Linearized asymptotic stability for fractional differential equations

• Published in: Electronic journal of qualitative theory of differential equations 2016-01, Vol.2016 (39), p.1-13
• Main Authors: Cong, Nguyen, Doan, Thai, Siegmund, Stefan, Tuan, Hoang
• Format: Article
• Language: English
• Subjects: fractional differential equations; linearized asymptotic stability; lyapunov’s first method
• ISSN: 1417-3875; 1417-3875
• DOI: 10.14232/ejqtde.2016.1.39

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 $\{\lambda \in \mathbb{C} : |\arg \lambda| > \frac{\alpha \pi}{2}\}$ where $\alpha > 0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.