Stability Of Two Fractional Nonlinear Systems With Delays

Fractional calculus as extension of integer order calculus in order describe the re-search objects in nature better. In recent years, with the rapid development of the com-puter technology, and the maturation of fractional calculus theories, fractional calculus have attracted the attention of mounting scholars both at home and abroad, and have been widely applied in many fields such as physics, biology, chemistry and engineering. Fractional calculus has become immediate research focus in mathematic study. Stability is the fundamental characteristic of the system, which is the precondition to make sure the system can run normally. Next, considering the nonlinear system could reflect the nature of the system well. Moreover, for nonlinear system, the stability is a key property. Therefore, the study of stability problem for nonlinear systems has important theoretical and practical significance. Especially, stability of the fractional-order nonlinear systems is the hot and difficult topic. This paper mainly considers stability of two fractional non-linear systems with delays, a series of conditions for ensuring stability are presented. The outline of this paper is organized as follows:In chapter I, we present the background of the topic, purpose and current state of the study, and briefly introduce the development of fractional calculus. Meanwhile the main work of this paper is introduced.In chapter II, some preliminary knowledge which is going to be used in this paper is given. Firstly, we present some common basic functions, which are used to study fractional calculus systems, such as Gamma function, Mittag-Leffler function. Then we introduce the three basic definitions of fractional calculus, which are the Riemann-Liouville definition, the Grunwald-Letnikov definition and the Caputo definition, and describe the relationship of the three definitions. Finally, some basic natures of the fractional calculus are introduced.In chapter III, we study the stabilization of a class of nonlinear fractional neutral singular systems. We extend Lyapunov direct method and Laplace transform to such systems and obtain several (asymptotical) stability and (generalized) Mittag-Leffier sta-bility conditions under Caputo and Riemann-Liouville derivatives. Then three illustrative examples verify the feasibility and effectiveness of the conclusions.In chapter IV, based on Holder inequality, Gronwall inequality as well as inequality scaling skills, we study the finite-time stability of a class of fractional delayed neural net-works of retarded-type. For fractional delayed neural models of retarded-type with order α∈(0,1/2] and α∈[1/2,1), sufficient conditions for the finite-time stability are presented, respectively. Numerical simulations also verify the theoretical results.Finally, the work of this paper is summarized with the outlook for the future.