The Study Of The Existence And Uniqueness And Stability For A Nonlinear Multi-variables Fractional Differential

Fractional calculus is not a subject of the calculus of a fraction, nor is it a part ofthe classical calculus (differentiation, integration, and variation). In fact it is a subjectof integration and differentiation to an arbitrary order, and it has a history of morethan 300 years. In the last long time, because of the lack of background and practicalapplication, it developed very slowly. In recent decades, many engineers pointed outthat fractional calculus is applied to describe the nature of various physical andchemical materials, such as the polymer. it has been proved to be very useful．In thereal world．applied scientists and engineers realized that such fractional differentialequations provided a natural framework for the discussion of various kinds ofquestions modeled by fractional equations．Such as viscoelastic systems, the electrode- electrolyte polarization, electrochemical, signal processing, diffusion process,control and so on. The fractional calculus and fractional differential equations attractmuch attention and increasing interest due to their potential applications in sciencesand engineering．As we all known, the problem of stability is a very essential andcrucial issue for control systems and it is open. While numerical simulation study isan important way to reduce costs. We try to investigate the existence and uniquenessfor a nonlinear multi-variables fractional differential equations and use Lyapunovdirect method to find stabilization condition for nonlinear fractional order system.This thesis mainly consists of four chapters. The first chapter reviews thedevelopment of fractional calculus history. The second chapter presents themathematical basis of fractional calculus, including the definitions and properties ofgamma function, beta function and the Mittag-Leffler function such as theGrünwald-Letnicov , Riemann-Liouville and Caputo definitions of fractional calculus,Introduces three definitions of fractional calculus, such as the Grünwald-Letnicov,Riemann-Liouville and Caputo fractional calculus. We also describe their propertiesand present the relationship between various definitions, Compared the differencefractional calculus and integral calculus.Next, In Chapter 3, using Schauder fixed point theorems and Global contractionmapping theorem, we prove existence and uniqueness theorems for some classes of nonlinear multi-variables fractional differential equations. And it generalizes andimproves some results of existing literature. In Chapter 4, we introduce the definitionof Mittag－Leffler stability and the generalized Mittag－Leffler stability. We alsofurther study its interesting properties．Then we investigate the research of Podlubny,Matignon and others for finding stabilization condition of nonlinear fractional ordersystem. Based on the research of them, We obtain stabilization condition for thegeneralized nonlinear fractional order system by using Lyapunov direct method,which extends the application of fractional calculus in nonlinear systems and enrichesthe knowledge of both the system theory and the fractional calculus.Lastly, we use MATLAB/SIMULINK software as a tool to discuss the stability ofthe new criterion by a computer simulation to resolve the effectiveness of theproposed method.