Stability Of Fractional Order Differential Systems

In the recent decades, fractional order differential systems have attracted great attention. It has been proved that fractional calculus and fractional order differ-ential equations are valuable tools in the modeling of various systems processes and systems in the areas of physics, chemistry, biology, economics, electrical engi-neering, etc. So, fractional order differential system is a research field which has the value of theory and the background of application. Stability of solution is an important research topic in the theory of fractional order differential system.This Ph.D thesis is composed of six chapters, which mainly studied the stabili-ty of fractional order differential systems with Caputo derivative and the finite time stability of fractional order neutral differential systems with Caputo derivative.In the first chapter, the research backgrounds and research status on stabili-ty of fractional order differential systems are introduced. It also provides the the main works of this dissertation. Chapter2gathers the needed preliminary results; in particular, it recalls the exact definitions of Riemann-Liouville fractional or-der integral, Riemann-Liouville fractional order derivative, Caputo fractional order derivative, as well as their basic properties.In Chapter3, firstly, we develop some fractional order integral/differential inequalities. Secondly, we study the existence of solution for a class of fractional order integral equation and give the expression of solution. Then, we obtain a generalized Gronwall inequality by the expression of solution and fractional order integral inequality. Finally, by using the fractional order differential inequality, we investigate the Lyapunov stability, Mittag-Leffler stability, generalized Mittag-Leffler stability and instability of nonlinear Caputo fractional order differential systems and obtain stability theorem and instability theorem for the nonlinear Caputo fractional order differential systems. In Chapter4, we study the Lyapunov stability of nonlinear fractional order delay differential systems equipped with the Caputo derivative. We extend the Lyapunov-Krasovskii approach for the nonlinear fractional order delay differential systems. By using the Lyapunov-Krasovskii approach and Lyapunov functional, we obtain the necessary and sufficient conditions of stability, uniform stability and asymptotic stability for the nonlinear fractional order delay differential systems.In Chapter5, we study the Lyapunov stability and finite time stability of nonlinear Caputo fractional order neutral differential systems. We extend the Lyapunov-Krasovskii approach for the nonlinear fractional order neutral differen-tial systems. Firstly, by using Laplace transform, we study the Lyapunov stability and Mittag-Leffler stability of nonlinear Caputo fractional order neutral differen-tial systems. Secondly, By using the Lyapunov-Krasovskii approach and Lyapunov functional, we obtain the necessary and sufficient conditions of stability, uniform stability and asymptotic stability. Furthermore, we develop a instability theorem for the nonlinear Caputo fractional order neutral differential systems. Finally, we study the existence and uniqueness for a class of linear Caputo fractional order neu-tral differential systems and develop a finite time stability theorem for the linear Caputo fractional order neutral differential systems.Finally, some concluding remarks are given in Chapter6.