The Analysis Of Generalized Mittag-Leffler Stability And Lp Stability Of Fractional Order Nonlinear Systems

In recent years, with the fast development of the computer, fractional calculus is used widely in optical system, viscous elastic mechanics, signal process, control system and so on. Stability is the fundamental characteristic of the system, which is the precondition to make sure the control system can run normally; Next, considering the nonlinear system could reflect the nature of the system well, so we analyse the stability of fractional nonlinear systems. This paper is mainly composed by four chapters as follows:The chapter1is the preface. In section§1.1, we present the background of the topic, and then put forward the question. In section§1.2, we introduce the development history of fractional calculus theory, and the definition of frac-tional integrals. Then we introduce Ricmman-Liouvelle fractional order oper-ator, Caputo fractional order operator and their individual properties. In sec-tion§1.3. list some special functions:Mittag-Lefflcr function、 Mittag-Lcfflcr function with two parameters-generalized Mittag-Leffler function-Gamma function, hypergeometric function, class-k function. These functions will be used in the subsequent chapters.In chapter2, we mainly discuss the method to judge its stability with-out solving nonlinear differential equation. In section§2.1. for the integer order nonlinear systems and present integer order Lyapunov second method. In section§2.2, we discuss the fractional order nonlinear systems. First, we introduce fractional order Lyapunov stability method; Second, we present the basic definitions and the theorem of Mittag-Lcfflcr stability and generalized Mittag-Leffler stability:At last, we introduce the Lp stability of fractional or-der nonlinear systems on the finite time interval. In section§2.3, the summary of the chapter is given out.In chapter3, we focus on the Lp stability of the Mittag-Leffier function with two parameters times a power law function lγ-1Eα,β(-λta) on the infi-nite time interval. In section§3.1, some fundamental Lemmas are presented, which will be used in the subsequent sections. In section§3.2, the necessary and sufficient conditions of Lp stability of lγ-1Eα,β(-λta) are derived on the infinite time interval. It is shown that lγ-1Eα,β(-λta) played an intermedi-ate process in between the power law and exponential phenomena. In section§3.3, a number of examples are illustrated to validate the theorem. Finally, the summary of the chapter is given out.chapter4is a conclusion with the summary of the thesis and prospect research.