Stability Analysis Of Fractional Calculus System

Fractional calculus has been a long period of development, but due to the lack of meaningful comparison of fractional calculus in practical applications, it has been limited so as to develop a short slow. The concept has to be shaped by year1965Professor-American mathematician Mandelbrot from Yale University proposed the concept of fractal and he study the Brownian motion analysis with the Riemann-Liouville fractional calculus theory. Until now, the theory of fractional calculus start to get attention and considerable progress in many disciplines and engineering fields, it has become one of the most popular disciplines.This paper totally has four chapters. The main contents are two parts, a simple analysis on the stability of the Riemann-Liouville fractional differential systems and Caputo integral differential systems. This article is mainly carried out by scaling the corresponding solution of the systems with some simple methods, and then use the relevant theorems deduced the nature of the zero solution asymptotically stable system and stable conditions.The first chapter is introduction. This chapter mainly introduces the theory of fractional order calculus systems research background、development situation and the main work of this article.The second chapter is preliminary knowledge. This chapter gives the definitions and basic properties of Riemann-Liouville fractional calculus and Caputo derivatives and their Laplace transform, in addition to a brief introduction to some properties and lemmas commonly used in this paper.The third chapter analyzes the stability of the system and their fractional differential perturbed systems. Mainly on the solution of the systems of simple scaling then apply the appropriate theorems to prove, given the stability of the Riemann-Liouville fractional differential systems and determine the conditions of perturbed systems.The fourth chapter mainly uses the Gronwall inequality on the system’s solution to scale assumptions and conditions by using the given zero solution analysis. Finally, the asymptotic solutions of the Caputo fractional Integral differential systems stability conclusions are given.