Solution, Stabiity And Control Problems Of Degenerate Differential Equations With Delay

In many practical systems, people have found that there were many degen-erate phenomena existing extensively, such as economic systems, power systems, engineering systems and so on. And this phenomenon has attracted many scholars' attention and lots of available and important results have been obtained. However, time delay is a common phenomenon in the objective world and engineering fields. In order to describe the system more accurately in many practical systems, we must take the influence of degradation and delay into consideration altogether. So, it has a practical significance to study the solution and its characteristics of degenerate differential systems with delay.In this dissertation, solution, stability and control problems of degenerate dif-ferential equations with delay are discussed. This doctoral dissertation is composed of four chapters, which mainly studies existence conditions of solution of fractional order functional differential equations and general solution of fractional order de-generate differential equations, and stability and guaranteed cost control problem of degenerate differential equations with delay.Firstly, the backgrounds and significance of the problems are given. The main work done in this dissertation is introduced.Secondly, the existence results of solution of fractional order functional differ-ential equation are derived based on Banach fixed point theorem, Schauder fixed point theorem and successive approximations technique, respectively. These results extend the corresponding ones of ordinary differential equations and functional dif-ferential equations of integer order. The expressions of the general solution for fractional order ordinary differential equations are obtained by defining the solv-able matrix pair, and these results extend the corresponding ones of integer order degenerate differential equations and fractional order linear ordinary differential equations.Further, some new stability criteria of degenerate differential equations with delay are obtained, and we can overcome the difficulties of solving roots of tran-scendental equation by applying these criteria, which are more convenient to check all-delay stability. The V-functional method for general singular differential de-lay system is proposed. The asymptotic stability criteria for degenerate differ-ential nonlinear systems with multiple time-varying delays are derived based on V-functional method and some analytical techniques, which are described as ma-trix equations or matrix inequalities. The results obtained are computationally flexible and efficient.Finally, we discuss guaranteed cost control problem of continuous-time uncer-tain degenerate system with state and control delays. Sufficient conditions for the existence of the guaranteed cost controller are derived based on the linear inequal-ity (LMI) approach, a parameterized characterization of the guaranteed cost laws is given in term of the feasible solutions to a certain LMI, and the cost function of guaranteed cost controller exists the upper bound.