Stability Of Solutions For Several Classes Of Fractional Differential Equations

Lyapunov stability theory is widely used in natural science and engineering technology,such as automatic control,dynamic system,biological population,etc.In this paper,we mainly discuss the asymptotic stability of solutions of fractional differential equations under different conditions by the first method of Lyapunov and the Mittag-Leffler stability of solutions of differential equations by the second method.I will discuss three main points.First,we investigates the theorem of linearized asymptotic stability for fractional differential equations with commensurate order 1 < ? < 2.More precisely,we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable.Second,we mainly study the necessary and sufficient conditions for the stability of planar fractional Cauchy problems with two different Caputo derivatives by the Lyapunov's first method.Solutions of planar fractional differential equations with two different Caputo derivatives are obtained by Laplace transformation and inverse transformation.As a consequence we extend Lyapunov's first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector {? ? C : | arg(?)| >??/2} where ? > 0denotes the order of the fractional differential equation,then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.And finally,we present a practically generalized Mittag-Leffler stability for fractionalorder nonlinear systems depending on a parameter.By using a Lyapunov function,we give a sufficient condition on practically generalized Mittag-Leffler stability.