Quasilinearization Method For A Class Of Nonlinear Singular Systems

While the study of modern control theory developing gradually and impregnating into many application areas of aviation, spaceflight, energy sources, chemical industry, electric power, communication and network etc. Scholars found a vast dynamic system, arose when Rosen-brock analyzed electrical networks. From then on the theory of singular systems started to form and gradually developed into many separate branches of modern control theory. Its models lie in many fields of social production such as optimal control problems and constrained control problems, some population growth models and singular perturbations, so researching theory of singular systems have far-reaching real significance. The basic theory of the solution of the linear singular systems have been given by S.L.Campbell, but theory of the nonlinear singular systems are imperfect, which are the focus of scholars, and the convergence of the solution is the hottest one, which play an important role of the qualitative theory's development. This paper will use quasilinearization method to study the convergence of the solution of the nonlinear sin-gular systems. This dissertation focuses on two sides:one is construct the monotone sequences for the nonlinear singular systems; the other side is using quasilinearization method for proving the convergence uniformly and quadratically to the unique solution of the problem. The paper is made up of four chapters. Main contents are as follows:In chapter one, we give a survey to the development and current state of singular system with deviating arguments, as well as the main work status. In chapter two, we discuss the initial value problem of nonlinear singular difference system, we transform the given nonlinear prob-lem into a corresponding linear singular difference system, to the linear problem we employ upper-lower solution method and monotone iterative technique to construct two approximate solution sequences. By using the Ascoli-Arzela's theory we show that those two approximate solution sequences both converge uniformly to the unique solution of the given nonlinear prob-lem, then by using the quasilinearazition method we obtain the result that the solution sequences converge quadratically to the unique solution. In chapter three, the quasilinearization method is extended to nonlinear singular system with control, we obtain two monotone iterative sequences converging uniformly and quadratically to the solution of the given problem. In chapter four, we study the boundary value problems of second order singular differential equations. At first, we reduce the BVPs to initial value problems of first order singular integro-differential equa-tions, then we employ the quasilinearization method in studying the IVPs, so we can obtain two monotone iterative sequences, which are converge uniformly and quadratically to the unique solution of the IVPs, finally we get the similar results for the given BVPs.