Linearized Asymptotic Stability for Fractional Differential Equations

Abstract

We prove the theorem of linearized asymptotic stability for fractional differential equations. 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. 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 $\{λ\in \C : |\arg λ| > \frac{απ}{2}\}$ where $\alpha >0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.

Cited by in corpus (2)

• An instability theorem for nonlinear fractional differential systems
• A Perron-type theorem for fractional linear differential systems