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Linearized Asymptotic Stability for Fractional Differential Equations
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 \C : |\arg \lambda| > \frac{\alpha \pi}{2}\} where
denotes the order of the fractional differential equation, then the equilibrium
of the nonlinear fractional differential equation is asymptotically stable
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