Stability And Control Of Delayed Fractional Differential Equation

The stability of fractional differential equations and their control are investigated in this paper. Based on the stability theory of FDE (fractional differential equations) and Laplace transform, we mainly consider the feasibility of Laplace transform in FDE (fractional differential equation), stability theory of FDE with single delay and their ap-plications, stability theory of FDE with multiple delays and their applications. The paper is organized as follows.In the introduction, it deals with the background and significance of fractional cal-culus, fundamental definitions of fractional calculus, and the history and recent results of stability theory of fractional differential equations.In Chapter 2, it is devoted to the feasibility of Laplace transform in fractional differ-ential equations with delay. First, the estimation analysis of solutions of fractional delayed differential equations and the analysis about Laplace transform of fractional derivatives are recalled. Then according to the theory of Laplace transform and its theoretical analy-sis, we show that under mild conditions the Laplace transform could be performed on the fractional delayed differential equations. The validity of analysis is verified through the solutions of linear fractional differential equations.Chapter 3 is concerned with (globally) asymptotic stability of fractional differential equations with single delay. By using comparison principle of fractional linear models and Lyapunov stability theory, the stability criterion of nonlinear fractional equation are derived. Furthermore, the results are applied to the synchronization of complex networks. Numerical simulation results show the effectiveness of theoretical analysis.In Chapter 4, (global) asymptotic stability of fractional differential equations with multiple delays is given. First, two Mittag-Leffler functions are showed to be nonnegative. Then comparison principles for two kinds of fractional-order systems with multiple time delays are established. Using the above results, stability conditions of fractional-order systems with multiple time delays are given。Furthermore, these results are applied to cellular neural networks. Numerical simulation results show the effectiveness of theoretical analysis.