The Expression Of Solutions And Periodic Solutions On Degenerate Differential Systems With Delays

One technique for studying physical and engineering problems consists of the design of processing mathematical models. A common model in the design of circuit problems is a state-space symmetric system.Symmetric systems in the standard case have been studied by different authors.For instance, structural properties for these systems have been analyzed by using Gramian matrices.Also,in such many practical fields, degenerate and impulsive phenomena are existing extensively. And this phenomenon has attracted many scholars' attention and lots of available and important results have been obtained. We notice that, in many practical systems, in order to describle the system more accurately, we must take the influence of delay.So, it is important practical meanings to study it.Firstly,we introduce symmetric singular systems as a generalization of the symmetric standard systems.Singular systems present certain difficulties because the model involves singular matrices.The solution to these systems can be given by generalized inverses. Some special properties of the matrices of the system permit us to obtain the solution in an easier form.Using the group inverse,an explicit solution to a singular system is described.The general explicit solution is derived when the symmetric singular system satisfies the regularity condition.Secondly,the existence to periodic solutions to nonlinear neutral equations of one form is considered,certain some theory of the form are presented and verified.In chapter one,some different names of the general systems are introduced in detail. And then, some properties different with ordinary differential systems are presented,and the general situation of researching singular systems are brief summarized.The main work done in this paper is introduced.In chapter two, using the solution theory of generalized systems,through the transformation and the inverse of the matrices.The solution of the continuous symmetric singular control system is derived and verified.In chapter three, by using the theory of matrix measure and the method of point theorem of functional analysis, some sufficient conditions that guarantee the existence of periodic solutions of systems are obtained.