, the necessary and sufficient conditions of asymptotically stable are obtained.Chapter 3 Used Lyapunov functional discussed the stability of singular uncertain differential systems with multiple time-varying delays, then a criterion theorem will be given. Chapter 4 Considered the initial problem of delay differential system, analysised the stability of its numerical solution, under certain conditions of Lagrange interpolation, gave the necessary and sufficient conditions and their proofs.Chapter 5 Furthermore, on the basis of Chapter4, we considered the asymptotic stability and numerical analysis for a kind of singular differential delay equations. Under certain conditions, gave the necessary and sufficient conditions and their proofs.